15 Model-Based Inference for Populations

Any estimator used in finite population inference should be based on the best model that can be conveniently constructed and, whenever possible, on inference based on both predictive models and the sampling design. Brewer (2002)

The previous chapters of this text focus on the search for a representative sampling strategy under the paradigm of design-based inference. It is worth noting that this approach, proposed in Neyman (1934), is widely used by government agencies around the world. However, for several decades it has been challenged by theoretical statisticians who express dissatisfaction with the philosophical foundations of this approach. In recent decades, other types of approaches have been proposed, the most important of them being the superpopulation-model-based approach. Before entering this topic, it is necessary to briefly review the evolution of inference for finite populations.

15.1 A Little History

According to Rao (2005), the first person interested in the representative method, later known as sampling theory, was the Norwegian statistician Kiaer (1901), who empirically demonstrated that selecting stratified samples produces better results in estimates of means and totals. Later, Bowley (1906) used normal-distribution approximations for estimating proportions and proposed the variance estimation formula for stratified sampling designs. By the 1920s, the representative method was widely used in the United States and around the world. Thus, in 1924, the ISI, the International Statistical Institute, created a committee to discuss this method. The results of this committee include the work of Bowley (1926), based on representative selection methods with equal inclusion probabilities. These theoretical advances and the publication of random number tables by Tippett (1927) facilitated the selection of probability samples. In that same year, Hubback (1927)¹ recognized the need to use this approach in agricultural studies because it:

Avoids possible personal biases.
Makes it possible to determine a sample size that satisfies a margin of error specified by the researcher.

Bowley’s work, together with the ISI report, led Neyman to examine the very foundations of inference in finite populations. In particular, the article by Neyman (1934) is considered one of the pillars on which the foundations of sampling as known today rest. In this regard, Leslie Kish, in a comment on Smith (1976)’s article, states that Neyman made seven major contributions to sampling:

He proposed Neyman allocation for sample size in stratified designs.
He discovered that cluster sampling can be carried out based on a probability scheme such that the variances of the resulting estimators can be calculated or estimated.
For the above to hold, a large sample of units is needed.
To select a large sample, it is crucial to define a random-number selection frame.
Subjective knowledge of population behavior can be used to form population subgroups or strata.
A probability selection scheme is better than a purposive selection scheme.
To convince skeptics of the validity of his claims, he set out to provide practical examples with real large-scale surveys.

Neyman’s new theory revolutionized the world of sampling and freed it from the paradigm of equal inclusion probabilities. In a single article, he introduced the ideas of efficiency, optimal allocation, generalization of Markov’s theorem, and cluster sampling, and he presented a clear case in which purposive sampling led to incorrect conclusions. Later, Neyman proposed two-phase sampling. Smith (1976) states that probability-proportional sampling and multistage sampling are results of Neyman’s ideas. He also proposed conducting large-sample inference based on confidence interval theory “without regard to the properties of the finite population, whatever they might be”. Any method satisfying the previous assumptions was called representative.

Cochran (1939) made several significant contributions: he introduced the use of ANOVA to estimate the efficiency gain due to stratification, proposed variance estimation for two-stage surveys, and brought together the components needed for regression estimation under two-phase sampling. He also introduced the concept of the superpopulation: “The finite population could be viewed as a random sample from an infinite population.” Later, Cochran (1940) introduced the ratio estimator and developed the theory of estimating totals and means using regression models. Shortly thereafter, Madow and Madow (1944) introduced the theory of systematic sampling.

Meanwhile in India, Mahalanobis founded the Indian Statistical Institute, where he made major contributions by formulating expressions for the variance of estimators as a function of survey cost. Several texts appeared after the 1940s addressing the problem of sample selection and parameter estimation in finite populations. One of the greatest developments in terms of current theory was made by Horvitz and Thompson (1952), who proposed a framework for the theory of proportional sampling without replacement and developed an elegant treatment of sampling, thereby completing the foundations of design-based inference.

Definition

Design-based inference (Särndal et al. 1992) This approach estimates the parameters depending on the sampling design chosen to select the sample, without taking into account the properties of the finite population. Thus, for example, the estimator of the population total $t_y$ will be given by:

\[ \hat{t}_y=\sum_{k\in m} d_ky_k \]

where $d_k$ is a weight induced by the sampling design. Under this perspective, the values $y_k$ are taken as the observation on individual $k$ of the characteristic of interest $y$. However, $y$ is not taken as a random variable, but as a fixed quantity.

From that point to the present day, advances, contributions, and new theories of sample selection and parameter estimation have appeared while maintaining the philosophy of design-based inference. Rao (2005) cites some of them, for example: sampling on several occasions, panel-type samples, estimation of distribution functions and quantiles, and small-domain estimation.

On the other hand, around the same time Godambe (1955) proved the following theorem, which calls into question the concept of efficiency to which Neyman referred, since it proves that, under design-based inference, there is no unbiased minimum-variance estimator.

Result

Let $p(\cdot)$ be a sampling design with sample size $n(S)<N$ such that $\pi_k>0 \ \ \forall k \in U$. Then there is no uniformly minimum-variance unbiased estimator in the class of all unbiased estimators.

Proof. Basu (1971) proposes the following proof: suppose that $\hat{t}$ is an unbiased estimator for the population total $t$. Therefore, $\hat{t}$ is unbiased for any population structure $\mathbf{y}=(y_1,\ldots,y_N)$. Note that $\mathbf{y} \in \mathcal{Y}$, with $\mathcal{Y}$ the set of all possible populations. In particular, this estimator is unbiased for $\mathbf{y}_0 \in \mathcal{Y}$. Therefore, $\hat{t}_0$ is unbiased for $t_0$. Now note that \[ \hat{t}^*=\hat{t}+t_0-\hat{t}_0 \] is also an unbiased estimator for $t$. Moreover, when $\mathbf{y}=\mathbf{y}_0$, we have $\hat{t}^*=t_0$ and therefore $Var(\hat{t}^*)=0$. In conclusion, for an unbiased estimator to be uniformly minimum variance for any population structure $\mathbf{y} \in \mathcal{Y}$, it must have zero variance. This is impossible because the sampling design does not consider the census. Therefore, the result is proved.

The previous theorem is a consequence of the generality of Neyman’s inferential approach, since inferences are made with respect to the sampling design without taking the population structure into account. Smith (1976) states that this approach allows a great deal of freedom for an inferential theory and therefore it is not possible to find an optimum for all population structures. This argument, together with Basu’s elephant fable (Basu 1971), led theoretical statisticians to reconsider continuing to make inferences based on the sampling design.

15.1.1 Basu’s Elephant Fable

As Brewer (2002) states, the following published fable shook the foundations of design-based inference.

The owner of a circus is planning to transport 50 adult elephants. For this purpose, he needs a good estimate of the total weight of the elephants. Since weighing an elephant is a very inconvenient task, the circus owner wants to estimate the total weight by weighing only one elephant. Which elephant should he weigh? The circus owner decides to look at his records and discovers an old list of the elephants’ weights, prepared three years earlier. He finds that, three years ago, Sambo, a medium-sized elephant, was the average, in weight, of the herd. The circus owner verifies the information with the trainer, who assures him that Sambo can still be considered the average of the herd.

Thus, the circus owner plans to weigh Sambo and take $50 \times y_{Sambo}$, where $y_{Sambo}$ is Sambo’s weight, as an estimate of the total weight $t_y=y_1+\ldots+y_{50}$ of the herd. But the circus statistician is horrified when he learns the owner’s sampling strategy, which uses a nonprobability sampling design.

How can you obtain an unbiased estimate of $t_y$? the statistician protests.

Thus, they work together to develop a sampling design. With the help of a random number table, they construct a plan that assigns Sambo an inclusion probability of 99/100 and assigns probabilities of 1/4900 to the rest of the herd. Naturally, Sambo is selected and the circus owner is happy.

How are you going to estimate $t_y$? asks the statistician.
Why? The estimate should be $50 \times y_{Sambo}$, of course, replies the owner.
Oh no, that is incorrect, replies the statistician. Recently, I read in an article in the Annals of Mathematical Statistics that the Horvitz-Thompson estimator is the only hyper-admissible estimator in the class of all generalized polynomial unbiased estimators.
What would the Horvitz-Thompson estimate be in this case? asks the impressed owner. Since Sambo’s inclusion probability was 99/100, says the statistician, the estimator is $\dfrac{100}{99} \times y_{Sambo}$.
And what would our estimate be if the sampling plan had selected Jumbo? asks the incredulous owner.
According to my understanding of the Horvitz-Thompson estimator, says the unhappy statistician, the estimator of $t_y$ would be $4900 \times y_{Jumbo}$, where $y_{Jumbo}$ is Jumbo’s weight.

In this way, the statistician lost his job, and perhaps became a professor.

Lohr (2000) asks whether it was fair to fire the statistician. One of the flaws in the sampling strategy used by the statistician appears in the construction of the sampling design, which induces a very large selection probability for an elephant whose value of the characteristic of interest, weight, is average relative to the rest of the herd. As seen in previous chapters, an efficient sampling strategy using the Horvitz-Thompson estimator is one whose sampling design induces inclusion probabilities proportional to the value taken by the characteristic of interest.

15.1.2 The Fable of the Two Statisticians

Lahiri (1968) expresses the difficulties that arise when trying to explain the finite population inference approach to an ordinary person through the following situation.

Suppose that two statisticians, or sampling specialists, are hired to select a sample of size $n$ from a given finite population. Both have the same information about the behavior of the population. This knowledge includes an auxiliary information characteristic for each unit belonging to the population. One decides to select a simple random sample and the other decides to select a sample with probability proportional to size. As a complement to the sampling strategy, both decide to use the estimator $\bar{y}=\sum_sy_k/n$. Incredibly, the two statisticians select exactly the same units in the sample of size $n$. Of course, both know that the typical deviation is given in terms of $\bar{y}- \bar{Y}$; however, they propose completely different measures for the precision of their estimators.

How can this situation be explained? This type of fable contributes greatly to the development of statistics. In fact, the previous story is a clear example of how much remains to be done in our statistical science. However, note that the same type of reasoning appears if the same statisticians face a frequentist problem and one decides that the likelihood of the data is normal while the other decides that it is beta. They would surely arrive at different estimates. Whoever proposes the sampling strategy is forced to make the same subjective decisions as whoever proposes a likelihood, in the frequentist case, or a prior distribution, in the Bayesian case. It is now the researcher’s duty to ensure that subjectivity is framed within certain limits. Of course, if you are going to measure the distance from the Earth to the Moon, you surely would not use a meter stick.

With the previous arguments, another type of inference for finite populations was born: the approach based on a superpopulation model, which assumes that the population structure follows a specific model. The distribution induced by the model provides the tools to predict specific values for the individuals in the population that were not selected.

Definition

Model-based inference (Valliant et al. 2000; Smith 1976) This approach assumes the use of auxiliary information and relates the characteristic of interest to the auxiliary information through a superpopulation model $\xi$. Under this perspective, the data are not required to come from a probability sample, since the way in which the sample is chosen is not taken into account for estimating the parameters of interest, and the observation of the characteristic of interest in the population units $y_k$ is defined as the realization of a random variable $Y_k$. Starting from the fact that the population total can be written as

\[ T_y=\sum_{k\in s} Y_k+\sum_{k\notin s} Y_k, \]

the task is to estimate, through the model $\xi$, the respective observations $y_k$ of the elements that were not selected in the sample. Denoting this estimate as $E(Y_k)$, a predictor for the total would be given by:

\[ \hat{T}_y=\sum_{k\in s} Y_k+\sum_{k\notin s} E_{\xi}(Y_k) \]

and therefore the realization of $\hat{T}_y$ with the specific data from the selected sample $s$ would be defined as

\[ \hat{t}_y=\sum_{k\in s} y_k + \sum_{k\notin s} \hat{E}_{\xi}(Y_k) \]

where $\hat{E}_{\xi}(Y_k)$ is an estimate of $E_{\xi}(Y_k)$ made with the data obtained from the selected sample $s$.

Godambe and Thompson (1977) suggested, during a discussion at the international statistical congress in New Delhi, that a way should be sought to find estimators that made sense under both types of inference. Later, Sarndal and Wright (1984) and Brewer (1999) implemented this suggestion.

Although the dominant type of inference after World War II was design-based inference, in the early 1970s Richard Royall, with the help of many coauthors, decisively changed that trend. He stated that design-based inference, although it makes no assumptions about probabilities and appears to be nonparametric and robust, was subject to important defects. Some of the limitations cited by Royall (1971) are:

The surprising complications encountered in the study and execution of probability-proportional-to-size designs, and
the awkwardness and mistakes in almost all probability estimates concerning ratio estimation.

Royall’s suggestion was even more radical. He proposed abandoning design-based inference in favor of estimators whose useful properties, such as unbiasedness, consistency, and optimality, were defined in terms of the appropriate predictive model. This means that concepts such as bias and variance are no longer defined as expectations over all possible samples, but as averages of the realizations of the population units, whether in the sample or not, under the established predictive model. From Royall’s point of view, the randomization process becomes irrelevant, and he proposes that the sample be chosen purposively, which in practice means choosing the largest units. However, this type of inference must be used very carefully because, as Box (1979) states:

All models are wrong, but some are useful. The fact that all models are wrong becomes clearer and clearer as sample size increases; therefore estimates resulting from an incorrect predictive model are deficient.

One thing is certain: inference based on predictive models and inference based on the sampling design should not be seen as competitors, but as points of view that can become complementary. This is how design-based inference assisted by predictive models, model-assisted survey sampling, is born. However, although these two types of inference can be combined, they cannot be reconciled because their philosophies are literally different.

Design-based inference differs radically from inference based on predictive models and perhaps from any other statistical model because it is based exclusively on the sample observations and makes no a priori assumptions. In addition, its direction of analysis runs counter to the direction of model-based inference. Kyburg (1987) writes in his article a defense and vindication of model-based inference and comments on the types of statistical inferences that exist; he states that:

Inverse inference proceeds from the particular to the general; direct inference proceeds from the general to the particular.

From this point of view, design-based inference is inverse and inference based on predictive models is direct. Note that Bayesian inference also belongs to the group of inverse inferences. Brewer (1999) argues that:

At present, the tendency is to use design-based inference for estimation in large domains and synthetic sampling, model-based inference, for estimation in small domains within the same study.

He also alludes to the use of cosmetic calibration estimators that combine the two types of inference simultaneously. The idea of cosmetic estimators originated with Sarndal and Wright (1984), and the argument for using that word is the fact that an estimator can be seen or interpreted as a predictor obtained from a regression, which makes it very attractive.

Finally, since the appearance of the classic sampling book by Särndal et al. (1992), the history of inference in finite populations has taken on another nuance, defining not only black and white but also a kind of spectrum between these two currents of inferential thought. Isaki and Fuller (1982) raise the problem of taking into account both the sample selection method and the relationship model $\xi$ between the characteristic of interest and the auxiliary information, but it is in Cassel et al. (1976) that a very controversial term is coined: model-assisted design-based inference. That is, the basis of inference is the sampling design, but the sampling strategy is complemented by taking a model $\xi$ into account in estimating the parameter of interest.

To finish the historical review, Brewer (2002) presents the following dialogue between two statisticians, called E and L, who use different approaches to finite population inference. One uses inference based on predictive models, which uses the sample data to build a model that makes it possible to predict the values not observed in the sample and thereby arrive at an estimate of the quantities of interest, without using inclusion probabilities. The other uses design-based inference. Each is a staunch defender of their point of view.

E: I think you are still living in the eighties. Have no doubt that things have changed a bit. Many academic statisticians are in favor of inference based on predictive models.

L: That is true, but that type of inference is not used in professional practice. Name at least one government agency that uses it!

E: Of course there are. When estimating parameters in small domains, synthetic estimators are used. Those estimators are based on predictive models.

L: Ah, but they are only used in small domains. Otherwise they are not used. Fine, if you are trying to estimate a parameter in a small domain, inference based on predictive models can be particularly useful.

E: No, it is more than that. The point is that design-based inference is particularly poor for small samples. Notice that with a probability sample you can select the largest units and leave the small ones aside; with design-based inference you would get poor estimates. A safer way to avoid that possibility is to divide the population into groups and select units in each group.

L: Like a kind of stratification?

E: Hmm, stratification, yes, let us say so. Stratification by unit size is very useful, but the point is that one should know the population very well.

L: Precisely, and if you do not know the population very well, you could fit a completely wrong model and end up with very poor predictions.

15.2 Some Predictive Models

Valliant et al. (2000) argues that there is no compelling reason why the principles of inference for finite populations should be so far removed from the rest of statistical theory. In this way, the design-based inference approach states that randomization of units into the sample is the only valid principle for making inferences about the finite population. However, this rigidity leaves the statistician without statistical foundations for making inferences if the data do not come from a sampling design. Of course, it is reasonable to think that statisticians have many tools that allow them to make inferences regardless of the nature of the data. One of these tools is the likelihood principle, which states the following (Gelman et al. 2004):

When making inferences or decisions about a parameter $\theta$ after the data have been observed, all relevant information is contained in the likelihood function for the observed data.

It is not difficult to verify that the likelihood function for any sampling design is the same and is given by an indicator function. Thus, Valliant et al. (2000) concludes that, although randomization is desirable, it is neither necessary nor sufficient for rigorous statistical inference. The validity of statistical inference remains standing with or without randomization. The following sections describe some of the many predictive models used for specific situations in finite population inference.

15.2.1 A Model for Simple Random Sampling

Suppose that $Y_1,\ldots,Y_N$ is a population of independent and identically distributed random variables. The probabilistic mechanism governing the population is given by a superpopulation model $\xi$ defined as

\[ Y_k=\beta+\varepsilon_k \]

where each $\varepsilon_k$, $k\in U$, is an independent and identically distributed random variable with zero mean and constant variance $\sigma^2$, such that:

\[ \begin{aligned} E_{\xi}(Y_k)&=\beta \\ Var_{\xi}(Y_k)&=\sigma^2. \end{aligned} \]

From this population, a sample $s$ of size $n$ is selected. This gives the following results.

Result

Under model 11.2.1, the best linear unbiased estimator of $\beta$ is given by \[ \hat{\beta}=\bar{Y}_s=\frac{1}{n}\sum_{k\in s}Y_k \]

Proof. The estimator of $\beta$ is obtained by minimizing the following dispersion function: \[ D=\sum_{k\in s} \frac{\left(y_k-\beta \right)^2}{\sigma^2}. \]

After differentiating and setting the result equal to zero, it is easy to find that $\hat{\beta}=\bar{Y}_s$. On the other hand, \[ \begin{aligned} E_{\xi}(\hat{\beta})=\frac{1}{n}\sum_{k\in s}E_{\xi}(Y_k)=\beta \end{aligned} \]

Using the Gauss-Markov theorem (Ravishanker and Dey 2002, Result 4.4.1), $\hat{\beta}$ is the best estimator because it has minimum variance.

Result

Under model 11.2.1, the best linear unbiased predictor of $T_y$ and its mean squared error ($MSE_{\xi}$) are given by \[ \hat{T}_{y}=\frac{N}{n}\sum_{k\in s}Y_k \] \[ MSE_{\xi}(\hat{T}_{y})=\frac{N^2}{n}\left(1-\frac{n}{N}\right)\sigma^2 \] respectively.

Proof. First note that \[ \begin{aligned} \hat{T}_y=\sum_{k\in s} Y_k+\sum_{k\notin s} \hat{\beta}=\sum_{k\in s} Y_k+(N-n) \bar{Y}_s=\frac{N}{n}\sum_{k\in s}Y_k \end{aligned} \]

$\hat{T}_y$ is unbiased because \[ \begin{aligned} E_{\xi}(\hat{T}_y-T_y)=E_{\xi}\left(\frac{N}{n}\sum_{k\in s}Y_k-\sum_{k\in U}Y_k\right)=\beta-\beta=0 \end{aligned} \]

Finally, \[ \begin{aligned} MSE_{\xi}(\hat{T}_y)&=E_{\xi}\left(\hat{T}_y-T_y\right)^2\\ &=E_{\xi}\left(\left[\frac{N}{n}-1\right]\sum_{k\in s}Y_k - \sum_{k\notin s}Y_k\right)^2\\ &=\left[\frac{N}{n}-1\right]^2E_{\xi}\left(\sum_{k\in s}Y_k\right)^2 -2\left[\frac{N}{n}-1\right]E_{\xi}\left(\sum_{k\in s}Y_k\right)E_{\xi}\left(\sum_{k\notin s}Y_k\right)\\ &\hspace{4cm} + E_{\xi}\left(\sum_{k\notin s}Y_k\right)^2\\ &=\left[\frac{N}{n}-1\right]^2E_{\xi}\left(\sum_{k\in s}Y_k\right)^2 -2(N-n)^2\beta^2 +E_{\xi}\left(\sum_{k\notin s}Y_k\right)^2\\ &=\left[\frac{N}{n}-1\right]^2E_{\xi}\left(\sum_{k\in s}Y_k-n\beta\right)^2 +E_{\xi}\left(\sum_{k\notin s}Y_k-(N-n)\beta\right)^2 \end{aligned} \]

Since $E_{\xi}\left(\sum_{k\in s}Y_k-n\beta\right)^2=Var_{\xi}\left(\sum_{k\in s}Y_k\right)$, it follows that \[ \begin{aligned} MSE_{\xi}(\hat{T}_y)&=\left[\frac{N}{n}-1\right]^2Var_{\xi}\left(\sum_{k\in s}Y_k\right) +Var_{\xi}\left(\sum_{k\notin s}Y_k\right)\\ &=\left[\frac{N}{n}-1\right]^2n\sigma^2+(N-n)\sigma^2\\ &=\frac{N^2}{n}\left(1-\frac{n}{N}\right)\sigma^2 \end{aligned} \]

Note that $S^2$ can be used to estimate $\sigma^2$. In this way, the two inference approaches appear to coincide because the expressions for the estimator and its estimated variance are identical, although the background and interpretation are different. In addition, Lohr (2000) states that the confidence intervals constructed from the two approaches also coincide, although their interpretation does not²

15.2.2 A Model for Stratified Random Sampling

Suppose that $Y_1,\ldots,Y_N$ is a population of random variables whose behavior differs across $H$ population groups, each of size $N_h$ ($h=1,\ldots,H$), defining a stratified population $U=\{U_1,\ldots,U_H\}$. Clearly, the overall population size is $N=N_1+\cdots,N_H$. The probabilistic mechanism governing the population is given by a superpopulation model $\xi$ defined as \[ Y_{hk}=\beta_h+\varepsilon_{hk} \]

where the subscript $hk$ refers to quantities associated with the $k$th element within the $h$th stratum. Each $\varepsilon_{hk}$ is an independent and identically distributed random variable with zero mean and constant variance $\sigma^2_h$ within stratum $h$, uncorrelated across strata, such that: \[ \begin{aligned} E_{\xi}(Y_{hk})&=\beta_h \\ Var_{\xi}(Y_{hk})&=\sigma^2_h\\ Cov_{\xi}(Y_{hk},Y_{gl})&=0 \ \ \ \ \text{if $h\neq g$}. \end{aligned} \]

From each stratum, a sample $s_h$ of size $n_h$ ($h=1,\ldots,H$) is drawn. The overall sample size is $n=n_1+\cdots,n_H$.

Result

Under model 11.2.6, the best linear unbiased estimator of $\beta_h$ ($h=1,\ldots,H$) is given by \[ \hat{\beta}_h=\bar{Y}_{s_h}=\frac{1}{n_h}\sum_{k\in s_h}Y_{hk} \]

Proof. The estimator of $\beta_h$ is obtained by minimizing the following dispersion function: \[ D=\sum_{k\in s_h} \frac{\left(y_{hk}-\beta_h \right)^2}{\sigma^2_h}. \]

After differentiating and setting the result equal to zero, it is easy to find that $\hat{\beta}_h=\bar{Y}_{s_h}$. On the other hand, \[ \begin{aligned} E_{\xi}(\hat{\beta}_h)=\frac{1}{n_h}\sum_{k\in s_h}E_{\xi}(Y_{hk})=\beta_h \end{aligned} \]

From one-way analysis of variance with fixed effects, it follows that it is the best estimator because it has minimum variance.

Result

Under model 11.2.6, the best linear unbiased predictor of $T_y$ and its mean squared error are given by \[ \hat{T}_{y}=\sum_{h=1}^H\frac{N_h}{n_h}\sum_{k\in s_h}Y_{hk} \] \[ MSE_{\xi}(\hat{T}_{y})=\sum_{h=1}^H\frac{N_h^2}{n_h}\left(1-\frac{n_h}{N_h}\right)\sigma^2_h \] respectively.

Proof. First note that the total random variable $T$ can be rewritten as \[ \begin{aligned} T_y=\sum_{h=1}^H\sum_{k\in U_h}Y_{hk}=\sum_{h=1}^HT_{yh} \end{aligned} \]

with $T_{yh}$ denoting the total random variable for stratum $h$. Using Result 11.2.2, $\hat{T}_{yh}=\frac{N_h}{n_h}\sum_{k\in s_h}Y_{hk}$ is an unbiased predictor for $T_{yh}$. Therefore, \[ \begin{aligned} E_{\xi}(\hat{T}_y-T_y)=\sum_{h=1}^HE_{\xi}\left(\hat{T}_{yh}-T_{yh}\right)=0 \end{aligned} \]

Finally, \[ \begin{aligned} MSE_{\xi}(\hat{T}_y)&=E_{\xi}\left(\sum_{h=1}^H(\hat{T}_{yh}-T_{yh})\right)^2\\ &=E_{\xi}\left(\sum_{h=1}^H(\hat{T}_{yh}-T_{yh})^2+ \sum_{h}\sum_{g\neq h}E_{\xi}(\hat{T}_{yh}-T_{yh})(\hat{T}_{yg}-T_{yg})\right)\\ &=E_{\xi}\left(\sum_{h=1}^H(\hat{T}_{yh}-T_{yh})^2\right)=\sum_{h=1}^HE_{\xi}(\hat{T}_{yh}-T_{yh})^2\\ &=\sum_{h=1}^H\frac{N_h^2}{n_h}\left(1-\frac{n_h}{N_h}\right)\sigma^2_h \end{aligned} \]

Analogously to the model for simple random sampling, it is possible to estimate $\sigma_h^2$ with $S^2_h$, in which case the same estimates would be obtained under both approaches.

15.2.3 A Model for Cluster Sampling

Suppose that $Y_1,\ldots,Y_N$ is a population of random variables grouped into $N_I$ clusters that induce a partition of the population and at the same time define a population of clusters $U_I=\{U_1,\ldots,U_{N_I}\}$. The size of the $i$th cluster is $N_i$ ($i=1,\ldots,N_I$). The overall population size is $N=N_1+\cdots,N_{N_I}$. The probabilistic mechanism governing the population is given by a superpopulation model $\xi$ defined as \[ Y_{ik}=\beta+\varepsilon_{ik} \]

where the subscript $ik$ refers to quantities associated with the $k$th element within the $i$th cluster. Each $\varepsilon_{ik}$ is an independent and identically distributed random variable with zero mean and constant variance $\sigma^2_i$ within the same $i$th cluster ($i=1,\ldots,N_I$), with autocorrelation structure $\sigma_i\rho_i$ for elements belonging to the same $i$th cluster and uncorrelated across clusters, such that \[ \begin{aligned} E_{\xi}(Y_{ik})&=\beta \\ Var_{\xi}(Y_{ik})&=\sigma^2_i\\ Cov_{\xi}(Y_{ik},Y_{jl})&=\sigma^2_i\rho_i \ \ \ \ \text{if $i\neq j$ and $k\neq l$}. \end{aligned} \]

The model indicates that all elements have a common mean. Within clusters, the elements have a common variance (which may differ from one cluster to another), and within the same cluster the elements share a correlation factor. Thus, a sample of clusters $s_I$ of size $n_I$ is selected and every element belonging to the selected clusters is observed.

Result

Under model 11.2.11, the best linear unbiased estimator of $\beta$ is given by \[ \hat{\beta}=\sum_{i\in s_I}v_i\bar{Y}_{U_i} \]

where \[ v_i=\frac{\left(N_i/\sigma^2_i[1+(N_i-1)\rho_i]\right)}{\sum_{i\in S_I}\left(N_i/\sigma^2_i[1+(N_i-1)\rho_i]\right)} \]

Proof. Using an argument similar to that used for the previous models and applying random-effects analysis of variance gives the proof of the result.

Result

Under model 11.2.11, the best linear unbiased predictor of $T_y$ and its mean squared error are given by \[ \hat{T}_{y}=\sum_{i\in s_I}\sum_{k=1}^{N_i}Y_{ik}+\sum_{i\notin s_I}N_i\hat{\beta} \] \[ MSE_{\xi}(\hat{T}_{y})=\sum_{i\notin s_I}N_i\sigma_i^2[1+(N_i-1)\rho_i]+ \dfrac{(N_I-n_I)^2}{\sum_{i\in s_I}N_i/\sigma_i^2[1+(N_i-1)\rho_i]} \] respectively.

Proof. The reader may consult the proof of this result in Royall (1976) and Scott and Smith (1969), noting that the total can be written as \[ T_y=\sum_{i\in s_I}\sum_{k=1}^{N_i}Y_{ik}+\sum_{i\notin s_I}\sum_{k=1}^{N_i}Y_{ik} \]

15.2.4 A Model for Multistage Sampling

Suppose the same model 11.2.11, but this time a sample of clusters $s_I$ of size $n_I$ is selected and, for each cluster $U_i\in s_I$, a subsample $s_i$ of size $n_i$ is selected.

Result

Under model 11.2.11 and two-stage selection, the best linear unbiased estimator of $\beta$ is given by \[ \hat{\beta}=\sum_{i\in s_I}v_i\bar{Y}_{s_i} \]

where \[ v_i=\frac{\left(n_i/\sigma^2_i[1+(n_i-1)\rho_i]\right)}{\sum_{i\in S_I}\left(n_i/\sigma^2_i[1+(n_i-1)\rho_i]\right)} \]

Proof. Using an argument similar to that used for the previous models and applying nested random-effects analysis of variance gives the proof of the result.

Result

Under model 11.2.11 and two-stage selection, the best linear unbiased predictor of $T_y$ is given by \[ \hat{T}_{y}=\sum_{i\in s_I}\sum_{k\in s_i}Y_{ik}+ \sum_{i\in s_I}(N_i-n_i)\left[w_i\bar{Y}_{s_i}+(1-w_i)\hat{\beta}\right]+ \sum_{i\notin s_I}N_i\hat{\beta} \] with $w_i=n_i\rho_i/[1+(n_i-1)\rho_i]$.

Proof. The reader may consult the proof of this result in Royall (1976) and Scott and Smith (1969), noting that the total can be written as \[ T_y=\sum_{i\in s_I}\sum_{k\in s_i}Y_{ik}+\sum_{i\in s_I}\sum_{k\notin s_i}Y_{ik}+\sum_{i\notin s_I}\sum_{k=1}^{N_i}Y_{ik} \]

Note that for cluster or multistage sampling, both the predictor and its variance differ significantly from the estimator constructed under the design-based inferential approach.

15.2.5 A Model for the Ratio Estimator

Suppose that $Y_1,\ldots,Y_N$ is a population of independent and identically distributed random variables and that $X_1,\ldots, X_N$ form a population of auxiliary variables whose realization for each population element, $x_1,\ldots,x_N$, is known. The probabilistic mechanism governing the population and defining the relationship between $Y_k$ and $X_k$ is given by a superpopulation model $\xi$ defined as

\[ Y_k=\beta X_k+\varepsilon_k \]

where each $\varepsilon_k$, $k\in U$, is an independent and identically distributed random variable with zero mean and nonconstant variance $\sigma^2X_k$, such that

\[ \begin{aligned} E_{\xi}(Y_k)&=\beta X_k \\ Var_{\xi}(Y_k)&=\sigma^2X_k. \end{aligned} \]

This model is valid only if the regression line passes through the origin and the variance increases as the auxiliary variable increases in magnitude. From this population, a sample $s$ of size $n$ is selected. This gives the following results.

Result

Under model 11.2.18, the best linear unbiased estimator of $\beta$ is given by \[ \hat{\beta}=\dfrac{\bar{Y}_s}{\bar{X}_s}=\dfrac{\sum_{k\in s}Y_k}{\sum_{k\in s}X_k} \]

Proof. The estimator of $\beta$ is obtained by minimizing the following dispersion function: \[ D=\sum_{k\in s} \frac{\left(y_k-\beta \right)^2}{\sigma^2X_k}. \]

After differentiating and setting the result equal to zero, it is easy to find that $\hat{\beta}=\dfrac{\sum_{k\in s}Y_k}{\sum_{k\in s}X_k}$. On the other hand, \[ \begin{aligned} E_{\xi}(\hat{\beta})=\frac{1}{\sum_{k\in s}X_k}\sum_{k\in s}E_{\xi}(Y_k)=\frac{1}{\sum_{k\in s}X_k}\sum_{k\in s}\beta X_k=\beta \end{aligned} \]

Result

Under model 11.2.18, the best linear unbiased predictor of $T_y$ and its mean squared error are given by \[ \hat{T}_{y}=\dfrac{\bar{Y}_s}{\bar{X}_s}T_x \] \[ MSE_{\xi}(\hat{T}_{y})=\dfrac{\sum_{k\notin s}X_k}{\sum_{k\in s}X_k}\sigma^2T_x \] respectively, with $T_x=\sum_{k\in U}X_k$.

Proof. First, the predictor takes the following form: \[ \begin{aligned} \hat{T}_y=\sum_{k\in s}Y_k+\sum_{k\notin s}\hat{\beta}X_k =\dfrac{\bar{Y}_s}{\bar{X}_s}\left[n\bar{X}_s+\sum_{k\notin s}X_k\right] =\dfrac{\bar{Y}_s}{\bar{X}_s}T_x \end{aligned} \]

$\hat{T}_y$ is unbiased because \[ \begin{aligned} E_{\xi}(\hat{T}_y-T_y)=E_{\xi}\left(\dfrac{\bar{Y}_s}{\bar{X}_s}T_x-\sum_{k\in U}Y_k\right)=\beta T_x-\beta T_x=0 \end{aligned} \]

Finally, because the predictor is unbiased, \[ \begin{aligned} MSE_{\xi}(\hat{T}_y-T_y)&=Var_{\xi}(\hat{T}_y-T_y)\\ &=Var_{\xi}\left(\sum_{k\in s}Y_k+\sum_{k\notin s}\hat{\beta}X_k-\sum_{k\in U}Y_k\right)\\ &=Var_{\xi}\left(\sum_{k\notin s}\hat{\beta}X_k-\sum_{k\notin s}Y_k\right)\\ &=Var_{\xi}\left(\sum_{k\notin s}\hat{\beta}X_k\right)+Var_{\xi}\left(\sum_{k\notin s}Y_k\right)\\ &=\left(\sum_{k\notin s}X_k\right)^2Var_{\xi}\left( \frac{\sum_{k\in s}Y_k}{\sum_{k\in s}X_k} \right) + Var_{\xi}\left(\sum_{k\notin s}(\beta X_k+\varepsilon_k)\right)\\ &=\left(\sum_{k\notin s}X_k\right)^2 \frac{\sum_{k\in s}Var_{\xi}(Y_k)}{\left(\sum_{k\in s}X_k\right)^2} + Var_{\xi}\left(\sum_{k\notin s}\varepsilon_k\right)\\ &=\left(\sum_{k\notin s}X_k\right)^2 \frac{\sigma^2\sum_{k\in s}X_k}{\left(\sum_{k\in s}X_k\right)^2} + \sigma^2\sum_{k\notin s}X_k\\ &=\sigma^2\left( \frac{\sum_{k\notin s}X_k}{\sum_{k\in s}X_k} \right) + \left[\sum_{k\notin s}X_k+\sum_{k\in s}X_k\right] =\frac{\sum_{k\notin s}X_k}{\sum_{k\in s}X_k}\sigma^2T_x \end{aligned} \]

15.2.6 A Model for the Regression Estimator

\[ Y_k=\beta_0 +\beta_1 X_k+\varepsilon_k \]

where each $\varepsilon_k$, $k\in U$, is an independent and identically distributed random variable with zero mean and nonconstant variance $\sigma^2$, such that:

\[ \begin{aligned} E_{\xi}(Y_k)&=\beta_0 + \beta_1 X_k \\ Var_{\xi}(Y_k)&=\sigma^2. \end{aligned} \]

Result

Under model 11.2.23, the best linear unbiased estimator of $\beta_0$ and $\beta_1$ is given by \[ \hat{\beta}_1=\dfrac{\sum_{k\in s}(x_k-\bar{X}_s)(y_k-\bar{Y}_s)}{\sum_{k\in s}(x_k-\bar{X}_s)^2} \] and \[ \hat{\beta}_0=\bar{Y}_s-\hat{\beta}_1\bar{X}_s \]

Proof. The estimators are found by minimizing the following dispersion function: \[ D=\sum_{k\in s} \frac{\left(y_k-\beta_0-\beta_1X_k \right)^2}{\sigma^2}. \]

After differentiating and setting the result equal to zero, the result is easily obtained.

Result

Under model 11.2.23, the best linear unbiased predictor of $T_y$ is given by \[ \hat{T}_{y}=N\left(\hat{\beta}_0+\hat{\beta}_1\bar{X}_U\right) \]

Proof. Note that the predictor can be written as: \[ \begin{aligned} \hat{T}_y&=\sum_{k\in s}Y_k+\sum_{k\notin s}(\hat{\beta}_0+\hat{\beta}_1X_k)\\ &=n\bar{Y}_s+\sum_{k\notin s}(\hat{\beta}_0+\hat{\beta}_1X_k)\\ &=n(\hat{\beta}_0+\hat{\beta}_1\bar{X}_s)+\sum_{k\notin s}(\hat{\beta}_0+\hat{\beta}_1X_k)\\ &=\sum_{k\notin s}(\hat{\beta}_0+\hat{\beta}_1X_k)+\sum_{k\notin s}(\hat{\beta}_0+\hat{\beta}_1X_k) =\sum_{k\notin U}(\hat{\beta}_0+\hat{\beta}_1X_k)=N(\hat{\beta}_0+\hat{\beta}_1\bar{X}_U) \end{aligned} \]

15.3 The General Prediction Theorem

Just as the general regression estimator is a general case of many other estimators, under the predictive-model-based inferential approach there is a general regression predictor that encompasses many predictors, including those seen in the previous section. However, in this section we will study not only predictions of population totals but also of any linear function of the variables of interest. The reader will note that the general result is based on linear model theory and, in particular, on the Gauss-Markov theorem. Although in this section we make no assumptions about parametrized distributions (such as the normal, gamma, or exponential family), it is possible to do so and obtain optimal results using statistical inference results such as Scheffe’s lemma or the Rao-Blackwell theorem (Shao 2003).

Suppose that the finite population consists of $N$ units. The vector of variables of interest is $\mathbf{Y}=(Y_1,Y_2,\ldots,Y_N)'$, and for each population element the realization of these random variables is $\mathbf{y}=(y_1,y_2,\ldots,y_N)'$. Suppose that the objective is to estimate a linear combination $\boldsymbol{\gamma}'\mathbf{y}$. For this purpose, a sample $s$ of size $n$ is selected. Note that both $\mathbf{y}$ and $\boldsymbol{\gamma}$ can be partitioned as follows: $\mathbf{y}=(\mathbf{y}_s',\mathbf{y}_r')'$ and $\boldsymbol{\gamma}=(\boldsymbol{\gamma}_s', \boldsymbol{\gamma}_r')'$; where the subscript $s$ indicates that the vector contains the $n$ elements of the selected sample, and the subscript $r$ indicates that the vector contains the $N-n$ elements that were not selected in the sample.

In this way, the linear combination to be estimated can be rewritten as $\boldsymbol{\gamma}'\mathbf{y}=\boldsymbol{\gamma}'_s\mathbf{y}+\boldsymbol{\gamma}'_r\mathbf{y}_r$, which is a realization of the random variable $\boldsymbol{\gamma}'\mathbf{Y}=\boldsymbol{\gamma}'_s\mathbf{Y}+\boldsymbol{\gamma}'_r\mathbf{Y}_r$. It is clear that the problem of estimating $\boldsymbol{\gamma}'\mathbf{y}$ reduces to the problem of predicting $\boldsymbol{\gamma}'_r\mathbf{y}_r$.

Definition

A linear estimator of $\theta=\boldsymbol{\gamma}'\mathbf{Y}$ is defined as $\hat{\theta}=\mathbf{g}_s'\mathbf{Y}_s$, where $\mathbf{g}_s=(g_1,g_2,\ldots,g_n)'$ is a vector of size $n$.

Definition

The estimation error of an estimator $\hat{\theta}$ is given by $\hat{\theta}-\theta=\mathbf{g}_s'\mathbf{Y}_s-\boldsymbol{\gamma}'\mathbf{Y}$ and can be rewritten as \[ \begin{aligned} \mathbf{g}_s'\mathbf{Y}_s-\boldsymbol{\gamma}'\mathbf{Y}&=(\mathbf{g}_s'\boldsymbol{\gamma}_s)\mathbf{Y}_s-\boldsymbol{\gamma}_r'\mathbf{Y}_r\\ &=\mathbf{a}'\mathbf{Y}_s-\boldsymbol{\gamma}_r'\mathbf{Y}_r \end{aligned} \] with $\mathbf{a}=\mathbf{g}_s-\boldsymbol{\gamma}_s$

Note that using $\mathbf{g}_s'\mathbf{Y}_s$ to estimate $\theta=\boldsymbol{\gamma}'\mathbf{Y}$ is equivalent to using $\mathbf{a}'\mathbf{Y}_s$ to predict $\boldsymbol{\gamma}'_r\mathbf{Y}_r$ and, consequently, finding an optimal vector $\mathbf{g}_s$ is equivalent to finding an optimal vector $\mathbf{a}$.

The problem addressed in this section is framed within the general linear model given by

\[ \mathbf{Y}=\mathbf{X}_k'\boldsymbol{\beta} +\varepsilon_k \]

where each $\varepsilon_k$, $k\in U$, is an identically distributed random variable with zero mean, variance $Var_{\xi}(\varepsilon_k)=\sigma^2_k$, and covariance $Cov_{\xi}(\varepsilon_k,\varepsilon_l)=\rho_{kl}\sigma_k\sigma_l$, with $\rho_{kl}$ a correlation factor between elements $k$ and $l$ ($k\neq l$), such that:

\[ \begin{aligned} E_{\xi}(Y_k)&=\mathbf{X}_k'\boldsymbol{\beta} \\ Var_{\xi}(Y_k)&=\sigma^2_k\\ Cov_{\xi}(Y_k,Y_l)&=\rho_{kl}\sigma_k\sigma_l \ \ \ \ \ \text{for $k\neq l$} \end{aligned} \]

In matrix form, the previous model is defined as

\[ \begin{aligned} E_{\xi}(\mathbf{Y})&=\mathbf{X}\boldsymbol{\beta} \\ Var_{\xi}(\mathbf{Y})&=\mathbf{V} \end{aligned} \]

where $X$ is a matrix of auxiliary variables of size $N\times p$, $\boldsymbol{\beta}$ is a vector of unknown regression coefficients of size $p\times 1$, and $\mathbf{V}$ is a positive definite covariance matrix. Note that, when the sample is selected, both $\mathbf{X}$ and $\mathbf{V}$ can be rewritten as

\[ \mathbf{X}= \begin{pmatrix} \mathbf{X}_s \\ \mathbf{X}_r \\ \end{pmatrix}, \ \ \ \ \ \ \mathbf{V}= \begin{pmatrix} \mathbf{V}_{ss} & \mathbf{V}_{sr} \\ \mathbf{V}_{rs} & \mathbf{V}_{rr} \\ \end{pmatrix} \]

where $\mathbf{X}_s$ is of size $n\times p$, $\mathbf{X}_r$ is of size $(N-n)\times p$, $\mathbf{V}_{ss}$ is of size $n\times n$, $\mathbf{V}_{rr}$ is of size $(N-n)\times (N-n)$, $\mathbf{V}_{sr}$ is of size $n\times (N-n)$, and $\mathbf{V}_{rs}=\mathbf{V}_{sr}'$, assuming that $\mathbf{V}_{ss}$ is a positive definite matrix.

Definition

An estimator $\hat{\theta}$ is unbiased if $E_{\xi}(\hat{\theta})=0$

Definition

The mean squared error of an estimator $\hat{\theta}$ is given by $MSE_{\xi}(\hat{\theta})=E_{\xi}(\hat{\theta}-\theta)^2$

Royall (1976)

The best linear unbiased estimator of $\theta$ is given by \[ \hat{\theta}=\boldsymbol{\gamma}_s'\mathbf{Y}_s+ \boldsymbol{\gamma}_r'\left[\mathbf{X}_r\hat{\boldsymbol{\beta}} + \mathbf{V}_{rs}\mathbf{V}_{ss}^{-1}(\mathbf{Y}_s)\mathbf{X}_s\hat{\boldsymbol{\beta}}\right] \]

where \[ \hat{\boldsymbol{\beta}}=(\mathbf{X}_s'\mathbf{V}_{ss}\mathbf{X}_s)^{-1}\mathbf{X}_s'\mathbf{V}_{ss}\mathbf{Y}_s \]

The mean squared error of $\hat{\theta}$ is given by

\[\begin{multline} MSE_{\xi}(\hat{\theta})=\boldsymbol{\gamma}_r'(\mathbf{V}_{rr}-\mathbf{V}_{rs}\mathbf{V}_{ss}^{-1}\mathbf{V}_{sr})\boldsymbol{\gamma}_r\\ +\boldsymbol{\gamma}_r'(\mathbf{X}_r-\mathbf{V}_{rs}\mathbf{V}_{ss}^{-1}\mathbf{V}_{sr}\mathbf{X}_s) (\mathbf{X}_s'\mathbf{V}_{ss}^{-1}\mathbf{X}_s)^{-1} (\mathbf{X}_r-\mathbf{V}_{rs}\mathbf{V}_{ss}^{-1}\mathbf{V}_{sr}\mathbf{X}_s)'\boldsymbol{\gamma}_r \end{multline}\]

Proof. First, the mean squared error is given by the following expression: \[ \begin{aligned} E_{\xi}(\hat{\theta}-\theta)^2&=E_{\xi}(\mathbf{a}'\mathbf{Y}_s-\boldsymbol{\gamma}_r\mathbf{Y}_r)^2\\ &=Var_{\xi}(\mathbf{a}'\mathbf{Y}_s-\boldsymbol{\gamma}_r\mathbf{Y}_r)+\left(E_{\xi}(\mathbf{a}'\mathbf{Y}_s-\boldsymbol{\gamma}_r\mathbf{Y}_r)\right)^2\\ &=\mathbf{a}'\mathbf{V}_{ss}\mathbf{a}-2\mathbf{a}'\mathbf{V}_{sr}\boldsymbol{\gamma}_r+\boldsymbol{\gamma}_r'\mathbf{V}_{rr}\boldsymbol{\gamma}_r +\left((\mathbf{a}'\mathbf{X}_s-\boldsymbol{\gamma}_r\mathbf{X}_r)\boldsymbol{\beta}\right)^2 \end{aligned} \]

On the one hand, an unbiased estimator is sought, so the last term must be zero. That is, $\mathbf{a}\mathbf{X}_s=\boldsymbol{\gamma}_r'\mathbf{X}_r$. On the other hand, the best unbiased estimator is sought; that is, the estimator with minimum $MSE$. This minimization is performed using the method of Lagrange multipliers. Therefore, the function to minimize, subject to unbiasedness of the estimator, is \[ \begin{aligned} \mathcal{L}(\mathbf{a},\boldsymbol{\lambda})=\mathbf{a}'\mathbf{V}_{ss}\mathbf{a}-2\mathbf{a}'\mathbf{V}_{sr}\boldsymbol{\gamma}_r+\boldsymbol{\gamma}_r'\mathbf{V}_{rr}\boldsymbol{\gamma}_r +2(\mathbf{a}'\mathbf{X}_s-\boldsymbol{\gamma}_r\mathbf{X}_r)\boldsymbol{\lambda} \end{aligned} \]

where $\boldsymbol{\lambda}$ is a vector of Lagrange multipliers. Differentiating with respect to $\boldsymbol{\lambda}$ and $\mathbf{a}$ and setting the results equal to zero, we obtain \[ \begin{aligned} \frac{\partial\mathcal{L}}{\partial\boldsymbol{\lambda}}&=\mathbf{a}'\mathbf{X}_s-\boldsymbol{\gamma}_r\mathbf{X}_r=0\\ \\ \frac{\partial\mathcal{L}}{\partial\boldsymbol{\lambda}}&=2\mathbf{V}_{ss}\mathbf{a}-2\mathbf{V}_{sr}\boldsymbol{\gamma}_r+2\mathbf{X}_s\boldsymbol{\lambda}=0 \end{aligned} \]

From the first equation, \[ \begin{aligned} \mathbf{a}'\mathbf{X}_s=\boldsymbol{\gamma}_r\mathbf{X}_r \end{aligned} \]

From the second equation, \[ \begin{aligned} \mathbf{a}&=\mathbf{V}_{ss}^{-1}(\mathbf{V}_{sr}\boldsymbol{\gamma}_r-\mathbf{X}_s\boldsymbol{\lambda}) \end{aligned} \]

and using restriction (11.3.7), we also have \[ \begin{aligned} \boldsymbol{\lambda}=\mathbf{A}_s^{-1}(\mathbf{X}_s'\mathbf{V}_{ss}^{-1}\mathbf{V}_{sr}-\mathbf{X}_r')\boldsymbol{\gamma}_r \end{aligned} \]

with $\mathbf{A}_s=\mathbf{X}_s'\mathbf{V}_{ss}^{-1}\mathbf{X}_{s}$. Replacing this last expression in (11.3.8), the optimal value of $\mathbf{a}$ is found to be \[ \begin{aligned} \mathbf{a}_{opt}&=\mathbf{V}_{ss}^{-1}(\mathbf{V}_{sr}-\mathbf{X}_s\mathbf{A}_s^{-1} (\mathbf{X}_s'\mathbf{V}_{ss}^{-1}\mathbf{V}_{sr}-\mathbf{X}_r'))\boldsymbol{\gamma}_r \end{aligned} \]

In this way, after some algebra, the best predictor of $\boldsymbol{\gamma}_r\mathbf{Y}_r$ is \[ \begin{aligned} \mathbf{a}_{opt}'\mathbf{Y}_s&=\boldsymbol{\gamma}_r'(\mathbf{V}_{rs}- (\mathbf{V}_{rs}\mathbf{V}_{ss}^{-1}\mathbf{X}_s-\mathbf{X}_r)\mathbf{A}_s^{-1}\mathbf{X}_s')\mathbf{V}_{ss}^{-1}\mathbf{Y}_s\\ &=\boldsymbol{\gamma}_r'(\mathbf{V}_{rs}\mathbf{V}_{ss}^{-1}\mathbf{Y}_s -\mathbf{V}_{rs}\mathbf{V}_{ss}^{-1}\mathbf{X}_s\boldsymbol{\beta}+\mathbf{X}_r\boldsymbol{\beta})\\ &=\boldsymbol{\gamma}_r'(\mathbf{V}_{rs}\mathbf{V}_{ss}^{-1}(\mathbf{Y}_s-\mathbf{X}_s\boldsymbol{\beta})+\mathbf{X}_r\boldsymbol{\beta}) \end{aligned} \]

From Definition 11.3.1, $\hat{\theta}=\mathbf{g}_s\mathbf{Y}_s$, and from Definition 11.3.2, $\mathbf{g}_s=\mathbf{a}+\boldsymbol{\gamma}_s$. Thus, $\hat{\theta}=\boldsymbol{\gamma}_s'\mathbf{Y}_s+\mathbf{a}'\mathbf{Y}_s$. Substituting appropriately gives the proof of the result. The $MSE$ of the unbiased estimator is given by \[ \begin{aligned} MSE_{\xi}(\hat{\theta})=\underbrace{\mathbf{a}'\mathbf{V}_{ss}\mathbf{a}}_{P1} -\underbrace{2\mathbf{a}'\mathbf{V}_{sr}\boldsymbol{\gamma}_r}_{P2}+\boldsymbol{\gamma}_r'\mathbf{V}_{rr}\boldsymbol{\gamma}_r \end{aligned} \]

Taking into account that $\mathbf{A}_s^{-1}\mathbf{X}_s'\mathbf{V}_{ss}^{-1}\mathbf{X}_{s}=\mathbf{I}$, with $\mathbf{I}$ the identity matrix, and after carrying out the necessary algebraic steps, the first part P1 is equivalent to \[ \begin{aligned} P1&=\boldsymbol{\gamma}_r'\mathbf{V}_{rs}\mathbf{V}_{ss}^{-1}\mathbf{V}_{sr}\boldsymbol{\gamma}_r\\ &-\boldsymbol{\gamma}_r'\mathbf{V}_{rs}\mathbf{V}_{ss}^{-1}\mathbf{X}_s\mathbf{A}_s^{-1}\mathbf{X}_s'\mathbf{V}_{ss}^{-1}\mathbf{V}_{sr}\boldsymbol{\gamma}_r\\ &-\boldsymbol{\gamma}_r'\mathbf{X}_r\mathbf{A}_s^{-1}\mathbf{X}_r'\boldsymbol{\gamma}_r \end{aligned} \]

and the second part P2 is equivalent to \[ \begin{aligned} P2&=2\boldsymbol{\gamma}_r'\mathbf{V}_{rs}\mathbf{V}_{ss}^{-1}\mathbf{V}_{sr}\boldsymbol{\gamma}_r\\ &-2\boldsymbol{\gamma}_r'\mathbf{V}_{rs}\mathbf{V}_{ss}^{-1}\mathbf{X}_s\mathbf{A}_s^{-1}\mathbf{X}_s'\mathbf{V}_{ss}^{-1}\mathbf{V}_{sr}\boldsymbol{\gamma}_r\\ &+2\boldsymbol{\gamma}_r'\mathbf{X}_r\mathbf{A}_s^{-1}\mathbf{X}_s'\mathbf{V}_{ss}^{-1}\mathbf{V}_{sr}\boldsymbol{\gamma}_r \end{aligned} \] Since the transpose of a number is the same number, adding the necessary parts gives the full proof of the theorem.

Note that all estimators, predictors, and models in the previous sections are particular cases of this result.

15.4 Ignoring the Sampling Design

Gelman et al. (2004) explain that one would have to be a naive statistician to claim that all inference should be conditional on the data, regardless of where or how they were selected. This is a mistaken conception of the likelihood principle. The notion that the sample selection method is irrelevant in inferential analysis can be contradicted with a very simple argument: suppose that ten data points are available from rolling ten dice, and all of them were sixes. The statistician’s attitude about the nature of the data would be different if (1) only ten rolls were made, (2) sixty rolls were made but only those that resulted in six were reported, or (3) six appeared ten times in five hundred rolls and these realizations were reported honestly. In such situations, it is clear that the distribution of the observed data follows a completely different pattern that should not be ignored.

In general terms, a sampling design is nothing more than a multivariate probability distribution defined over a set of samples that belong to a support. But a probability distribution is only an assumed model; in this case, it is a model that enables the selection of probability samples. A sample $s$ induces an inclusion vector given by \[ \mathbf{I}(s)=(I_1(s),\ldots,I_k(s),\ldots,I_N(s))' \]

where $I_k(s)$ is defined by (2.1.8). Given the previous scheme, another way to denote the sampling design is $f_{\mathbf{I}}(\mathbf{I})$, which is known for all possible values of $\mathbf{I}$ in all possible samples $s$. On the other hand, if it is assumed that measuring the characteristic of interest $y_k$ in population individuals is subject to error, then these measurements should be viewed as realizations of random variables $Y_k$. Thus, it is necessary to define a model for the population values that may depend on a certain parameter. In this case, if $Y=(Y_1,\ldots,Y_k,\ldots,Y_N)'$ is the population vector of the characteristic of interest, then $f_{\mathbf{Y}}(\mathbf{Y};\theta)$ will define such a model.

To make any kind of inference about the parameter $\theta$, it is necessary to work with a joint probability distribution of ($\mathbf{I},\mathbf{Y}$) that makes it possible to unify the previous scheme into a single process. The question for the statistician is the following: how can this joint distribution be expressed in terms of $f_{\mathbf{I}}(\mathbf{I})$ and $f_{\mathbf{Y}}(\mathbf{I};\theta)$? Chambers and Skinner (2003) answer this question by motivating the assumption that $\mathbf{Y}$ is independent of $\mathbf{I}$. In some cases, as in Chambers and Skinner (2003), the sampling design depends on the values of the characteristic of interest; for example, in a case-control study, the response $y_k$ is binary, indicating whether the $k$th unit is a case or a control. In turn, cases and controls induce strata whose samples are selected independently. In this case, the sampling design depends directly on the values of the characteristic of interest. Therefore, the relationship between $\mathbf{I},\mathbf{Y}$ must be expressed as \[ f_{\mathbf{I},\mathbf{Y}}(\mathbf{I},\mathbf{Y};\theta)= f_{\mathbf{I}|\mathbf{Y}}(\mathbf{I}|\mathbf{Y})f_{\mathbf{Y}}(\mathbf{Y};\theta) \]

In this case, the sampling design is said to be informative and cannot be ignored in terms of inference for $\theta$. On the other hand, if the sampling design is noninformative, the relationship between $\mathbf{I},\mathbf{Y}$ must be expressed as \[ f_{\mathbf{I},\mathbf{Y}}(\mathbf{I},\mathbf{Y};\theta)= f_{\mathbf{I}}(\mathbf{I})f_{\mathbf{Y}}(\mathbf{Y};\theta) \]

and clearly, the sampling design can be ignored. Chambers and Skinner (2003) state that sampling designs that depend directly on the variable of interest are not rare in practice. However, sampling designs implemented when the sampling frame is very poor, such as two-phase sampling, where a first sample is selected and the strategy for a second subsample is designed based on its results, cannot be classified as noninformative and therefore cannot be ignored. On the other hand, it is more common to find that the sampling design depends on other auxiliary information variables, as in stratified designs or probability-proportional-to-size designs. The following presents the general framework given by Valliant et al. (2000) for jointly modeling the sampling design and the probabilistic mechanism that gives rise to the variable of interest.

Suppose that the sampling design depends on the variable of interest $\mathbf{Y}$, on some auxiliary information variables collected in a matrix $\mathbf{X}$, and on some vector of parameters $\boldsymbol{\phi}$; then it is rewritten as: \[ f_{\mathbf{I}|\mathbf{X},\mathbf{Y}}(\mathbf{I}|\mathbf{X},\mathbf{Y};\boldsymbol{\phi}). \]

In turn, the probability distribution of $\mathbf{Y}$ depends on $\mathbf{X}$ and its relationship is governed by a vector of parameters $\boldsymbol{\beta}$; then it is rewritten as

\[ f_{\mathbf{Y}|\mathbf{X}}(\mathbf{Y}|\mathbf{X};\boldsymbol{\beta}). \]

Result

A model for $\mathbf{I},\mathbf{Y}$ is given by \[ f_{\mathbf{I},\mathbf{Y}|\mathbf{X}}(\mathbf{I},\mathbf{Y}|\mathbf{X};\boldsymbol{\phi},\boldsymbol{\beta})= f_{\mathbf{Y}|\mathbf{X}}(\mathbf{Y}|\mathbf{X};\boldsymbol{\beta}) f_{\mathbf{I}|\mathbf{Y},\mathbf{X}}(\mathbf{I}|\mathbf{Y},\mathbf{X};\boldsymbol{\phi}) \]

Proof. Applying the definition of joint and conditional distributions gives the result because \[ \begin{aligned} f_{\mathbf{I},\mathbf{Y}|\mathbf{X}}(\mathbf{I},\mathbf{Y}|\mathbf{X};\boldsymbol{\phi},\boldsymbol{\beta})&= \dfrac{f_{\mathbf{I},\mathbf{Y},\mathbf{X}}(\mathbf{I},\mathbf{Y},\mathbf{X};\boldsymbol{\phi},\boldsymbol{\beta})} {f_{\mathbf{X}}(\mathbf{X};\boldsymbol{\beta})}\\ &=\dfrac{f_{\mathbf{I}|\mathbf{Y},\mathbf{X}}(\mathbf{I}|\mathbf{Y}|\mathbf{X};\boldsymbol{\phi}) f_{\mathbf{Y},\mathbf{X}}(\mathbf{Y},\mathbf{X};\boldsymbol{\beta})} {f_{\mathbf{Y},\mathbf{X}}(\mathbf{Y},\mathbf{X};\boldsymbol{\beta})/f_{\mathbf{Y}|\mathbf{X}}(\mathbf{Y}|\mathbf{X};\boldsymbol{\beta})}\\ &=f_{\mathbf{Y}|\mathbf{X}}(\mathbf{Y}|\mathbf{X};\boldsymbol{\beta}) f_{\mathbf{I}|\mathbf{Y},\mathbf{X}}(\mathbf{I}|\mathbf{Y},\mathbf{X};\boldsymbol{\phi}) \end{aligned} \]

Of course, unless this is a census, we will never observe all elements of the vector $\mathbf{Y}$. That is, in a sense, model (11.4.3) is useless for inference. When a sample $s$ is selected, the vector $\mathbf{Y}$ is immediately partitioned into $\mathbf{Y}_s,\mathbf{Y}_r$. In this way, the relationship between $\mathbf{I},\mathbf{Y}_s$ is given by the following result.

Result

The joint distribution of $\mathbf{I},\mathbf{Y}_s$ is given by \[ f_{\mathbf{I},\mathbf{Y}_s|\mathbf{X}}(\mathbf{I},\mathbf{Y}_s|\mathbf{X};\boldsymbol{\phi},\boldsymbol{\beta})= \int f_{\mathbf{Y}|\mathbf{X}}(\mathbf{Y}_s,\mathbf{Y}_r|\mathbf{X};\boldsymbol{\beta}) f_{\mathbf{I}|\mathbf{Y},\mathbf{X}}(\mathbf{I}|\mathbf{Y}_s,\mathbf{Y}_r,\mathbf{X};\boldsymbol{\phi})\ d\mathbf{Y}_r \]

Proof. This proof is based on the definition of joint and marginal density functions (Mood et al. 1974, 141), which states that if $V$ and $W$ are two random variables with joint density given by $f_{V,W}(V,W)$, then the marginal density of $V$ is given by $\int f_{V,W}(V,W)dW$. In our conditional context, note that the vector $\mathbf{Y}$ has been partitioned; therefore, applying the previous principle and using the previous result, we have \[ \begin{aligned} f_{\mathbf{I},\mathbf{Y}_s|\mathbf{X}}(\mathbf{I},\mathbf{Y}_s|\mathbf{X};\boldsymbol{\phi},\boldsymbol{\beta})&= \int f_{\mathbf{I},\mathbf{Y}|\mathbf{X}}(\mathbf{I},\mathbf{Y}_s,\mathbf{Y}_r|\mathbf{X};\boldsymbol{\phi},\boldsymbol{\beta})\ d\mathbf{Y}_r\\ &=\int f_{\mathbf{Y}|\mathbf{X}}(\mathbf{Y}_s,\mathbf{Y}_r|\mathbf{X};\boldsymbol{\beta}) f_{\mathbf{I}|\mathbf{Y},\mathbf{X}}(\mathbf{I}|\mathbf{Y}_s,\mathbf{Y}_r,\mathbf{X};\boldsymbol{\phi})\ d\mathbf{Y}_r \end{aligned} \]

Note that if the sampling design is ignorable, then the probabilistic mechanism governing sample selection does not depend on the configuration of the population values of the variable of interest; this would mean that \[ f_{\mathbf{I}|\mathbf{Y},\mathbf{X}}(\mathbf{I}|\mathbf{Y},\mathbf{X};\boldsymbol{\phi})= f_{\mathbf{I}|\mathbf{X}}(\mathbf{I}|\mathbf{X};\boldsymbol{\phi}). \]

If this were to happen, then (11.4.4) would become

\[ \begin{aligned} f_{\mathbf{I},\mathbf{Y}_s|\mathbf{X}}(\mathbf{I},\mathbf{Y}_s|\mathbf{X};\boldsymbol{\phi},\boldsymbol{\beta})&= f_{\mathbf{I}|\mathbf{X}}(\mathbf{I}|\mathbf{X};\boldsymbol{\phi}) \int f_{\mathbf{Y}|\mathbf{X}}(\mathbf{Y}_s,\mathbf{Y}_r|\mathbf{X};\boldsymbol{\beta})\ d\mathbf{Y}_r\\ &=f_{\mathbf{I}|\mathbf{X}}(\mathbf{I}|\mathbf{X};\boldsymbol{\phi}) f_{\mathbf{Y}_s|\mathbf{X}}(\mathbf{Y}_s|\mathbf{X};\boldsymbol{\beta}) \end{aligned} \]

In terms of statistical inference for the parameter vector $\boldsymbol{\beta}$, the following comments apply:

Note that (11.4.7) is composed of two terms that are multiplied as follows: \[ \begin{aligned} f_{\mathbf{I},\mathbf{Y}_s|\mathbf{X}}(\mathbf{I},\mathbf{Y}_s|\mathbf{X};\boldsymbol{\phi},\boldsymbol{\beta})= \underbrace{f_{\mathbf{I}|\mathbf{X}}(\mathbf{I}|\mathbf{X};\boldsymbol{\phi})}_{h(\mathbf{X})} \underbrace{f_{\mathbf{Y}_s|\mathbf{X}}(\mathbf{Y}_s|\mathbf{X};\boldsymbol{\beta})}_{g(T;\boldsymbol{\beta})} \end{aligned} \] The above implies that if there were a sufficient statistic $T$ for $\boldsymbol{\beta}$, then, by the Neyman Factorization Criterion (Mood et al. 1974, 306), $T$ would be contained in the conditional density $f_{\mathbf{Y}_s|\mathbf{X}}(\mathbf{Y}_s|\mathbf{X};\boldsymbol{\beta})$. For this reason, in terms of statistical inference for $\boldsymbol{\beta}$, the distribution $f_{\mathbf{I}|\mathbf{X}}(\mathbf{I}|\mathbf{X};\boldsymbol{\phi})$ would contain no information.
A measure of how well the data support a parameter $\boldsymbol{\beta}_2$ compared with a parameter $\boldsymbol{\beta}_1$ is the likelihood ratio criterion (Mood et al. 1974, 419), which is given by \[ \begin{aligned} \dfrac{f_{\mathbf{I},\mathbf{Y}_s|\mathbf{X}}(\mathbf{I},\mathbf{Y}_s|\mathbf{X};\boldsymbol{\phi},\boldsymbol{\beta}_2)} {f_{\mathbf{I},\mathbf{Y}_s|\mathbf{X}}(\mathbf{I},\mathbf{Y}_s|\mathbf{X};\boldsymbol{\phi},\boldsymbol{\beta}_1)} &=\dfrac{f_{\mathbf{I}|\mathbf{X}}(\mathbf{I}|\mathbf{X};\boldsymbol{\phi}) f_{\mathbf{Y}_s|\mathbf{X}}(\mathbf{Y}_s|\mathbf{X};\boldsymbol{\beta}_2)} {f_{\mathbf{I}|\mathbf{X}}(\mathbf{I}|\mathbf{X};\boldsymbol{\phi}) f_{\mathbf{Y}_s|\mathbf{X}}(\mathbf{Y}_s|\mathbf{X};\boldsymbol{\beta}_1)}\\ &=\dfrac{f_{\mathbf{Y}_s|\mathbf{X}}(\mathbf{Y}_s|\mathbf{X};\boldsymbol{\beta}_2)} {f_{\mathbf{Y}_s|\mathbf{X}}(\mathbf{Y}_s|\mathbf{X};\boldsymbol{\beta}_1)} \end{aligned} \] Once again, in terms of inference for $\boldsymbol{\beta}$, the distribution $f_{\mathbf{I}|\mathbf{X}}(\mathbf{I}|\mathbf{X};\boldsymbol{\phi})$ would contain no information.

The previous arguments suggest that it is possible not to take the distribution $f_{\mathbf{I}|\mathbf{X}}(\mathbf{I}|\mathbf{X};\boldsymbol{\phi})$ into account. If this were to happen, then, if a sampling design is ignorable, (11.4.7) would become \[ \begin{aligned} f_{\mathbf{I},\mathbf{Y}_s|\mathbf{X}}(\mathbf{I},\mathbf{Y}_s|\mathbf{X};\boldsymbol{\phi},\boldsymbol{\beta}) =f_{\mathbf{Y}_s|\mathbf{X}}(\mathbf{Y}_s|\mathbf{X};\boldsymbol{\beta}) \end{aligned} \]

which leads to the conclusion that the sample selection mechanism truly can be disregarded. Sudgen and Smith (1984) state that sampling designs such as simple random sampling, stratified random sampling, probability-proportional-to-size sampling, convenience sampling, or balanced sampling correspond to cases in which it is possible to ignore the selection mechanism. They also conclude that although sampling designs can sometimes be ignored in terms of inference for $\boldsymbol{\beta}$, it is wrong to think that they can always be ignored in terms of predictive inference for the population total $T_y$.

In conclusion, the choice of approach (design-based or predictive-model-based) should be based on the adequacy of the model for the population; that is, on whether the assumed model is correct. Thus, if the predictive-model-based approach is chosen and the model is not correct, the estimates will be biased away from reality. On the other hand, estimates based on the sampling design are robust and unbiased under any model.

15.5 Exercises

Suppose that the following model $\xi$ fits the population:

\[ Y_k=\mu+\sqrt{X_k} \varepsilon_k \ \ \ \ \ \ \ \ k=1,\ldots,N. \]

where $E_{\xi}(\varepsilon_k)=0$, $Var_{\xi}(\varepsilon_k)=\sigma^2$, and the errors are considered independent. Suppose the following predictors for the population total $T_Y$:

\[ \begin{aligned} \hat{T}_{Y,1}&=N\frac{\sum_{k\in s} Y_k/X_k}{\sum_{k\in s} 1/X_k} \\ \hat{T}_{Y,2}&=\sum_{k\in s} Y_k + (N-n)\frac{\sum_{k\notin s} Y_k/X_k}{\sum_{k\notin s} 1/X_k} \end{aligned} \]

Show that both $\hat{T}_{Y,1}$ and $\hat{T}_{Y,2}$ are unbiased under the predictive model, so that $E_{\xi}(\hat{T}_{Y}-T_Y)=0$.
Assuming that the sample $s$ was selected using a simple random sampling design, show that none of the previous predictors is unbiased with respect to this sampling design.
Suppose that the sample $s$ was selected using a $\pi$PS sampling design, with $\pi_k=nX_k/T_X$. Show that $\hat{T}_{Y,1}$ is unbiased with respect to this sampling design, assuming that the sample size is large enough to state that the expectation of the ratio is approximately equal to the ratio of expectations.
Generate a normal population of $N=40$ units with mean $2 X_k$ and variance 4, with $X_k$ varying between 10 and 20. Select 10 simple random samples of size $n=5$ from this population. Calculate $\bar{X}_s$ and $\hat{T}_0=N\bar{Y}_s$ for these 10 samples. Is there any correspondence between $\bar{X}_s-\bar{X}_U$ and $\hat{T}_0-T_Y$?
Suppose that a sample of size $n=10$ was selected from a population of $N=393$ hospitals. The characteristic of interest is the number of patients served in a specific period of time. In addition, an auxiliary information characteristic is known, corresponding to the number of hospital beds. In the hospital population, the total number of beds is 107956 and the total number of patients served is 320159. Assume that the collected values are the following:

$Y$:	41	92	297	377	95	231	601	1063	1645	1894
$X$:	15	25	80	96	111	125	242	275	551	937

Make a scatterplot of $Y$ against $X$.
Suppose that a model of the form $Y_k=\beta X_k+\varepsilon_k$, with $\varepsilon_k \sim (0, \sigma^2X_k)$, is to be fitted. Calculate the best estimator for $\beta$ and plot the estimated regression line on the scatterplot from part (a).
Suppose that a model of the form $Y_k=\beta_0+\beta_1X_k+\varepsilon_k$, with $\varepsilon_k \sim (0, \sigma^2)$, is to be fitted. Calculate the best estimator for $\boldsymbol{\beta}=(\beta_0,\beta_1)$ and plot the estimated regression line on the scatterplot from part (a).
Calculate the expansion predictor $\hat{T}_0=\frac{N}{n}\sum_sY_k$, the ratio predictor $\hat{T}_r=N\bar{Y}_s\frac{\bar{X}_U}{\bar{X}_s}$, and the linear regression predictor $\hat{T}_{lr}=\frac{N}{n}\sum_sY_k+(T_X-\frac{N}{n}\sum_sX_k)\hat{\beta}_1$. Calculate the estimation error for each estimate. What is the effect of using the auxiliary information characteristic?
Suppose that a sample of size $n=10$ was selected from a population of $N=393$ hospitals. The characteristic of interest is the number of patients served in a specific period of time. In addition, an auxiliary information characteristic is known, corresponding to the number of hospital beds. In the hospital population, the total number of beds is 107956 and the total number of patients served is 320159. Assume that the collected values are the following:

$Y$:	78	315	594	778	410	754	1166	1632	1547	2818
$X$:	38	70	113	156	227	279	347	437	549	860

Make a scatterplot of $Y$ against $X$.
Suppose that a model of the form $Y_k=\beta X_k+\varepsilon_k$, with $\varepsilon_k \sim (0, \sigma^2X_k)$, is to be fitted. Calculate the best estimator for $\beta$ and plot the estimated regression line on the scatterplot from part (a).
Suppose that a model of the form $Y_k=\beta_0+\beta_1X_k+\varepsilon_k$, with $\varepsilon_k \sim (0, \sigma^2)$, is to be fitted. Calculate the best estimator for $\boldsymbol{\beta}=(\beta_0,\beta_1)$ and plot the estimated regression line on the scatterplot from part (a).
Calculate the expansion predictor $\hat{T}_0=\frac{N}{n}\sum_sY_k$, the ratio predictor $\hat{T}_r=N\bar{Y}_s\frac{\bar{X}_U}{\bar{X}_s}$, and the linear regression predictor $\hat{T}_{lr}=\frac{N}{n}\sum_sY_k+(T_X-\frac{N}{n}\sum_sX_k)\hat{\beta}_1$. Calculate the estimation error for each estimate. What is the effect of using the auxiliary information characteristic?
Write the following simulation program:
Generate a population of size $N=200$ following the model $Y_k= X_k+\varepsilon_k\sqrt{X_k}$, with independent $\varepsilon_k$ following a standard normal distribution and $X_k$ varying between 10 and 20.
Select 50 simple random samples without replacement of size $n=30$, and calculate the ratio predictor for each sample.
Calculate the regression predictor under the model $Y_k= X_k+\varepsilon_k$.
Compare, empirically, the biases and mean squared errors of the regression and ratio predictors.
How much efficiency is lost by using incorrect variance specifications?
Consider the following hypothetical situations for some studies:
A stratified random sample of hospitals is selected using population stratification by the types of service provided by each hospital. Estimates are needed for average length of stay per patient, classified by disease type for a particular quarter. Since the largest hospitals in the region tend not to respond, even if they are selected randomly, they will not be taken into account.
Suppose that in part (a), all selected hospitals (both large hospitals and the others) are contacted for an interview, but half of the selected hospitals refuse to respond.
Suppose that in part (a), all selected hospitals agree to respond to the interview, but large hospitals provide information only during the third week of each month.

Discuss whether the previous sampling mechanisms are ignorable or not.

Show that the general regression predictor $\hat{T}_{Y, greg}=\hat{T}_{Y,\pi}+(\mathbf{T_X}-\mathbf{\hat{T}_{X,\pi}})'\hat{\boldsymbol{\beta}}$ is unbiased under the model $Y_k= \mathbf{X}'_k\boldsymbol{\beta}+\varepsilon_k$. Suppose that the errors have zero mean and constant variance.

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R. A. Fisher’s early work was influenced by Hubback. Note that Fisher justified data analysis without considering the selection method only in cases where the results were very close when considering randomization of the units (Smith 1976).↩︎
Under the design-based inferential approach, the interpretation is as follows: if all possible samples of size $n$ from the support $Q$ induced by the sampling design are considered and 95% confidence intervals for the mean are constructed, then 95% of those intervals are expected to contain the parameter $\mu$. In contrast, the predictive-model-based inferential approach must be interpreted in terms of model 11.2.1. In this way, the procedure induces two random variables LS and LI such that $Pr(LI \leq \mu \leq LS)=0.95$.↩︎