The Generalized Regression (GREG) Estimator in Sample Survey Statistics
1. Introduction to the Generalized Regression (GREG) Estimator
The Generalized Regression (GREG) estimator is a cornerstone of modern sample survey statistics, representing a sophisticated model-assisted technique for estimating finite population parameters, most commonly population totals or means.1 Its fundamental characteristic is the incorporation of auxiliary information—data variables for which population-level information (such as totals or means) is known—to enhance the precision of estimates derived from a sample.3 The GREG estimator is formulated within a model-assisted framework, which means it utilizes a "working" statistical model to guide the estimation process while crucially retaining desirable properties rooted in the probability sampling design itself.5 This approach allows it to generalize and extend simpler estimation methods like post-stratification and raking.1
The primary purpose of employing the GREG estimator is to achieve a significant increase in the precision of survey estimates, translating to a reduction in their sampling variance, when compared to estimators that do not leverage auxiliary information, such as the basic Horvitz-Thompson (HT) estimator.2 Beyond variance reduction, the GREG estimator is also motivated by its potential to reduce bias, particularly non-response bias. This is achieved by calibrating the sample weights such that the weighted sample estimates of auxiliary variable totals match their known population totals, thereby aligning the sample more closely with the population structure.5 The impetus for its use often stems from the practical reality that auxiliary data are frequently available from sources like administrative registers, censuses, or previous survey waves, and the GREG provides a systematic way to harness this information for more reliable and efficient statistical inference.3
The conceptual origins of the GREG estimator can be traced back to discussions by Cassel, Särndal, and Wretman in 1976.7 However, its theoretical framework was extensively developed and popularized by Carl-Erik Särndal, Bengt Swensson, and Jan Wretman in their seminal 1992 book, "Model Assisted Survey Sampling".3 This work is widely regarded as a cornerstone reference that solidified the model-assisted approach and the associated theory of calibration weighting in survey sampling. Earlier foundational work by Särndal in the early 1980s also contributed significantly to its development.7 Furthermore, the application of GREG principles to estimation for specific population subgroups, or domains, was pioneered by researchers including Särndal, and Michael Hidiroglou, in the 1980s.12
The introduction and formalization of the GREG estimator marked a significant evolution in survey estimation methodology. Traditionally, survey estimation relied heavily on purely design-based approaches, such as the Horvitz-Thompson estimator, where the statistical properties of an estimator are derived solely from the probabilistic mechanism of the sample selection process.5 At the other end of the spectrum, purely model-based estimators depend critically on the assumed statistical model for the population data. The GREG estimator forged a middle path.5 It strategically employs a "working model" – typically a linear regression model – not as an assertion of absolute truth about the population structure, but as an instrumental tool to improve the efficiency of estimates. The genius of this approach, largely crystallized by Särndal, Swensson, and Wretman (1992), lies in the fact that key desirable properties of the GREG estimator, most notably its asymptotic design-unbiasedness, are maintained even if the chosen working model is misspecified.5 This hybrid nature, which leverages the insights from statistical models for efficiency gains while retaining the robustness and validity conferred by the sampling design, represented a paradigm shift. It provided survey practitioners with a powerful and theoretically sound method to make more effective use of auxiliary data without abandoning the foundational principles of design-based inference.
2. Core Principles and Mathematical Formulation
The GREG estimator operates within a model-assisted framework, distinguishing it from purely design-based or purely model-based estimators. This framework posits a "working" or "assisting" statistical model, commonly a linear regression model, to describe the anticipated relationship between the study variable of interest, denoted as y, and a vector of auxiliary variables, denoted as x.4 A crucial tenet of this framework is that the desirable design-based properties of the GREG estimator, particularly its asymptotic design-unbiasedness, do not hinge on the perfect correctness of this assisting model.2 Statistical inference is primarily anchored to the stochastic structure imposed by the sampling design rather than the model itself.5
The general mathematical formula for the linear GREG estimator of a population total, Ty​=∑k∈U​yk​ (where U denotes the finite population), is typically expressed as 4:
T^y,GREG​=T^y,HT​+B^T(Tx​−T^x,HT​)
Where:
●	T^y,HT​=∑k∈s​πk​yk​​ is the Horvitz-Thompson (HT) estimator of the total of the study variable y. Here, yk​ is the value of the study variable for unit k, s denotes the sample, and πk​ is the first-order inclusion probability of unit k in the sample.15
●	Tx​=∑k∈U​xk​ is the known vector of population totals for the auxiliary variables x.4
●	T^x,HT​=∑k∈s​πk​xk​​ is the HT estimator of the vector of population totals for the auxiliary variables x.4
●	B^ is the vector of estimated regression coefficients from the assisting linear model yk​=xkT​B+ϵk​. These coefficients are typically estimated using weighted least squares (WLS), incorporating the survey design weights. The formula for B^ is: $$ \hat{\mathbf{B}} = \left( \sum_{k \in s} \frac{\mathbf{x}_k \mathbf{x}k^T}{\pi_k v_k} \right)^{-1} \left( \sum{k \in s} \frac{\mathbf{x}_k y_k}{\pi_k v_k} \right) $$ Here, vk​ is a known constant related to the variance of the error term ϵk​ in the assisting model, i.e., Vξ​(ϵk​)=vk​σ2, where ξ denotes expectation with respect to the model.4 Often, vk​ is set to 1, implying a homoscedastic model, or it can be chosen to model heteroscedasticity (e.g., vk​=xjk​ for some auxiliary variable xjk​).4 The term πk​vk​ in the denominator can also be seen as πk​/ck​, where ck​ are positive constants specified by the statistician.17
The GREG estimator formula comprises two main components:
1.	The Horvitz-Thompson (HT) part (T^y,HT​): This is a design-unbiased estimator of the population total Ty​ that would be used if no auxiliary information were incorporated or if the sample perfectly mirrored the population with respect to the auxiliary variables (i.e., Tx​=T^x,HT​).4
2.	The regression adjustment term (B^T(Tx​−T^x,HT​)): This term adjusts the initial HT estimate. The adjustment is proportional to the difference between the known population totals of the auxiliary variables (Tx​) and their sample-based HT estimates (T^x,HT​). The magnitude and direction of this adjustment are determined by the strength and nature of the estimated relationship (B^) between the study variable y and the auxiliary variables x.4
The design weights, dk​=1/πk​ (also denoted ak​=1/πk​ in 18), are fundamental to the GREG estimator. They are incorporated not only in the calculation of the HT estimates but also critically in the estimation of the regression coefficient vector B^, ensuring that the estimator retains its desirable design-based properties.4
An alternative but equivalent representation of the GREG estimator is as a linearly weighted sum of the sample yk​ values 4:
T^y,GREG​=k∈s∑​wk​yk​
where wk​ are the GREG weights, often referred to as calibrated weights. These weights can be expressed as wk​=dk​gk​=(1/πk​)gk​, where gk​ are known as "g-weights" or calibration factors. For the linear GREG estimator, the g-weight for unit k is given by 4:
$$ g_k = 1 + (\mathbf{T}x - \hat{\mathbf{T}}{x,HT})^T \left( \sum_{j \in s} \frac{\mathbf{x}_j \mathbf{x}_j^T}{\pi_j v_j} \right)^{-1} \frac{\mathbf{x}_k}{v_k} $$
It is often observed that these gk​ values are close to 1, especially in large samples or when the sample is well-balanced with respect to the auxiliary variables.13
The derivation of the GREG estimator can be approached by considering the prediction of non-sampled yk​ values using the assisting model. The sum of these predictions for non-sampled units is added to the sum of observed yk​ values for sampled units, followed by adjustments to ensure good design-based properties. A common alternative form that illuminates this predictive aspect is 13:
T^y,GREG​=k∈U∑​y^​k​+k∈s∑​πk​yk​−y^​k​​
where y^​k​=xkT​B^ are the model-predicted values for all units k in the population U. The first term, ∑k∈U​y^​k​=TxT​B^, represents a synthetic-type estimate based on the model. The second term is a design-weighted sum of the sample residuals (ek​=yk​−y^​k​), which corrects the synthetic estimate for model deficiencies.
This structure inherently balances model-based prediction with design-based rigor. The term ∑k∈U​y^​k​ can be viewed as a model-based prediction of the population total if the model were to hold perfectly for the entire population. The term ∑k∈s​(yk​−y^​k​)/πk​ effectively adjusts this prediction using the observed discrepancies in the sample, weighted by the inverse inclusion probabilities. If the model is perfectly specified such that yk​=y^​k​ for all k∈s, then the adjustment term vanishes, and the estimator relies on the model predictions calibrated by the auxiliary totals Tx​. Conversely, if the model is poor, the residuals yk​−y^​k​ will be substantial, and their design-weighted sum will dominate the correction, ensuring that the estimator's properties are primarily driven by the sampling design. This elegant interplay between model prediction and design-weighted empirical correction is central to the "model-assisted" philosophy of the GREG estimator. If the sample estimate of auxiliary totals (T^x,HT​) happens to perfectly match the known population totals (Tx​), the adjustment term in the primary formula becomes zero, and the GREG estimator reduces to the HT estimator.4
3. Statistical Properties of the GREG Estimator
The GREG estimator possesses several key statistical properties that make it attractive for survey estimation. These properties are primarily evaluated with respect to the sampling design (design-based properties).
Design Consistency:
GREG estimators are constructed to be design consistent.3 This implies that as the sample size n increases and approaches the population size N (under appropriate asymptotic settings for finite populations), the GREG estimator T^y,GREG​ converges in probability to the true population total Ty​. This property ensures that with sufficient sample information, the estimator will yield a value close to the true parameter.
Asymptotic Unbiasedness:
A cornerstone property of the GREG estimator is its asymptotic design unbiasedness (ADU).2 This means that the design bias of the estimator, Ep​(T^y,GREG​−Ty​) (where Ep​ denotes expectation over all possible samples under the given design), tends to zero as the sample size becomes large. Crucially, this ADU property holds irrespective of how well the assisting model actually describes the true relationship between the study variable y and the auxiliary variables x.2 This robustness to model misspecification is a defining feature of the model-assisted approach and a primary reason for the GREG's widespread appeal.5
The term "nearly design unbiased" is also frequently used, particularly in contexts where the bias ratio (defined as the bias divided by the standard deviation of the estimator) approaches zero at a rate of O(n−1/2) as the total sample size n tends to infinity.3 For estimators exhibiting this property, the design bias makes an asymptotically insignificant contribution to the estimator's Mean Squared Error (MSE) under mild conditions.3
Efficiency and Variance Considerations:
The primary motivation for using the GREG estimator is to improve the efficiency of estimation, which translates to a reduction in the sampling variance.
If the assisting model provides a good fit to the data—meaning the auxiliary variables x are strongly correlated with the study variable y—the GREG estimator can achieve substantial reductions in design variance compared to estimators that do not use auxiliary information, like the HT estimator.1 The stronger the linear relationship captured by the assisting model, the greater the potential gain in efficiency.13
The design variance of the GREG estimator can be approximated by the design variance of the sum of weighted population residuals:
Vp​(T^y,GREG​)≈Vp​(k∈s∑​πk​ek​​)
where ek​=yk​−xkT​B are the population residuals based on the true population regression coefficient vector B.13 This means that if the model explains a large portion of the variation in y, the residuals ek​ will be small, leading to a smaller variance for T^y,GREG​.
While model misspecification does not invalidate the ADU property, it can lead to an increase in the design variance.5 Therefore, careful model specification remains important, particularly in practical applications with highly skewed data, such as those encountered in many business surveys.20 Furthermore, if the design weights are widely dispersed, the GREG estimator might be less efficient than a Best Linear Unbiased (BLU) estimator, especially if the assumptions required for the BLU estimator (correct model specification and ignorable design) hold.19
The ADU property of the GREG estimator serves as a powerful theoretical safeguard, ensuring that, for sufficiently large samples, bias stemming from the choice of the assisting model becomes negligible.2 However, this asymptotic comfort does not fully translate to all finite sample situations. In practice, especially when dealing with small sample sizes overall, or within specific domains 3, a poorly fitting or misspecified assisting model can lead to an inflated variance 5 or even a non-negligible finite sample bias.6 As noted in 20, the ADU property "is not a substitute for a careful model specification search, especially when dealing with the highly variable and outlier prone populations that are the focus of many business surveys." This highlights a practical tension: while the theory provides assurance for large samples, practitioners must still exercise diligence in selecting, fitting, and validating their assisting models to ensure good performance in the finite sample context they face. The robustness of GREG to model choice is primarily an asymptotic property concerning bias, not necessarily a guarantee of optimal Mean Squared Error performance in all scenarios.
Moreover, the efficiency gains offered by the GREG estimator are fundamentally linked to the actual predictive power of the chosen auxiliary variables for the study variable, not merely their availability. The variance reduction is achieved because the adjustment term in the GREG formula effectively diminishes the component of variability that needs to be estimated directly from the sample observations of y.11 If the auxiliary variables x are not strongly correlated with the study variable y, the estimated regression coefficients B^ will tend to be small or statistically insignificant. In such cases, the adjustment term (Tx​−T^x,HT​) might not effectively compensate for sampling variations or reduce overall variance.1 As emphasized in 21, for estimators like the ratio and regression estimators (which GREG generalizes), the auxiliary variable must be "strongly correlated with the variable of interest" to achieve greater precision. Thus, the mere presence of auxiliary data is insufficient; its relevance and predictive capacity for the study variable are paramount for the GREG estimator to offer substantial efficiency improvements over simpler estimators like the Horvitz-Thompson estimator. This underscores the critical importance of thoughtful model selection and validation in the application of GREG.
4. The Role of Auxiliary Information and Calibration Weighting
The GREG estimator's effectiveness is intrinsically tied to its ability to leverage auxiliary information and its formulation as a calibration estimator.
Mechanism of Incorporating Auxiliary Data:
The GREG estimator utilizes auxiliary variables xk​ for which reliable population-level information, typically population totals Tx​, is available from external sources such as censuses, administrative registers, or prior surveys.3 This information is integrated into the estimation process in two principal ways:
1.	Model Fitting: The sample data (xk​,yk​) for k∈s are used to estimate the parameters (e.g., regression coefficients B^) of the assisting model that describes the relationship between y and x.
2.	Adjustment Term: The known population totals Tx​ are directly used in the adjustment component of the GREG formula, B^T(Tx​−T^x,HT​), to correct the initial HT estimate of Ty​ based on discrepancies between sample estimates and population benchmarks for the auxiliary variables.4
Auxiliary information can be available at different levels of aggregation. Unit-level (micro-data) auxiliary information, where xk​ values are known for each unit in the sample and potentially for all units in the population, allows for more flexible and powerful modeling approaches, such as the use of Generalized Linear Mixed Models (GLMMs).3 Aggregate-level auxiliary data, where only population totals Tx​ are known, is also commonly used.
GREG as a Calibration Estimator:
The GREG estimator is a prominent example and, in many ways, a foundational member of the broader family of calibration estimators.1
The calibration principle in survey sampling involves adjusting the initial design weights, dk​=1/πk​, to a new set of weights, wk​, often called calibrated weights. This adjustment is performed such that two conditions are met:
1.	The new weights wk​ should be "close" to the original design weights dk​, according to some chosen distance measure.
2.	The new weights wk​ must satisfy a set of constraints known as calibration equations.1
The calibration equations typically require that the sum of the calibrated weights multiplied by the auxiliary variable values for the sampled units equals the known population totals for those auxiliary variables. That is, for each auxiliary variable xj​ in the vector x:
k∈s∑​wk​xk​=Tx​
This ensures that the weighted sample, using the calibrated weights, perfectly reproduces the known population totals for the auxiliary variables, thereby achieving consistency with these benchmarks.1
The linear GREG estimator specifically arises when the distance measure chosen for minimizing the difference between wk​ and dk​, subject to the calibration equations, is a chi-squared type distance function, such as D(w,d)=∑k∈s​dk​qk​(wk​−dk​)2​, where qk​ are pre-specified positive constants (often qk​=1 or related to the model variance vk​).14 The resulting calibrated weights wk​ (also denoted wGREG,i​ in 4) are then used to compute the GREG estimate of the total as T^y,GREG​=∑k∈s​wk​yk​.
Relationship to the Broader Family of Calibration Estimators:
The work of Särndal, Swensson, and Wretman (1992) demonstrated that the GREG estimator (derived from the chi-squared distance) serves as a first-order approximation to all other members of the family of calibration estimators that can be derived using different distance measures (e.g., raking ratio distance, logit distance).11 Importantly, all such calibration estimators are asymptotically equivalent to the GREG estimator. This implies that, for large samples, their statistical properties (like variance) are similar, and the variance estimator developed for the GREG can typically be used for these other calibration estimators as well.11 The choice among different calibration estimators in practice often hinges on considerations such as the numerical properties of the resulting weights (e.g., the desire to avoid negative or excessively large weights) and computational convenience.11 For instance, raking (iterative proportional fitting) is another well-known calibration method, and the GREG framework can be seen as generalizing it.1
The calibration perspective offers a powerful alternative justification for the GREG estimator, distinct from its derivation via an explicit regression model, and underscores its crucial role in ensuring sample representativeness concerning the auxiliary variables. While the GREG is frequently introduced through the lens of an assisting regression model 4, its formulation via weight calibration 11 provides an equally valid and insightful understanding. The calibration equations, ∑wk​xk​=Tx​ 11, guarantee that the weighted sample accurately reflects the known population aggregates for the auxiliary variables. This property is often described as a "consistency check" 11 and is highly appealing to survey practitioners. The rationale is that if the calibrated weights perform well in estimating the totals of the auxiliary variables (which are often chosen for their presumed relevance to the study variables), they are also likely to perform well for estimating the total of the study variable y.11 This dual pathway to the GREG estimator—one through model assistance and the other through calibration—strengthens its conceptual underpinnings and helps explain its widespread adoption and utility in survey practice.
Furthermore, the principle of ensuring the "closeness" of the calibrated weights wk​ to the original design weights dk​ is a critical, albeit sometimes implicit, optimality criterion embedded within the GREG methodology. This aims to minimize the loss of information contained in the original sampling design. The design weights dk​=1/πk​ encapsulate all the information derived from the (potentially complex) sampling procedure used to select the sample.17 Deviating substantially from these original weights without strong justification could discard valuable design information and potentially lead to increased variance or bias, especially if the assisting model is poorly specified. The calibration process, by seeking new weights wk​ that satisfy the calibration constraints ∑wk​xk​=Tx​ while simultaneously minimizing a chosen distance D(w,d) to the original design weights d 11, implies a constrained optimization. The GREG estimator, therefore, attempts to achieve consistency with known population benchmarks with the "least possible disturbance" to the original design weights, as quantified by the specific distance measure employed. This inherent characteristic helps explain why the GREG estimator retains its fundamental design-based properties, such as asymptotic design-unbiasedness; it does not abandon the design weights but rather adjusts them in an optimal and constrained manner.
5. Advantages and Practical Applications
The GREG estimator offers several distinct advantages over simpler estimation techniques and has found wide application across various fields of survey research.
Improvements in Precision over Simpler Estimators:
A primary advantage of the GREG estimator is its potential to yield estimates with higher precision (i.e., lower sampling variance) compared to the Horvitz-Thompson (HT) estimator, provided that the auxiliary variables used are correlated with the study variable.1 By effectively utilizing the information contained in the auxiliary variables, the GREG estimator can explain a portion of the variability in the study variable, thereby reducing the unexplained variance that contributes to sampling error.
Compared to the classical ratio estimator, the GREG estimator is more general. The ratio estimator can be viewed as a special case of GREG under specific model assumptions (a linear relationship through the origin and a particular error variance structure).3 The GREG estimator is generally more efficient than the ratio estimator unless these restrictive assumptions hold perfectly. Similarly, the GPEG generalizes the classical regression estimator by incorporating design weights and allowing for multiple predictors and more complex error structures.
Potential for Bias Reduction (e.g., non-response bias):
The calibration property inherent in the GREG estimator can contribute to a reduction in bias, particularly bias arising from unit non-response.1 If the auxiliary variables used for calibration are related to both the likelihood of response and the study variable itself, adjusting the weights of responding units to match known population totals for these auxiliaries can help compensate for systematic differences between respondents and non-respondents. This re-weighting aims to make the respondent sample more representative of the target population with respect to the auxiliary characteristics, thereby mitigating potential non-response bias in the estimates of the study variable.
Flexibility in Handling Different Data Types and Sampling Designs:
The GREG framework is notably flexible. The assisting model is not restricted to simple linear regression for continuous study variables. It can be adapted to suit various types of study variables, including:
●	Continuous variables (using a linear regression model).
●	Binary or polytomous categorical variables (using logistic or multinomial logistic regression models).
●	Count variables (using Poisson regression models or other count data models).3 This adaptability allows the GREG estimator to be applied to a wide array of survey data. Furthermore, the GREG estimator is suitable for use with diverse probability sampling designs, including equal probability sampling (e.g., simple random sampling), stratified sampling, cluster sampling 4, and, importantly, complex unequal probability sampling designs.3 The incorporation of design weights (1/πk​) is integral to its formulation for such designs. For multipurpose surveys, where many different variables are of interest, a single set of calibrated GREG weights can often be derived and used for estimating totals or means for all these variables, which is a practical advantage.5
Illustrative Examples:
The GREG estimator is widely employed in various practical survey settings:
●	Official Statistics: National Statistical Institutes (NSIs) around the world routinely use GREG estimators for producing official statistics on economic activity, social conditions, and demographic characteristics.5 For instance, Statistics Canada developed the Generalized Estimation System (GES), a software system based on GREG theory, for its survey programs.24
●	Economic and Business Surveys: GREG is used for estimating characteristics of businesses, such as total revenue, employment, or investment. Administrative data (e.g., tax records, business registers) often serve as powerful auxiliary information in these contexts.20 These surveys often deal with highly skewed distributions and outliers, which has spurred the development of robust versions of the GREG estimator.
●	Environmental Monitoring and Natural Resource Surveys: In fields like forestry, GREG estimators are used to estimate attributes such as total timber volume, basal area, or biomass. Auxiliary information can come from remote sensing data (e.g., satellite imagery providing tree canopy cover) or previous inventories.2
●	Household Surveys: In surveys of individuals and households, GREG can be used to improve estimates of income, expenditure, health status, or employment. A specific application involves the development of "two-level GREG estimators" to ensure consistent estimates between person-level characteristics (e.g., individual income) and household-level characteristics (e.g., total household income) when data are collected at both levels.4
●	Financial Data Analysis: The GREG estimator is valuable for analyzing financial and economic data, which are often incomplete due to missing values or non-response. The ability of GREG to incorporate auxiliary information helps in producing more reliable estimates in such situations.8
An interesting aspect of GREG's application is the potential for synergistic efficiency gains when there is a "double use" of auxiliary information. This occurs when the same auxiliary variable(s) are employed both in the sampling design stage (e.g., for Probability Proportional to Size (PPS) sampling, where a size measure is used to determine inclusion probabilities) and subsequently in the estimation stage via the GREG estimator. For example, in Monte Carlo experiments involving PPS samples, it was observed that the accuracy of the GREG estimator "improved when including the PPS size variable in the assisting model".23 This "double-use" implies that the auxiliary variable first contributes to selecting a more "informative" sample (by giving larger or more relevant units higher chances of selection) and then further helps to refine the estimates derived from that sample by accounting for remaining discrepancies. This suggests a powerful integrated strategy for survey design and estimation: selecting auxiliary variables for the sampling design that are also strong predictors suitable for inclusion in the GREG assisting model.
6. Limitations, Challenges, and Considerations
Despite its strengths, the GREG estimator is not without limitations and challenges that practitioners must consider.
Impact of Model Misspecification:
While a key feature of the GREG estimator is its asymptotic design unbiasedness (ADU) regardless of the correctness of the assisting model, a severely misspecified model can still have adverse consequences in finite samples.1 If the chosen model poorly represents the true relationship between the study variable and the auxiliary variables, the GREG estimator may exhibit increased variance compared to what could be achieved with a better model, or even compared to simpler estimators in some cases. It has been strongly argued that the ADU property "is not a substitute for a careful model specification search," particularly when dealing with highly variable and outlier-prone data common in business surveys.20 The choice of the assisting model, including how to account for potential differences across domains (e.g., using fixed versus mixed effects), remains a critical decision for the practitioner.3
Sensitivity to Small Sample Sizes, Particularly in Domains:
The performance of the GREG estimator can degrade when sample sizes are small, a situation often encountered when producing estimates for specific population subgroups or domains.3 In domains with very few sample observations, the variance of the GREG estimator can become unacceptably large. Moreover, if the linear model does not fit well and the sample size is small, the GREG estimator may exhibit non-negligible bias in finite samples.6 For estimation in very small domains (small area estimation), model-based estimators like the Empirical Best Linear Unbiased Predictor (EBLUP) might be preferred over model-assisted GREG estimators.3
Issues with Outliers and Influential Observations:
The standard GREG estimator, being based on weighted least squares for estimating the regression coefficients, is vulnerable to the presence of outliers or influential observations in the sample data.19 Outliers can disproportionately affect the estimated regression coefficients and, consequently, the final GREG estimate, potentially leading to inflated variance and biased results. This sensitivity is a significant concern, especially in surveys of skewed populations (e.g., business revenues), and has motivated the development of robust GREG estimators (discussed in Section 8).
Variance of GREG Weights and Potential for Extreme Weights:
The calibration process, which adjusts design weights to satisfy constraints based on auxiliary information, can sometimes lead to a wide dispersion in the final GREG weights. In some instances, these calibrated weights can become very large, very small, or even negative.1 Highly variable or extreme weights (particularly negative weights, which lack direct interpretation) can inflate the variance of the GREG estimates and may be problematic for practitioners.1 To mitigate this issue, techniques such as weight trimming (truncating weights at certain bounds) or using bounded calibration functions within the calibration algorithm are often employed.1
Complexity in Variance Estimation:
Estimating the variance of the GREG estimator is generally more complex than for simpler estimators like the HT estimator.1 The variance estimation must account for the sampling design, the use of auxiliary information, and the estimation of the regression coefficients. Common techniques for GREG variance estimation include:
●	Linearization (Taylor series expansion): This method approximates the variance of the GREG estimator by linearizing it around the true population values.1
●	Replication methods: These involve drawing multiple sub-samples (replicates) from the full sample according to a specific rule (e.g., jackknife, bootstrap, balanced repeated replication (BRR)) and calculating the GREG estimate for each replicate. The variability among the replicate estimates is then used to estimate the variance of the full-sample GREG estimate.1 Classical approaches to variance estimation might underestimate the true variance by ignoring the uncertainty stemming from the estimation of the working model parameters, a bias that can be more pronounced with limited sample sizes.10 Furthermore, if the control totals for the auxiliary variables are themselves estimated from another survey (rather than being fixed, known population values), this introduces an additional component of variance that needs to be accounted for, further complicating the variance estimation procedure.30
An inherent tension exists within the GREG framework concerning its attempt to satisfy calibration constraints perfectly while maintaining the stability of its resulting weights. The calibration process forces the weighted sample to precisely match known population totals for the chosen auxiliary variables.11 However, if the selected sample happens to be "unlucky" and deviates substantially from the population structure with respect to these auxiliary variables, or if there are issues such as multicollinearity among the auxiliary variables, the calibration algorithm might be forced to produce extreme weights—weights that are very large (either positive or negative) or exhibit high variability—to satisfy the imposed constraints.1 While these extreme weights achieve the goal of calibration, they can, in turn, inflate the variance of the GREG estimator for the primary study variable y.1 This situation leads to a practical trade-off for the survey statistician: whether to insist on perfect adherence to calibration benchmarks or to accept some degree of tolerance in meeting these benchmarks to maintain more stable and intuitively reasonable weights. This is precisely why techniques like bounded calibration, which restrict the range of acceptable weight adjustments 1, and the careful, judicious selection of auxiliary variables are important practical considerations in the application of GREG estimators.
7. GREG in Relation to Other Survey Estimators
Understanding the GREG estimator is enhanced by comparing it with other commonly used estimators in survey sampling, particularly the Horvitz-Thompson estimator and classical ratio and regression estimators.
Comparison with the Horvitz-Thompson (HT) Estimator:
●	The Horvitz-Thompson (HT) estimator is a direct, design-unbiased estimator. It calculates an estimate of the population total by summing the values of the study variable y for the sampled units, each weighted by the inverse of its inclusion probability (1/πk​). Crucially, the HT estimator does not make use of any auxiliary information beyond what is needed to determine the inclusion probabilities.13 While enjoying the property of design unbiasedness, the HT estimator can be inefficient (i.e., have high variance), especially if the sample size is small or if the design weights are highly variable.13
●	The GREG estimator, in contrast, is a model-assisted estimator that explicitly incorporates auxiliary information to improve precision and potentially reduce bias. As shown in its formula, the GREG estimator can be viewed as the HT estimator plus an adjustment term that depends on the auxiliary variables and their relationship with the study variable. If this adjustment term is zero (e.g., if the sample estimates of auxiliary totals perfectly match the known population totals, or if there is no relationship between y and x), the GREG estimator reduces to the HT estimator.4 Generally, if the auxiliary variables are correlated with the study variable, the GREG estimator will be more efficient (have lower variance) than the HT estimator.7
Comparison with Classical Ratio and Regression Estimators:
The GREG estimator can be seen as a generalization of the classical ratio and regression estimators.
●	The Ratio Estimator is typically used when the study variable y is approximately proportional to a single positive auxiliary variable x, and the regression of y on x is assumed to be a straight line passing through the origin (yk​=βxk​+ϵk​). Specific assumptions about the variance of the error term (e.g., V(ϵk​)∝xk​ or V(ϵk​)∝xk2​) are often made. The ratio estimator is a special case of the GREG estimator under these model assumptions.3
●	The Classical Linear Regression Estimator, in its simplest form (e.g., for simple random sampling, SRS), assumes a linear model yk​=α+βxk​+ϵk​ with a constant error variance (homoscedasticity) and typically does not explicitly incorporate design weights in the estimation of α and β if derived purely from a model-based perspective for SRS.
The GREG estimator generalizes these classical forms in several important ways:
1.	Incorporation of Design Weights: GREG explicitly incorporates the survey design weights (1/πk​) into the estimation of the regression coefficients (B^), making it suitable for complex sampling designs beyond SRS.13
2.	Multiple Auxiliary Variables: GREG readily accommodates multiple auxiliary variables (i.e., x can be a vector), whereas classical ratio and simple regression estimators are often presented with a single auxiliary variable.11
3.	General Model Variance Structure: GREG allows for more general assumptions about the variance of the error terms in the assisting model through the vk​ (or ck​) terms, enabling it to handle heteroscedasticity more flexibly.4 As stated in 7, the GREG framework "includes a wide range of estimators, notably the ratio estimator and the classical regression estimator." The GREG is generally preferred when dealing with complex survey designs, when multiple auxiliary variables are available, or when the more restrictive assumptions of the simpler ratio or classical regression estimators are unlikely to be met.13
The following table provides a comparative summary:
Table 7.1: Comparative Analysis of Survey Estimators
Feature	Horvitz-Thompson (HT)	Ratio Estimator	Classical Regression Estimator (SRS)	Generalized Regression (GREG)	
Use of Auxiliary Info (X)	No	Single X	Single X (typically)	Single or Multiple X	
Assumed Model for Y vs X	None	Y∝X (through origin)	Linear (Y=α+βX)	General linear (Y=XTβ), adaptable	
Design Weights Used	Yes (1/πk​)	Yes (implicitly or explicitly)	No (for basic form in SRS)	Yes (1/πk​ in B^ and HT parts)	
Bias Property	Design Unbiased	Approx. Unbiased (large n)	Approx. Unbiased (large n, SRS)	Asymptotically Design Unbiased (ADU)	
Efficiency	Baseline	Higher if model holds & X,Y correlated	Higher if model holds & X,Y correlated (SRS)	Higher if model good & X,Y correlated	
Calibration Property	No	Yes (to Tx​ implicitly)	No (basic form)	Yes (to Tx​ explicitly)	
Flexibility (Design)	Any	Complex designs need care	Primarily SRS (basic form)	Handles complex designs well	
Flexibility (Model)	N/A	Rigid	Rigid (basic form)	Flexible (GLMs, GLMMs possible)	
This table synthesizes the key distinctions. The Horvitz-Thompson estimator serves as a foundational design-unbiased estimator but does not leverage auxiliary information. The ratio and classical regression estimators introduce the use of a single auxiliary variable, typically under more restrictive model assumptions and often simpler sampling designs like SRS for the classical regression estimator. The GREG estimator emerges as a more comprehensive and flexible framework. It can handle single or multiple auxiliary variables, incorporates design weights systematically making it suitable for complex sampling designs, and allows for a broader class of assisting linear models. Its key properties of asymptotic design unbiasedness and calibration to known auxiliary population totals make it a powerful and widely adopted tool in modern survey practice, offering efficiency gains when relevant auxiliary information is available and correctly utilized.
8. Extensions and Robust Variants of the GREG Estimator
The basic GREG framework has been extended and adapted to address various practical challenges and specific estimation needs in survey sampling. These include estimation for domains, handling different types of response variables, and improving robustness against outliers.
GREG for Domain Estimation:
GREG estimators are frequently employed for producing estimates of totals or means for specific population subgroups, known as domains.3 When estimating for domains, the assisting model can be specified in several ways:
●	A separate model can be fitted for each domain of interest.
●	A single model can be fitted to the entire sample, with domain membership incorporated through indicator variables (fixed effects).
●	A mixed model can be used, where domain differences are accounted for through random effects.3 The performance of GREG estimators can degrade in domains with small sample sizes, leading to increased variance or instability.3 An "extended GREG family" of estimators utilizes Generalized Linear Mixed Models (GLMMs) as assisting models, particularly for domain estimation, offering more flexibility in modeling complex data structures.3
GREG with Generalized Linear Models (GLMs) and Generalized Linear Mixed Models (GLMMs):
While the "classical GREG" estimator typically relies on a fixed-effects linear regression model as the assisting model 12, significant extensions have been developed to accommodate a wider range of data types and relationships. These extensions allow the assisting model to be a member of the Generalized Linear Model (GLM) family. For example:
●	For binary (yes/no) or polytomous (multiple category) study variables, a logistic regression or multinomial logistic regression model can be used.3
●	For count study variables (e.g., number of occurrences), a Poisson regression model might be appropriate.3 Furthermore, Generalized Linear Mixed Models (GLMMs) can be employed, especially when unit-level auxiliary data are available and there is a need to model hierarchical structures or account for domain-specific random effects.3 These advanced models provide greater flexibility in capturing complex dependencies in the data.
Robust GREG Estimators (e.g., M-estimators, Winsorization) for Handling Outliers:
A significant practical challenge with the standard GREG estimator is its sensitivity to outliers in the data, as it is based on weighted least squares, which can be heavily influenced by extreme observations.19 This has led to the development of robust GREG estimators designed to mitigate the impact of such outliers:
●	M-estimators: This class of robust estimators modifies the standard least squares procedure by down-weighting the influence of observations that have large residuals with respect to the fitted model. This is achieved by replacing the quadratic loss function of least squares with a loss function that increases less rapidly for large residuals, often involving a bounded "influence function" (ψ-function). Alternatively, M-estimation techniques can be used to identify and modify the values of influential units.19
●	Winsorization: This technique involves identifying extreme values of the study variable (or residuals from a model fit) and truncating them at certain pre-defined cut-off points. For example, values above an upper cut-off are replaced by the cut-off value itself. Winsorization can be one-sided (addressing only high or low outliers) or two-sided. The GREG procedure is then applied to this modified ("Winsorized") dataset.26 Robust GREG estimators aim to provide estimates that are less distorted by the presence of outliers, leading to more stable and reliable results, particularly in applications involving skewed populations, such as business surveys.19 These methods often seek an optimal trade-off between the bias introduced by modifying data or weights and the variance reduction achieved by controlling outlier influence, with the goal of minimizing the overall Mean Squared Error (MSE).24
Jackknife Bias-Corrected GREG (GREG-JK):
Even with its ADU property, the GREG estimator can exhibit non-negligible bias in finite samples, especially if the sample size is small or the assisting linear model does not fit the data well.6 The Jackknife Bias-Corrected GREG (GREG-JK) estimator has been proposed to address this issue. It employs the jackknife resampling procedure to estimate the bias of the standard GREG estimator and then subtracts this bias estimate from the original GREG estimate. Simulation studies have indicated that the GREG-JK can effectively reduce the empirical bias, sometimes to negligible levels, across various sample sizes and population structures.6
The development and refinement of these GREG extensions and robust variants reflect a continuous endeavor within the survey statistics community to adapt a powerful theoretical tool to the often "messy" realities of practical survey data and diverse analytical objectives. The "classical GREG" 12 provided a robust and versatile foundation. However, its application in real-world scenarios quickly highlighted certain limitations, such as suboptimal performance in small domains 3, challenges with non-continuous response variables if a simple linear model was assumed 3, and, very notably, a significant vulnerability to outliers.19 The extensions incorporating GLMs and GLMMs 3 broadened GREG's applicability to a much wider range of data types (binary, count, categorical) and complex data structures, such as hierarchical data or situations requiring the modeling of domain-specific effects. Simultaneously, the development of robust estimation methods, primarily M-estimation and Winsorization 19, was a direct and necessary response to the pervasive problem of outliers, a particular concern in business surveys where distributions are often highly skewed.20 Furthermore, techniques aimed at bias correction, such as the GREG-JK 6, address concerns about the finite sample performance of the estimator. This ongoing evolutionary path demonstrates how the GREG framework is not static but is iteratively refined and enhanced to improve its practicality, robustness, and applicability across an expanding range of survey estimation problems.
The following table summarizes key GREG variants:
Table 8.1: Overview of GREG Variants

GREG Variant	Key Feature / Modification	Primary Purpose / Use-Case	Snippet Examples	
GREG for Domains	Model specification accounts for domain differences (fixed/mixed effects)	Estimation for population subgroups	3	
Extended GREG (GLM/GLMM based)	Uses GLM or GLMM as assisting model	Handles non-continuous y; complex variance/correlation structures	3	
Robust GREG (M-estimators)	Down-weights/modifies influential observations via ψ-function	Reduces impact of outliers on estimates	19	
Robust GREG (Winsorization)	Truncates extreme values of y or residuals	Reduces impact of outliers (often simpler to implement)	26	
Jackknife Bias-Corrected GREG	Uses jackknife to estimate and remove bias	Improves accuracy in small samples or with poor model fit	6	
Two-Level GREG	Constrains consistency for common variables across person/household levels	Consistent estimation in household surveys	4	
This table illustrates that GREG is not a single, monolithic estimator but rather a flexible framework. Each variant tailors the core GREG principles to address specific data characteristics or estimation goals, thereby enhancing its overall utility.
9. Software and Implementation
The practical application of the GREG estimator is facilitated by its availability in several major statistical software packages.
R:
The R statistical computing environment, widely used by statisticians and researchers, offers robust support for GREG estimation through various contributed packages:
●	The survey package, a comprehensive suite for complex survey data analysis, provides the calibrate() function. This function implements GREG estimation by calibrating survey weights to known population totals for auxiliary variables specified in a model formula. It supports different calibration functions including linear (which yields the standard GREG), raking, and logit, and also allows for bounded calibration to control the range of weight adjustments.1
●	The robsurvey package is specifically designed to implement various robust survey estimators, including robust versions of the GREG estimator, aimed at handling outliers.31
●	The mase (Model-Assisted Survey Estimation) package includes a greg() function that can compute GREG estimates using either a linear or logistic assisting model. It also offers features like model selection using LASSO regression to choose predictors from a larger set of auxiliary variables and provides options for several variance estimation methods, including Taylor linearization and bootstrap techniques.32
SAS:
SAS is another widely used statistical software, particularly in official statistics and large organizations.
●	Statistics Canada developed the Generalized Estimation System (GES), which is a SAS-based application specifically designed for implementing GREG estimators.24 GES can handle stratified one-stage element or cluster sampling designs, as well as two-phase designs. Its main functionalities include the calculation of sample design weights, the computation of g-weights (calibration factors), the production of calibration estimates (GREG estimates), and the generation of synthetic estimates. It supports estimation of totals, averages, and ratios for the entire population or for specified domains, using auxiliary variables within the GREG framework.
Other statistical software packages may also offer functionalities for calibration weighting or model-assisted estimation that can be used to implement GREG or closely related estimators.
The widespread availability of the GREG estimator in major statistical software packages like R and SAS is a testament to its mainstream acceptance within the survey statistics community and significantly facilitates its practical application by survey methodologists and analysts. This accessibility lowers the barrier to entry for using these advanced techniques. However, it is crucial for users to recognize that "GREG" represents a flexible framework, and specific implementations within different software packages (or even different functions within the same package) can vary in their default settings and available options. For example, the choice of the default distance function for calibration 1, the manner in which model variance components (vk​ or qk​) are handled, the specific algorithms used for variance estimation 25, and the approaches to robustification or weight bounding 1 can differ. Therefore, while software makes implementation easier, practitioners must diligently consult the relevant documentation (e.g., the detailed options for the calibrate function in R's survey package 1) to ensure that the chosen settings and methods align with their theoretical understanding of the GREG estimator and the specific requirements of their analytical goals. A naive or default use of software functions might not always yield the most appropriate or optimal GREG estimate for a given situation.
10. Conclusion and Future Directions
Summary of the GREG Estimator's Significance in Survey Statistics:
The Generalized Regression (GREG) estimator stands as a highly significant and versatile tool in the field of sample survey statistics. It successfully bridges the gap between purely design-based estimation and model-dependent approaches by offering a model-assisted framework. Its primary contribution lies in its ability to enhance the precision of survey estimates and potentially reduce biases (such as non-response bias) through the systematic incorporation of auxiliary information via calibration weighting. The GREG estimator achieves a commendable balance, leveraging the efficiency gains offered by statistical modeling while retaining the fundamental design-based properties of asymptotic unbiasedness and consistency. This robustness to model misspecification, coupled with its flexibility in handling various data types and complex sampling designs, has led to its widespread adoption in official statistics, economic surveys, social surveys, and environmental studies.
Recommendations for Practitioners:
When applying the GREG estimator, practitioners should adhere to several best practices:
1.	Model Selection and Validation: Despite the GREG's asymptotic robustness to model misspecification, the careful selection and validation of auxiliary variables and the form of the assisting model remain crucial for achieving good performance in finite samples. The strength of the correlation between auxiliary variables and the study variable directly impacts efficiency gains.
2.	Awareness of Limitations: Practitioners must be mindful of potential issues such as the possibility of extreme or negative calibrated weights, reduced performance with very small sample sizes (especially in domains), and the sensitivity of the standard GREG to outliers.
3.	Consideration of Alternatives: In situations with highly influential outliers, robust versions of the GREG (e.g., using M-estimation or Winsorization) or bounded calibration techniques should be considered to ensure the stability and reliability of estimates. For very small domains, model-based small area estimators might be more appropriate.
4.	Variance Estimation: The choice of variance estimation method (linearization or replication) should be appropriate for the sampling design and the GREG variant used. Users should understand the assumptions and potential limitations of the chosen variance estimator, particularly concerning the treatment of model parameter estimation uncertainty.
Potential Areas for Ongoing Research and Development:
The GREG estimator continues to be an active area of research, with several promising directions for future development:
1.	High-Dimensional Auxiliary Data: With the increasing availability of "big data" from administrative sources, paradata, and other non-survey origins, there is a growing need for GREG methods that can effectively handle a large number of potential auxiliary variables. Research into integrating GREG with variable selection techniques (e.g., LASSO, as mentioned in 7) or dimension reduction methods (e.g., principal components 7) is crucial for leveraging such rich data sources without issues of multicollinearity or overfitting in the assisting model. This signals an adaptation of GREG to the evolving data landscape.
2.	Advanced Variance Estimation: Developing more accurate and robust variance estimators for GREG, especially for its more complex variants (e.g., with GLMMs or robust methods) or when dealing with high-dimensional data, remains an important research area.7 Methods that better account for the uncertainty due to the estimation of working model parameters in finite samples are particularly needed.10
3.	Robust Estimation and Tuning: Further research into adaptive and data-driven methods for robust GREG estimation, including the optimal selection of tuning parameters for M-estimators or Winsorization cut-offs, could enhance their practical utility and performance across diverse datasets.26
4.	Integration with Non-Probability Samples and Multiple Data Sources: Exploring the adaptation of GREG principles or calibration weighting for estimation with non-probability samples, or in contexts involving the integration of multiple data sources with potential linkage errors, represents a frontier for survey methodology.33
A persistent and fundamental challenge in the application of GREG, and indeed in model-assisted estimation more broadly, lies in optimally navigating the trade-off between the complexity and fit of the assisting model and the overall robustness and stability of the resulting estimator, particularly within the constraints of finite sample sizes. More complex models, such as GLMMs or those incorporating numerous parameters, might offer a superior fit to the observed sample data and hold the potential for greater variance reduction if they accurately capture the underlying population structure.3 However, such complex models are inherently more challenging to specify correctly. They can also be less stable when sample sizes are limited, and their parameters are subject to greater estimation error, which can, in turn, adversely affect the GREG estimator's performance.10 While the asymptotic design-unbiasedness property of GREG provides a degree of comfort regarding model choice for bias in large samples 5, the finite-sample variance and the overall Mean Squared Error are still heavily influenced by the adequacy of the chosen model.20 Robust estimation methods 19 aim to provide protection against specific types of model failure, such as the presence of outliers, but the selection of their tuning parameters often involves its own intricate bias-variance trade-off considerations.26 This suggests that there is no universally "best" GREG specification; the optimal approach remains highly context-dependent, demanding careful statistical judgment, thorough diagnostic checks, and a deep understanding of both the data and the underlying principles of the estimator.
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