Alternative Mean Square Error Estimators and Condence Intervals for Prediction of Nonlinear Small Area Parameters

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Alternative Mean Square Error Estimators

and Conﬁdence Intervals for Prediction of

Nonlinear Small Area Parameters

Yanghyeon Cho and Emily Berg

Department of Statistics, Iowa State University, 2438 Osborn Dr., Ames, IA 50011, USA

Abstract

A diﬃculty in MSE estimation occurs because we do not specify a full distribution

for the survey weights. This obfuscates the use of fully parametric bootstrap proce-

dures. To overcome this challenge, we develop a novel MSE estimator. We estimate

the leading term in the MSE, which is the MSE of the best predictor (constructed

with the true parameters), using the same simulated samples used to construct the

basic predictor. We then exploit the asymptotic normal distribution of the parameter

estimators to estimate the second term in the MSE, which reﬂects variability in the

estimated parameters. We incorporate a correction for the bias of the estimator of

the leading term without the use of computationally intensive double-bootstrap pro-

cedures. We further develop calibrated prediction intervals that rely less on normal

theory than standard prediction intervals. We empirically demonstrate the validity of

the proposed procedures through extensive simulation studies. We apply the methods

to predict several functions of sheet and rill erosion for Iowa counties using data from

a complex agricultural survey.

1 Introduction

Small area estimation refers to the practice of using model-based estimators for domains

where direct estimators are considered unreliable. Many small area parameters are

nonlinear functions of the response variable in the model. Important examples that

occur in the domain of poverty mapping include the Gini coeﬃcient and the proportion

of the population with income below the poverty line. Nonlinear small area parameters

also occur when the parameter of interest is the mean, and the model is speciﬁed in

arXiv:2210.12221v1 [stat.ME] 21 Oct 2022

a transformed scale. For instance, a log transformation is commonly used for skewed,

positive response variables. Molina & Rao (2010) propose a simulation-based procedure

that approximates the empirical best predictor of general small area parameters that

may be nonlinear functions of the model response variable. We call the method of

Molina & Rao (2010) the EBP method. Molina & Rao (2010) focus on frequentist

inference for the unit-level linear model. The method of Molina & Rao (2010) has been

extended to Bayesian inference (Molina et al. 2014), complex sampling (Guadarrama

et al. 2018), two-level models (Marhuenda et al. 2017), generalized linear mixed models

(Hobza & Morales 2016), data-driven transformations (Rojas-Perilla et al. 2020), and

skew-normal models (Diallo & Rao 2018).

Molina & Rao (2010) deﬁne a parametric bootstrap estimator of the mean square

error (MSE) of their small area predictor. The bootstrap MSE estimator of Molina &

Rao (2010) does not incorporate a correction for the bias of the estimator of the leading

term, where the leading term in the MSE is the conditional variance of the small area

parameter given the data. The double-bootstrap is commonly used to estimate the

bias of the estimator of the leading term (Hall & Maiti 2006a,b). As noted in Molina

& Rao (2010), use of the double-bootstrap is often computationally prohibitive.

We propose an alternative way to construct MSE estimators for predictors

obtained using the EBP method of Molina & Rao (2010). Our MSE estimators incor-

porate a correction for the bias of the estimator of the leading term, without requiring

the double bootstrap. We claim that this is possible because the EBP method furnishes

samples from the conditional distribution of the population parameter given the data.

These samples from the conditional distribution can be used to obtain an estimator of

the leading term in the MSE without implementing a bootstrap. We then use boot-

strap procedures to estimate the second term in the MSE, which reﬂects the variation

due to parameter estimation. We also use the parametric bootstrap to estimate the

bias of the estimator of the leading term. The parametric bootstrap that we propose

is less computationally expensive than the parametric bootstrap method of Molina

& Rao (2010) because we only require generating bootstrap versions of elements in

the sample. In contrast, the procedure of Molina & Rao (2010) requires generating a

bootstrap version of the entire population.

A further beneﬁt of our proposed MSE estimation procedure is that it lends itself

naturally to the construction of calibrated prediction intervals. Molina & Rao (2010)

do not consider prediction intervals explicitly. One can construct a normal-theory pre-

diction interval as ˆ

θi±1.96√ˆmsei, where ˆ

θidenotes the predictor and ˆmseidenotes the

MSE estimator. The normal-theory prediction interval may have poor coverage if the

standardized statistic deﬁned as Ti=√ˆmsei

−1(ˆ

θi−θi) does not have an approximately

normal distribution, where θidenotes the true parameter. We use the basic ingredients

deﬁning the MSE estimator to construct calibrated prediction intervals that do not re-

quire normal theory. The proposed prediction intervals adapt the calibration procedure

of Carlin & Gelfand (1991) to the small area context. We use the same simulated sam-

ples used to estimate the leading term in the MSE to deﬁne a preliminary conﬁdence

interval. The preliminary interval is then calibrated using the bootstrap. The calibra-

tion procedure is similar to a small area prediction interval proposed in Section 2.8 of

Hall & Maiti (2006b). Our procedure is tailored more speciﬁcally toward construction

of intervals for nonlinear parameters under unit-level models than the method of Hall

& Maiti (2006b). The prediction interval of Hall & Maiti (2006b) requires a bootstrap

version of the population parameter. For our procedure, we only generate bootstrap

versions of sampled elements and do not construct a bootstrap version of the popula-

tion parameter. Therefore, the procedure of Hall & Maiti (2006b) is not tenable for

use in combination with our proposed bootstrap procedure.

A further innovation of our work is that we consider MSE estimation and conﬁ-

dence interval construction in the context of an informative design. The estimator of

the leading term in the MSE that we propose extends directly to an informative sam-

ple design. Estimation of the variance due to parameter estimation presents unique

challenges in the context of informative sampling. In our framework, we specify only

the ﬁrst moment of the sample distribution of the survey weight, instead of postulating

a full distribution for the survey weight. As a result, the parametric bootstrap used for

the noninformative design does immediately apply in the context of informative sam-

pling. To overcome this challenge, we simulate bootstrap parameter estimates from a

nonparametric estimate of the asymptotic covariance matrix of the vector of parameter

estimators. Our use of the large sample distribution of the parameter estimators allows

us to circumvent the problem of specifying a full distribution for the survey weight. We

also evaluate the proposed prediction intervals in the context of informative sampling.

In contrast, Hall & Maiti (2006b) restrict attention to noninformative designs. Extend-

ing the procedure of Hall & Maiti (2006b) to informative sampling is nontrivial because

the procedure of Hall & Maiti (2006b) requires a bootstrap version of the population

parameter.

Variations of the proposed procedures have been used elsewhere. Sun et al. (2021)

implements a version of the proposed MSE estimator in the speciﬁc context of a bivari-

ate small area model with discrete and continuous components. Berg (2022) adapts

the proposed procedure for the purpose of constructing a database for small area es-

timation. The studies of Sun et al. (2021) and Berg (2022) are very speciﬁc to the

frameworks that they consider and are not easily generalizable. Further, Sun et al.

(2021) and Berg (2022) do not consider estimation of the bias of the estimator of the

leading term. In this work, we generalize the procedures with the aim of reaching a

broad audience. We also provide empirical and theoretical support for the methodol-

ogy. We conduct a thorough empirical evaluation of several estimators of the bias of

the estimator of the leading term in the MSE.

Upon completing this work, we learned that the estimator of the leading term that

we propose is in current use for production of poverty indicators at the World Bank.

The World Bank MSE estimator, however, does not appropriately reﬂect variability due

to parameter estimation. One of the contributions of our study is to provide rigorous

support for the estimator of the leading term in the MSE that is currently in use at

the World Bank (Isabelle Molina, Personal Communication, 7-6-22). The estimator

of the second term in the MSE that we propose has potential use of inference about

poverty measures at the World Bank. The relevance of the proposed procedures to the

current practice at the World Bank demonstrates that the methodology in this paper

is of salient importance for statistical practice.

Many other works propose MSE estimators that incorporate corrections for the

bias of the estimator of the leading term. Hall & Maiti (2006a) and Hall & Maiti

(2006b) propose parametric and non-parametric double-bootstrap based MSE estima-

tors that are very computationally intensive to implement. We do not consider the

bootstrap procedures of Hall & Maiti (2006a) or Hall & Maiti (2006b) as a result of

the computational burden. To reduce the computational demands, Erciulescu & Fuller

(2016) develop a fast double bootstrap. The fast double bootstrap MSE estimator of

Erciulescu & Fuller (2016) can result in negative estimates. In a study of small area

estimation based on the gamma distribution, Cho and Berg (in prepration) ﬁnd that

the prevalence of negative estimates from the method of Erciulescu & Fuller (2016) is

nontrivial. Because one cannot construct a conﬁdence interval from a negative MSE

estimate, we do not consider the method of Erciulescu & Fuller (2016). Erciulescu &

Fuller (2019) develop calibrated conﬁdence intervals for small area means. It is not

immediately obvious to us that the method of Erciulescu & Fuller (2019) extends to

nonlinear small area parameters. Lahiri et al. (2007) develop positive MSE estimates

that incorporate a correction for the bias of the estimator of the leading term in the

context of an area-level model. An extension of their method to prediction of nonlinear

parameters in the context of a unit-level model is not straightforward and is beyond the

scope of our work. Further, their MSE estimator is much more diﬃcult to implement

than the MSE estimator that we propose. An alternative to the bootstrap is to use

the jackknife to estimate the bias of the estimator of the leading term (Lohr & Rao

2009). The jackknife MSE estimator of Lohr & Rao (2009) is developed for an area-

level model. Because we focus on unit-level models, we do not consider the jackknife

MSE estimator of Lohr & Rao (2009). The SUMCA method (Jiang & Torabi 2020)

is an alternative way to construct a bias correction. We do not consider the SUMCA

method because it is not clear to us that SUMCA appropriately reﬂects the variance

of parameter estimators for nonlinear parameters, as we explain in Appendix A of the

supplementary material (SM).

We propose inference procedures that are computationally simple to implement

when used in combination with the EBP procedure of Molina & Rao (2010). The

procedures lend themselves naturally to construction of calibrated prediction intervals

and informative sampling. In Section 2, we deﬁne the proposed method for non-

informative and informative sample designs. We also deﬁne the conﬁdence intervals and

corrections to the bias of the estimator of the leading term in Section 2. In Section 3,

we evaluate the proposed procedure through simulations that use both noninformative

and informative designs. In Section 4, we present two data analyses: one for non-

informative sampling and a second for informative sampling.

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AlternativeMeanSquareErrorEstimatorsandCondenceIntervalsforPredictionofNonlinearSmallAreaParametersYanghyeonChoandEmilyBergDepartmentofStatistics,IowaStateUniversity,2438OsbornDr.,Ames,IA50011,USAAbstractAdicultyinMSEestimationoccursbecausewedonotspecifyafulldistributionforthesurveyweights.Thisobf...

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Alternative Mean Square Error Estimators and Condence Intervals for Prediction of Nonlinear Small Area Parameters

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