Model-assisted optimal allocation for planned domains using composite estimation 1. IntroductionModel-assisted optimal allocation for planned domains using composite estimation 1. Introduction

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Sample surveys have long been used as cost-effective means for data collection but it is also the case that general purpose surveys will often not achieve adequate precision for statistics for subpopulations of interest (often called domains or areas). Domains may be geographically based areas such as states. They may also be cross-classifications of a small geographic area and a specific demographic or social group. A domain is regarded as small if the domain-specific sample is not large enough to produce a direct estimator with satisfactory precision.

In this paper, we suppose that stratified sampling is used with $H$ strata defined by the small areas, indexed by $h\in U^1.$ The set of all $N$ units in the population is denoted $U$ and the set of $H$ strata is denoted $U^1.$ This effectively assumes that small areas can be identified in advance, which is not always the case (Marker 2001). Even so, the survey designer may be able to make an educated guess at areas of interest, which should still result in an improved design even if new requirements for area statistics emerge after the survey has been run. The population of $N_h$ units in stratum $h$ is written $U_h$ and the sample of $n_h$ units selected by simple random sampling without replacement (SRSWOR) from stratum $h$ is $s_h$ . Let $Y_j$ be the value of the characteristic of interest for the $j^{\text{th}}$ unit in the population. The small area population mean for stratum $h$ is ${\overline{Y}}_h$ and the national mean is $\overline{Y}.$ The corresponding sample estimators are ${\overline{y}}_h$ and $\overline{y},$ respectively; ${\overline{y}}_h=n_h^{-1}{\displaystyle {\sum }_{j\in s_h}{y}_j}$ and $\overline{y}={\displaystyle {\sum }_{h\in U^1}{P}_h{\overline{y}}_h},$ where $P_h=N_h/N.$ Let the sampling variances be $v_h={\text{var}}_p\left({\overline{y}}_h\right)$ and $v={\text{var}}_p\left(\overline{y}\right).$

Longford (2006) considers the problem of optimal sample sizes for small area estimation for this design. The approach is based on minimizing the weighted sum of the mean squared errors of the planned small area mean estimators and an overall estimator of the mean. The weight attached to each area is proportional to the area population raised to the $q^{\text{th}}$ power, so the value of $0\le q\le 2$ specifies the relative importance of larger compared to smaller areas. The mean squared error of the all-strata mean estimator is multiplied by $G,$ where $G$ reflects the perceived priority of this estimator. An analytical solution exists for the case where $G=0,$ but it has undesirable practical properties, and may sometimes result in zero or minimum sample sizes for some strata. When $G\text{>}0,$ Longford (2006) suggests numerical optimization.

Choudhry, Rao and Hidiroglou (2012) investigate the use of nonlinear programming (NLP) to efficiently allocate sample to strata, when there may be bounds on stratum sample sizes, and priority on overall, stratum and cross-strata domain estimators of multiple variables. The paper mostly concentrates on design-based direct area estimators, but they also consider the objective criterion of Longford (2006) for composite estimation. For the Canadian Monthly Retail Trade Survey, they show that the Longford allocation gives extremely unequal sample sizes by strata, for $q$ equal to 0.5, 1 and 1.5. For example, when $q=1.5,$ the highest stratum coefficient of variation (CV) is 112%, and even for $q=0.5,$ the highest coefficient of variation is 24%, which was deemed too high. It is not clear whether these CV%s refer to direct or composite estimators - such high CVs would be surprising for composite estimators, as their CVs are bounded above even as the sample size tends to zero. Choudhry et al. (2012) did not investigate whether other designs such as power allocations can give low values of Longford’s criteria.

The aim of this paper is to find the best allocation to strata for a linear combination of the mean squared errors of small area composite estimators and of an overall estimator of the mean, similar to Longford (2006). In Section 2 we reformulate the objective in model-assisted terms, introduce the use of regressor variables, and derive a model-assisted composite estimator. Section 3 is devoted to optimizing the design. In Subsection 3.1 we discuss direct optimization, for example by NLP. Subsection 3.2 describes power allocation with the exponent chosen to numerically minimize the objective criterion. Section 4 is a numerical study of the various methods using the Swiss canton data of Longford (2006) and Section 5 contains conclusions.

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