Impact of Sampling on Small Area Estimation in Business Surveys
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1 Impact of Sampling on Small Area Estimation in Business Surveys Jan Pablo Burgard, Tomas Zimmermann and Ralf T. Münnic University of Trier, Economic and Social Statistics Department, Universitätsring 15, Trier, Germany Corresponding autor: Ralf T. Münnic, Abstract Modern Business statistics often faces te difficulty tat an increasing demand of information on sub-levels defined by regions or cross-classifications of variables suc as industry classes and business size can be observed, eg for measures of competitiveness by policy makers. In order to enable data producers to provide estimates on tose sub-levels, sopisticated stratifications are implemented in te sampling design. Tese detailed stratifications may produce two difficulties. First, many strata contain only very few elements and, ence, make it difficult to derive optimal sample sizes. Second, statistical model building may suffer from te survey weigts derived under tese constraints. Additionally, optimization of sampling designs may ave a strong impact on te accuracy of different estimation strategies. Te aim of te paper is to evaluate different sampling designs in te context of estimation on sub-levels by regions and cross-classifications and teir impact on tese domain estimates. As estimators of interest, te Horvitz-Tompson- and generalized regression estimator as design-based metods as well as te Battese-Harter-Fuller-, te You-Rao-, and te augmented estimators are considered. Te analysis is performed by means of a Monte Carlo study based on Italian business data. Keywords: optimal sampling design, model-based estimation, designbased estimation 1 Introduction Starting wit papers by Fay and Herriot (1979) and Battese et al. (1988) modern small area estimation tecniques gained popularity in data production in several fields of statistics. However, in business statistics te small area estimation tecniques were not applicable for a long time because teir assumptions are strongly violated by te underlying population of interest. Te specific itc of business statistics is tat te distribution of key variables, suc as return or turnover are igly skewed wit many outliers tat need to be taken into account in te estimation process. Classical large sample approximations may be questioned as te assumption of a normally distributed estimate often does not old even for considerably large sample sizes. Specially, wen distributions wit fat-tails or extreme outliers are present, te distribution of design-based point and variance estimates, resemble more a multimodal distribution. In small area estimation te assumption of normally distributed model errors and random effects is often violated, in particular wen te covariates at and are not explaining enoug of te variation of te dependent variable. Furter, for model-based small area estimation te sampling design may impose critical difficulties. Tis is eavily criticized by Gelman (2007) wit te words weigting is a mess.
2 In recent years tere are some developments of model-based small area estimation tecniques wic try to cope wit complex survey designs and weigting. Tese seem attractive for te application in business statistics. Terefore, it is of interest to study te performance of bot classical design-based and model-based small area estimator and te model-based small area estimators coping for complex survey designs. Capter 2 is devoted to explaining survey designs and estimation tecniques for small domain estimation in business statistics. Capter 3 comprises te set-up for our simulation study wic is used to compare different small domain estimators under a variety of survey designs. Capter 4 gives an outlook on present researc of te paper. 2 Design and estimation in business statistics Te sampling designs used in business statistics typically use stratified random sampling at some stage, were te strata are often constructed as cross-classifications of industry classes, enterprise size and a geograpical information. In te following we briefly describe some of te stratified allocations used in our study. A more detailed account on tese allocation procedures is given in Bernardini Papalia et al. (2013, C. 4.3). A very basic procedure is te equal allocation wic allocates te sample size to all te strata, i.e. (1) n Equal = n L. Note tat (1) may lead to igly different sampling fractions in te strata, if te stratum sizes are igly variable. If a constant sampling fraction witin te strata is desired and a variation of te sample sizes is accepted, te proportional allocation may be used. It is given by (2) n Proportional = n N N. Wile te equal allocation may be suitable for domain estimation due to a guaranteed sample size in te strata, te proportional allocation may be more efficient for te estimation at national level as it allocates more sample size to te larger strata. Tus, a convex combination of te allocations (1) and (2) as proposed by Costa et al. (2004) may give a reasonable balance between tese goals. If te sole purpose of a survey is to produce efficient estimates at te national level and te stratum specific variances are known, te optimal allocation due to Neyman and Tscuprow may be preferred as it minimizes te variance under stratified random sampling. Te optimal allocation assuming large N so tat te finite population correction may be ignored, follows as (3) n Optimal = n N σ, L N k σ k k=1 were σ refers to te standard deviation in stratum of our variable of interest. Te optimality of allocation (3) for national design-based estimation may come at te expense of reliable estimation on te domain-level as very small sample sizes may
3 result. Furter, te optimal allocation may result in igly different design weigts, wic may be an issue for model-based small domain estimation strategies. Tese disadvantages of te optimal allocation are dealt wit by introducing box-constraints for te stratum-specific sample sizes into te optimization procedure as proposed by Gabler et al. (2012). Tey consider te following optimization problem: min RRMSE < > (ˆµ) 2 = L 2 RRMSE(ˆµ <> ) n (4) =1 s.t. m n M, = 1,..., L L n n, =1 were m and M denote te lower and upper bound of te sample size in stratum. In te solution of problem (4), te set of stratum indices is split into tree parts. For te first group of strata, te unconstrained optimal sample size n Optimal would be smaller tan te lower bound m. As tis violates te condition m n, te resulting sample size is set to te lower bound. For te second group of strata n Optimal > M, and to fulfil te constraint n M, n is set to M. In te remaining strata, te optimal allocation (3) is applied using te remaining sample size. In addition to tese allocations, also sampling strategies may be considered, were te allocation is determined as to be optimal for a given design-based estimation procedure. Examples for tis approac in te context of stratified sampling are te allocations due to Longford (2006) and Coudry et al. (2012). Small domain estimators are used to produce estimates of a domain-specific quantity, suc as te domain mean or te domain total for a variable of interest. In te following we focus on estimating te domain mean, wic is defined as µ d = 1 N d y dj, d = 1,..., D, were y dj denotes te response of unit j in domain d and N d refers to te population size in domain d. Te different small domain estimators may be derived from a design-based perspective or a model-based perspective. A widely used design-based estimator in survey sampling is te weigted sample mean, wic is given by (5) ˆµ d,direct = n d n d w dj y dj, w dj were w dj denotes te design weigt of unit j in domain d. Estimator (5) as good design-based properties, but it may not be efficient for te estimation of small domains as it does not use any auxiliary information. Te family of generalized regression estimators (GREG) allows to incoporate auxiliary information troug an assisting model to reduce te variance compared to te direct estimator (5). In a small domain setting, te general form of te GREG is given by (6) ˆµ d,greg = 1 N d n d ŷ dj + w dj (y dj ŷ dj ), N d
4 were te ŷ dj refer to te predicted values under te assumed assisting model. It can be seen tat te sum of te predicted values for all units in a particular domain is corrected by te weigted residuals from all sampled units in tat domain. Note tat formula (6) accounts for various specifications of te assisting model (cf. Letonen and Veijanen, 2009). In te case of model-based estimators we assume tat te following unit-level mixed model olds (7) y dj = x T dj β + u d + ε dj, d = 1,..., D, j = 1,..., N d, were x dj refers to te p-dimensional vector of covariates for unit j in domain d, β iid denotes te p-dimensional vector of estimated fixed effects, u d N(0, σu) 2 iid, ε dj N(0, σε) 2 and te domain specific random effects u d are independent from te sampling errors ε dj. An empirical best linear unbiased predictor (EBLUP) under model (7) as been derived by Battese et al. (1988). Teir predictor is given by ] ˆµ ULEBLUP d = ˆγ d [y d + (X d x d ) T ˆβ + (1 ˆγ d )X T ˆβ d (8) = X T ˆβ d + û d. It can be seen [ tat (8) is a convex ] combination of te unweigted survey regression estimator y d + (X d x d ) T ˆβ and te regression syntetic component X T ˆβ. d As te local sample size n d increases, ˆγ d tends to 1 and more weigt is attaced to te survey regression estimator. For small n d and small estimated variances of te random effect, ˆσ u, 2 te srinkage coefficient ˆγ d is close to 0, and te estimator tends to its syntetic component. Te estimator is not design-consistent unless te sampling design is simple random sampling witin te domains. Tis assumption, owever, is violated in most business surveys as discussed above. An estimator based on model (7) wic is noneteless design-consistent is te pseudo EBLUP due to You and Rao (2002). Tey transform te unit-level mixed model (7) to a survey-weigted area-level model, were te weigts are normalized witin eac area. Te pseudo EBLUP under te survey-weigted area-level model follows as: (9) ˆµ d,y R = X T ˆβ d Y R + û d,y R wit ( û d,y R = ˆγ dw ȳ dw x T ˆβ ) dw Y R, were ˆβ Y R, ˆγ dw, ȳ dw and x dw are estimated under te survey-weigted area-level model. In addition to being design-consistent, estimator (9) also satisfies te bencmarking property automatically (cf. You and Rao, 2002). An implicit assumption wen deriving estimators (8) and (9) is tat te sampling design is non-informative, i.e. tat te model wic olds for te population olds for te sample as well (cf. Pfeffermann and Sverckov, 2009). In business surveys, tis assumption may be violated as te sampling weigts migt be related to te variable of interest after conditioning on te covariates. In some situations te bias due to an informative sampling design migt be overcome by including te design weigts as an additional covariate in te statistical model. Verret et al. (2010) consider using augmented models for te estimators (8) and (9).
5 3 Outlook on te simulation study Our design-based simulation study is conducted using te fully syntetic TRItalia dataset described in Bernardini Papalia et al. (2013, C. 5). TRItalia focuses on small and medium enterprises from te Italian population of businesses in Our variable of interest is te mean of labour costs in eac domain. Te domains are defined as cross-classifications of te first digit of te industry code and te twenty Italian provinces (NUTS 2), yielding D = 180 domains. Our sampling design is stratified random sampling, were te strata are defined as cross-classifications of te domain and te four size classes in terms of te number of employees, giving us L = 720 strata. As allocations we consider te equal allocation, te proportional allocation, te optimal allocation and box-constraint optimal allocations wit Gelman factors (cf. Burgard et al., 2012) constrained to 10 and 50. Due to te fact tat about 90% of all enterprises belong to te group wit one to five employees, te stratum sizes are extremely variable. In order to facilitate proper variance estimation for design-based metods, we require at least two units to be drawn from eac stratum giving rise to large sampling fractions in very small strata as indicated by Figure 1. Sampling fraction PROP EQ OPT BCOpt_50 BCOpt_ Percentiles of empirical cumulative distribution function Figure 1: Sampling fractions Acknowledgements Te researc was conducted witin te EU-FP7 project BLUE-ETS (ttp://www. blue-ets.eu). Te autors express teir gratitude to ISTAT for kindly providing te original data wic were used to build TRItalia.
6 References G. E. Battese, R. M. Harter, and W. A. Fuller. An error component model for prediction of county crop areas using survey and satellite data. Journal of te American Statistical Association, 83 (401):28 36, R. Bernardini Papalia, C. Bruc, T. Enderle, S. Falorsi, A. Fasulo, E. Fernandez- Vazquez, M. Ferrante, J. P. Kolb, R. Münnic, S. Pacei, R. Priam, P. Rigi, T. Scmid, N. Slomo, F. Volk, and T. Zimmermann. Best practice recommendations on variance estimation and small area estimation in business surveys. Tecnical report, BLUE-ETS, deliverable D6.2, J. P. Burgard, R. Münnic, and T. Zimmermann. Small area modelling under complex survey designs for business data. In Proceedings of te Fourt International Conference of Establisment Surveys, June 11-14, 2012, Montréal, Canada, G. H. Coudry, J. N. K. Rao, and M. A. Hidiroglou. On sample allocation for efficient domain estimation. Survey Metodology, 38(1):23 29, A. Costa, A. Satorra, and E. Ventura. Improving bot domain and total area estimation by composition. Statistics and Operations Researc Transactions, 28(1): 69 86, R. E. Fay and R. A. Herriot. Estimation of income for small places: An application of james-stein procedures to census data. Journal of te American Statistical Association, 74 (366): , S. Gabler, M. Ganninger, and R. Münnic. Optimal allocation of te sample size to strata under box constraints. Metrika, 75(2): , February A. Gelman. Struggles wit survey weigting and regression modeling. Statistical Science, 22: , R. Letonen and A. Veijanen. Design-based metods of estimation for domains and small areas. In D. Pfeffermann and C. R. Rao, editors, Handbook of Statistics, volume 29B, capter 31, pages Elsevier, New York, N. T. Longford. Sample size calculation for small area estimation. Survey Metodology, 32(1):87 96, D. Pfeffermann and M. Sverckov. Inference under informative sampling. In D. Pfeffermann and C. R. Rao, editors, Handbook of Statistics, volume 29B, capter 39, pages Elsevier, New York, F. Verret, M. A. Hidiroglou, and J. N. K. Rao. Small area estimation under informative sampling. In Proceedings of te Survey Metods Section SSC Annual Meeting, May 2010, Y. You and J. N. K. Rao. A pseudo-empirical best linear unbiased prediction approac to small area estimation using survey weigts. Te Canadian Journal of Statistics, 30: , 2002.
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