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In this study, efficiencies of Horvitz-Thompson (HT) and Generalized Regression (GREG) estimators are investigated in small domains namely Domain 1, 2 and 3. Efficiency criteria include the standard error of the estimates (SE) and coefficient of variation (CV) Computed for the study population of size 1052 households consisting of domain size of 885, 151 and 16 and domain samples sizes of 167, 29 and 3 respectively for Domains 1, 2 and 3. The estimates produced by HT estimator GREG estimator performs better than HT estimator with domainSE1(GREG) < SE1(HT), SE2(GREG) < SE2(HT) and SE3(GREG) < SE3(HT). Similarly the domain Coefficient of Variation (CVdth) ;CV1(GREG) < CV1(HT),CV2(GREG) < CV2(HT) and CV3(GREG) < CV3(HT). The study concluded that GREG estimator is more efficient than the HT estimator for small domain when it is clear that sample sizes of domains are small. The distribution of the study population sampled showed that Domain 1 with 84.1%of the respondents have an average monthly expenditure below₦21691.80(or less than about$2 per day World Bank poverty line). Nigerian Annals of Pure and Applied Science Vol. 2, 2019 |215 Introduction Small-Area Estimation (SAE) is a collection of statistical techniques designed for improving sample survey estimates through the use of auxiliary information. SAE offers the opportunity to improve accuracy of the current surveys through the use of statistical models, while avoiding additional respondents burden or survey costs, (Ghosh and Rao, 1994). Of course, SAE is of great importance in survey sampling due to a growing demand for reliable small area statistics from both public and private sectors for several reasons such as effective planning of health, mobilization and apportionment of resources and revenue and allocation of legislative seats as the case may be (Ghosh and Rao, 1994; Rao, 2003). Sample sizes relating to small domains of interest turn out to be so small in practice that estimates based on them under classical sampling theory happen to be inaccurate and unreliable (especially with large measures of error in terms of large standard errors and coefficients of variation). Consequently, ways of achieving higher serviceability requires modifying the classical methods that utilizes various auxiliary pieces of information. There are several estimators that can be used in SAE when auxiliary variables are available. These include Hansen-Hurwitz estimator (HH), RaoHartley-Cochran estimator, Horvitz-Thompson (HT) estimator, Regression Estimator, ModelDependent Empirical Best Linear Unbiased Predictor (EBLUP), synthetic, generalised regression (GREG) estimator etc. In this study, we utilized HorvitzThompson (HT) estimator developed by Horvitz and Thompson (1952) and generalised regression (GREG) estimator developed by Sarndal, Swensson and Wretman (1992) to investigate and compare their efficiencies in estimating household expenditure in Makurdi-Nigeria considering income as auxiliary variable. Makurdi is the capital city of Benue State and is located along the Benue River on latitude 7.73 degrees and longitude 8.52 degree. The town is mainly domiciled by those involved in Agriculture, Civil Servants and traders among others. Literature Review There are several studies in the literature (especially in Small Area) about GREG and HT estimators. Lehtonen (2006) investigated estimation for population subgroups or domains for model-assisted generalized regression (GREG) and Model-Dependent Empirical Best Linear Unbiased Predictor (EBLUP) estimators under different model choices and under unequal probability sampling. Merkouris (2009), on the other hand, explored the possibility of enhancing the precision of domain estimators by combining comparable information collected in multiple surveys of the same population. He proposed a regression method of estimation that is essentially an extended calibration procedure whereby comparable domain estimates from the various surveys are calibrated to each other. Santamarίa et al. (2004) studied small area estimation with auxiliary variables, complex sampling designs and highlighted methods for small-area estimation including direct, synthetic and some other indirect estimators. Costa et al. (2006) worked on improving small-area estimation by combining surveys and investigating how to integrate information from an auxiliary survey in small-area estimation. Pfeffermann (2013) presented some important developments in small area estimation methods. Rai and Pandey (2013) discussed the generalized class of synthetic estimators for estimating the population mean of small domains under the information of two auxiliary variables and the results of the numerical illustration for the two auxiliary variables were compared for the synthetic ratio estimator under single and two auxiliary variables. Münnich et al. (2014) developed a design which adequately supports the application of small area estimation methods that foster an accurate estimation of total population counts. Ken (2001) showed that, under mild conditions, the variance of the Horvitz-Thompson (HT) estimator depends almost entirely on first order 216| Investigating the Performance of Horvitz-Thompson and Generalized Regression Estimators in ... inclusion probabilities. Using both purely design-based and model-assisted arguments, approximate expressions and estimators were derived for this "natural variance" of the HT estimator and tentative formulae for the HT variance estimator were also provided for the most important case where the HT variance does not take its natural value, namely when sampling is systematic from a deliberately ordered population. Grakoviča(2006) compared HorvitzThompson (HT) estimator with GREG estimator in farm survey. According to him, GREG estimator depends on correlation among X (auxiliary variable which was total land area of farm) and Y (study variable denoting the number of cattle) and if correlation is small, GREG estimator is not better compared to HT estimator. Milda (2009) used generalized regression, synthetic, empirical best linear predictor and Horvitz-Thompson estimators to examine estimation for domains and small areas for enter-prise surveys. Münnich et al. (2011), conducted the first census after the re-unification in Germany where a register-assisted census was implemented using population register data and an additional sample. They concluded that in terms of design-based methods, the design had considerable impact on the accuracy of the estimators and simple random sampling, as expected, is very inefficient using the HT and much better for the GREG. Haslett et al. (2012) produced small-area estimates of poverty and malnutrition in Cambodia at district and sub-district levels by combining survey data with auxiliary data derived from the 2008 census. Wiese (2013) investigated the best design for the Swedish Investment Survey stratifying on number of employees, Neyman allocation, Stratified Simple Random Sampling (STSRS) and HT-estimation and the alternative method was to stratify on turnover, πρs-sampling and GREG-estimation. Bikauskaite (2014) analyzes several different types of techniques which produce small area estimates of poverty. Methods Let the population ℧ = u1, u2. . . uN be partitioned into domains D = 1,2, . . . d and let yi,d be the observation on the i th unit of the population in the d domain. Also let Pi = xi X ; X = ∑ xi N i=1 be the selection probability of the i unit in the population The unbiased Horvitz-Thompson estimator for domain total of Ydis given as tH = ∑ yd,iwiId,i 1

nigerian journals of pure and applied sciences (benue online)

Investigating the Performance of Horvitz-Thompson and Generalized Regression Estimators in Small Domains with Application to Income and Expenditure of Households in Makurdi-Nigeria