A note on regression estimation with unknown population size 1. IntroductionA note on regression estimation with unknown population size 1. Introduction

• Previous
• Next

Regression estimation has been increasingly used in large survey organizations as a means to improve the reliability of the estimators of parameters of interest (such as totals or means) when auxiliary variables are available in the population. A comprehensive overview of the regression estimator in survey sampling can be found in Cassel, Särndal and Wretman (1976) and Fuller (2009) among others. We next illustrate how the regression estimator can be used to estimate the total, $Y\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in U}}{y}_i$  where $U\mathrm{<mo>=</mo>}\left\{\mathrm{1,}\dots \mathrm{,}N\right\}$  denotes the target population. A sample $s$  of expected size $n$  is selected according to a sampling plan $p\left(s\right)$  from $U,$  where ${\pi }_i$  is the resulting probability of inclusion of the first order. In the absence of auxiliary variables, we use the Horvitz-Thompson estimator given by ${\widehat{Y}}_{\pi }\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in s}}{d}_i{y}_i$  (Horvitz and Thompson 1952) where $d_i\mathrm{<mo>=</mo>}1/{\pi }_i$  is referred to as the weight survey associated with unit $i.$  The regression estimator is given by

$${\widehat{Y}}_{\text{REG}}\mathrm{<mo> = </mo>}{\widehat{Y}}_{\pi }+{\left(X-{\widehat{X}}_{\pi }\right)}^{{\rm T}}\widehat{B}\mathrm{,}(1.1)$$

where $X\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in U}}{x}_i,$ ${\widehat{X}}_{\pi }\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in s}}{d}_i{x}_i,$ $x_i\mathrm{<mo>=</mo>}{\left(\mathrm{1,}{x}_{2i}\mathrm{,}\dots \mathrm{,}{x}_{pi}\right)}^{{\rm T}},$ and $\widehat{B}$ is a $p-$ dimensional vector of estimated regression coefficients, which is computed as a function of the observed variables ${\left(y_i\mathrm{,}{x}_i^{{\rm T}}\right)}^{{\rm T}}$ in the sample $s.$

Note that the components of the vector of population total $X$ are known for each of the corresponding components variables in the vector $x_i\mathrm{<mo>=</mo>}{\left(\mathrm{1,}{x}_{2i}\mathrm{,}\dots \mathrm{,}{x}_{pi}\right)}^{{\rm T}}$ used to compute $\widehat{B}.$ However, there are instances when we have more observed auxiliary variables in the sample than in the population. Assume that the sample has $q$ observed variables $\left(q\mathrm{<mo>></mo>}p\right),$ and that the $p$ variables in the population are a subset of the $q$ variables observed in the sample. Furthermore, suppose that some of the extra $q-p$ variables in the sample are well correlated with the variable of interest $y.$ Can these extra variables be incorporated in the regression estimator so as to make it more efficient? Singh and Raghunath (2011) attempted to respond to that question for the case where $q\mathrm{<mo>=</mo>}p+1.$ Their extra variable in the sample was the intercept. They used it to estimate the unknown population size $N$ by $\widehat{N}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in s}}{d}_i.$

In this article, we compare the estimator proposed by Singh and Raghunath (2011) to other regression estimators when $N$ is known or unknown. In Section 2, we describe standard regression estimators for estimating totals when $N$ is known as well as the regression proposed by Singh and Raghunath (2011) when $N$ is unknown. In Section 3, an alternative estimator is proposed for the case where $N$ is unknown. A simulation study is carried out in Section 4, to illustrate the performance of the various estimators studied in terms of bias and mean square error. Overall conclusions and recommendations are given in Section 5.

• Previous
• Next