On a new estimator for the variance of the ratio estimator with small sample corrections
Section 1. Introduction

• Previous
• Next

Consider a population of $N$ distinct units with values $\left(x_i,y_i\right)$ $\left(i=1,\dots ,N\right)$ of the variables $x$ and $y$ $\left(x_i>0\right).$ Denote the corresponding population means by $\overline{X}$ and $\overline{Y},$ that is $\overline{X}={\displaystyle {\sum }_{i=1}^N{x}_i}/N$ and $\overline{Y}={\displaystyle {\sum }_{i=1}^N{y}_i}/N.$ Define $R$ by $R=\overline{Y}/\overline{X}.$ Suppose that a simple random sample of size $n$ is selected from the population. When $\overline{X}$ is known, $\overline{Y}$ can be estimated by the ratio estimator

$${\widehat{\overline{Y}}}_R=\widehat{R}\overline{X},(1.1)$$

where $\widehat{R}={\overline{y}}_s/{\overline{x}}_s$ with ${\overline{y}}_s={\displaystyle {\sum }_{i=1}^n{y}_i}/n$ and ${\overline{x}}_s={\displaystyle {\sum }_{i=1}^n{x}_i}/n;$ see Cochran (1977, page 151). For large $n,$ the well-known approximation for the variance of ${\widehat{\overline{Y}}}_R$ is

$$\text{var}\left({\widehat{\overline{Y}}}_R\right)\approx \frac{1-f}{n}{S}_e^2,(1.2)$$

where $f=n/N,$ $S_e^2={\displaystyle {\sum }_{i=1}^N{e}_i^2}/\left(N-1\right)$ and $e_i=y_i-R{x}_i$ $\left(i=1,\dots ,N\right);$ note that $\overline{E}={\displaystyle {\sum }_{i=1}^N{e}_i}/N=0.$ When $n$ is small, the approximation error of (1.2) can be considerable; see Koop (1968). Moreover, this error may increase when, in practice, $S_e^2$ in (1.2) is replaced by its standard estimator $s_{\widehat{e}}^2=\frac{1}{n-1}{\displaystyle {\sum }_{i=1}^n{\widehat{e}}_i^2}$ where ${\widehat{e}}_i=y_i-\widehat{R}{x}_i$ $\left(i=1,\dots ,n\right);$ see Cochran (1977, page 163). As stated by Koop (1968), the cause of the discrepancy relative to the true variance lies in neglecting terms in $1/n^2$ and $1/n^3,$ and perhaps also those of higher orders.

The three main aims of this paper are: (i) to improve approximation (1.2) for small values of $n$ by using a second-order Taylor series expansion of $1/{\overline{x}}_s\text{};$ (ii) to derive a new estimator for $S_e^2$ that is less biased than $s_{\widehat{e}}^2;$ and (iii) to derive a new variance estimator for the ratio estimator. Although a normal distributional approximation might be imprecise at the sample sizes considered in this paper, such a more accurate variance estimator is useful in order to get some more insight into the precision of the ratio estimator in comparison with that of other estimators. For instance, in case of small samples from small strata the combined ratio estimate for $\overline{Y}$ is to be recommended rather than the separate ratio estimate certainly when the ratios (say, $R_h)$ are constant from stratum to stratum; see Cochran (1977, page 167).

The outline of the paper is as follows. Using some results of Nath (1968), we derive in Section 2 an alternative approximation formula for the variance of $\widehat{R}$ with an error of order $1/n^3\text{}.$ In addition, we derive a new approximation formula for the bias of the residual sampling variance $s_{\widehat{e}}^2$ of order $1/n.$ Furthermore, we propose two new estimators for the mean square error (MSE) of ${\widehat{\overline{Y}}}_R.$ In Section 3 we carry out a simulation study in order to compare the standard variance estimator with the new estimators proposed in Section 2. Section 4 summarizes the main conclusions.

• Previous
• Article main page
• Next