On a new estimator for the variance of the ratio estimator with small sample corrections
Section 2. A new variance estimator

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Noting that $\widehat{R}-R={\overline{e}}_s/{\overline{x}}_s,$ where ${\overline{e}}_s$ is the sample mean of $e_i,$ and using the second-order Taylor series expansion

$$\frac{1}{{\overline{x}}_s}=\frac{1}{\overline{X}}-\frac{1}{{\overline{X}}^2}\left({\overline{x}}_s-\overline{X}\right)+\frac{1}{{\overline{X}}^3}{\left({\overline{x}}_s-\overline{X}\right)}^2+O_p\left(\frac{1}{n^{1.5}}\right),$$

it is seen that the third-order Taylor series expansion of $\widehat{R}-R$ is

$$\widehat{R}-R=\frac{{\overline{e}}_s}{\overline{X}}-\frac{1}{{\overline{X}}^2}\left({\overline{x}}_s-\overline{X}\right){\overline{e}}_s+\frac{1}{{\overline{X}}^3}{\left({\overline{x}}_s-\overline{X}\right)}^2{\overline{e}}_s+O_p\left(\frac{1}{n^2}\right).(2.1)$$

Hence, using ${\widehat{\overline{Y}}}_R=\overline{X}\widehat{R},$ we obtain

$$\begin{array}{ll}\text{var}\left({\widehat{\overline{Y}}}_R\right)\hfill & =\text{var}\left({\overline{e}}_s\right)+\frac{1}{{\overline{X}}^2}\text{var}\left\{\left({\overline{x}}_s-\overline{X}\right){\overline{e}}_s\right\}-\frac{2}{\overline{X}}\text{cov}\left\{{\overline{e}}_s, \left({\overline{x}}_s-\overline{X}\right){\overline{e}}_s\right\}\hfill \\ \hfill & +\frac{2}{{\overline{X}}^2}\text{cov}\left\{{\overline{e}}_s,{\left({\overline{x}}_s-\overline{X}\right)}^2{\overline{e}}_s\right\}+O\left(\frac{1}{n^3}\right).(2.2)\hfill \end{array}$$

In (2.2) we omitted one variance and one covariance because the underlying fifth and sixth moments are of order $1/n^3\text{};$ see David and Sukhatme (1974). All (co)variances in (2.2) can be evaluated by using the following results on product moments of four arbitrary sample means, say ${\overline{x}}_{sa},{\overline{x}}_{sb},{\overline{x}}_{sc}$ and ${\overline{x}}_{sd},$

$$E\left({\overline{x}}_{sa}{\overline{x}}_{sb}{\overline{x}}_{sc}\right)=\left(1-f\right)\left(1-2f\right)S_{abc}/n^2+O\left(n^{-3}\right)(2.3)$$

$$E\left({\overline{x}}_{sa}{\overline{x}}_{sb}{\overline{x}}_{sc}{\overline{x}}_{sd}\right)=\gamma \left(S_{ab}{S}_{cd}+S_{ac}{S}_{bd}+S_{ad}{S}_{bc}\right)+O\left(n^{-3}\right)(2.4)$$

$$\text{cov}\left({\overline{x}}_{sa}{\overline{x}}_{sb},{\overline{x}}_{sc}{\overline{x}}_{sd}\right)=\gamma \left(S_{ac}{S}_{bd}+S_{ad}{S}_{bc}\right)+O\left(n^{-3}\right)(2.5)$$

$$E\left({\overline{x}}_{sa}^2{\overline{x}}_{sb}^2\right)=\gamma \left(S_{aa}{S}_{bb}+2S_{ab}^2\right)+O\left(n^{-3}\right),(2.6)$$

where $\gamma ={\left(1-f\right)}^2/n^2\text{},$ $S_{ab}={\displaystyle {\sum }_{i=1}^N{x}_{ia}{x}_{ib}}/\left(N-1\right)$ and $S_{abc}={\displaystyle {\sum }_{i=1}^N{x}_{ia}{x}_{ib}{x}_{ic}}/\left(N-1\right).$ Without loss of generality, it is assumed for expediency that the population means are zero, that is, ${\overline{X}}_a={\overline{X}}_b={\overline{X}}_c={\overline{X}}_d=0.$ Formulas (2.3) and (2.4) follow from Theorems 1 and 2 of Nath (1968) while (2.5) and (2.6) follow from (2.4). From (2.2)-(2.6) it follows that

$$\begin{array}{ll}\text{var}\left({\widehat{\overline{Y}}}_R\right)\hfill & =\frac{1-f}{n}{S}_e^2+{\left(\frac{1-f}{n\overline{X}}\right)}^2\left(S_x^2{S}_e^2+S_{xe}^2\right)-\frac{2}{n^2\overline{X}}\left(1-f\right)\left(1-2f\right)S_{xee}\hfill \\ \hfill & +2{\left(\frac{1-f}{n\overline{X}}\right)}^2\left(S_x^2{S}_e^2+2S_{xe}^2\right)+O\left(n^{-3}\right)\hfill \\ \hfill & =\frac{1-f}{n}{S}_e^2\left\{1+3\left(\frac{1-f}{n{\overline{X}}^2}\right)S_x^2\right\}+5{\left(\frac{1-f}{n\overline{X}}\right)}^2{S}_{xe}^2-2\frac{\left(1-f\right)\left(1-2f\right)}{n^2\overline{X}}{S}_{xee}+O\left(n^{-3}\right),(2.7)\hfill \end{array}$$

where

$$\begin{array}{ll}{S}_x^2\hfill & =\frac{1}{N-1}{\displaystyle \sum _{i=1}^N}{\left(x_i-\overline{X}\right)}^2\text{},S_{xe}=\frac{1}{N-1}{\displaystyle \sum _{i=1}^N}\left(x_i-\overline{X}\right)e_i\hfill \\ S_{xee}\hfill & =S_{ge}=\frac{1}{N-1}{\displaystyle \sum _{i=1}^N{e}_i^2\left(x_i-\overline{X}\right)},g_i=\left(x_i-\overline{X}\right)e_i.\hfill \end{array}$$

Similar formulas in terms of cumulants are derived by Tin (1965) using some results from Kendall and Stuart (1958). Unfortunately, the numerous cumulants in Tin’s formulas give little insight into the structure of $\text{var}\text{(}{\widehat{\overline{Y}}}_R\text{)}$ and, consequently, small sample corrections for the variance estimator require somewhat tedious calculations. In contrast, from (2.7) it is seen that for sufficiently large $n$ approximation (1.2) leads to an underestimate unless $S_{xee}(=S_{ge})$ is very positive. In addition, Tin also discusses three alternative estimators for a ratio but small sample corrections when estimating the various variances are ignored by him.

It follows from (2.1) and (2.3) that

$$\text{bias}\left({\widehat{\overline{Y}}}_R\right)=-\frac{1-f}{n\overline{X}}{S}_{xe}+O\left(\frac{1}{n^2}\right);(2.8)$$

also see Cochran (1977, page 161). Subsequently, using $S_{xee}=S_{ge},$ it follows from (2.7) and (2.8) that the mean square error of ${\widehat{\overline{Y}}}_R$ is

$$\text{MSE}\left({\widehat{\overline{Y}}}_R\right)=\frac{1-f}{n}{S}_e^2\left\{1+3\left(\frac{1-f}{n{\overline{X}}^2}\right)S_x^2\right\}+6{\left(\frac{1-f}{n\overline{X}}\right)}^2{S}_{xe}^2-2\frac{\left(1-f\right)\left(1-2f\right)}{n^2\overline{X}}{S}_{ge}+O\left(n^{-3}\right).(2.9)$$

When the variation coefficient $C_x\left(\equiv S_x/\overline{X}\right)$ is known, it is useful to write (2.9) as

$$\text{MSE}\left({\widehat{\overline{Y}}}_R\right)=\frac{1-f}{n}{S}_e^2\left\{1+3\left(\frac{1-f}{n}\right)C_x^2\left(1+2{\rho }_{xe}^2\right)-2\frac{\left(1-2f\right)}{n}{C}_x{\rho }_{ge}{S}_g/\left(S_x{S}_e\right)\right\},(2.10)$$

where ${\rho }_{xe}=S_{xe}/S_x{S}_e\text{},$ ${\rho }_{ge}=S_{ge}/S_g{S}_e$ and $S_g^2=\frac{1}{N-1}{{\displaystyle {\sum }_{i=1}^N\left(g_i-\overline{G}\right)}}^2.$ In practice, $\text{MSE}\text{(}{\widehat{\overline{Y}}}_R\text{)}$ in (2.10) can be estimated by

$${\widehat{\text{MSE}}}_1\left({\widehat{\overline{Y}}}_R\right)=\frac{1-f}{n}{s}_{\widehat{e}}^2\left\{1+3\left(\frac{1-f}{n}\right)C_x^2\left(1+2{\widehat{\rho }}_{x\widehat{e}}^2\right)-2\frac{\left(1-2f\right)}{n}{C}_x{\widehat{\rho }}_{\widehat{g}\widehat{e}}{s}_{\widehat{g}}/\left(s_x{s}_{\widehat{e}}\right)\right\},(2.11)$$

where ${\widehat{\rho }}_{x\widehat{e}}=s_{x\widehat{e}}/s_x{s}_{\widehat{e}}\text{},$ ${\widehat{\rho }}_{\widehat{g}\widehat{e}}=s_{\widehat{g}\widehat{e}}/s_{\widehat{g}}{s}_{\widehat{e}}$ and

$$\begin{array}{ll}{s}_{\widehat{g}}^2\hfill & =\frac{1}{n-1}{\displaystyle \sum _{i=1}^n}{\left({\widehat{g}}_i-{\overline{\widehat{g}}}_s\right)}^2\left[\text{note:}{\widehat{g}}_i=\left(x_i-{\overline{x}}_s\right){\widehat{e}}_i\right]\hfill \\ s_{x\widehat{e}}\hfill & =\frac{1}{n-1}{\displaystyle \sum _{i=1}^n}\left(x_i-{\overline{x}}_s\right){\widehat{e}}_i,s_{\widehat{g}\widehat{e}}=s_{x\widehat{e}\widehat{e}}=\frac{1}{n-1}{\displaystyle \sum _{i=1}^n}\left(x_i-{\overline{x}}_s\right){\widehat{e}}_i^2.\hfill \end{array}$$

However, the estimator in (2.11) does not take into account the bias of $s_{\widehat{e}}^2$ defined above.

In order to examine the bias of $s_{\widehat{e}}^2={\displaystyle {\sum }_{i=1}^n{\widehat{e}}_i^2}/\left(n-1\right),$ we use some additional symbols

$$\begin{array}{ll}{s}_x^2\hfill & =\frac{1}{n-1}{\displaystyle \sum _{i=1}^n}{\left(x_i-{\overline{x}}_s\right)}^2\text{},s_e^2=\frac{1}{n-1}{\displaystyle \sum _{i=1}^n}{\left(e_i-{\overline{e}}_s\right)}^2\hfill \\ s_{xe}\hfill & =\frac{1}{n-1}{\displaystyle \sum _{i=1}^n}\left(x_i-{\overline{x}}_s\right)e_i,q_i={\left(x_i-\overline{X}\right)}^2\left(i=1,\dots ,N\right).\hfill \end{array}$$

Now we can write $E\left(s_{\widehat{e}}^2\right)$ as

$$\begin{array}{ll}E\left(s_{\widehat{e}}^2\right)\hfill & =E\frac{1}{n-1}{\displaystyle \sum _{i=1}^n}{\left\{y_i-{\overline{y}}_s-R\left(x_i-{\overline{x}}_s\right)-\left(\widehat{R}-R\right)\left(x_i-{\overline{x}}_s\right)\right\}}^2\hfill \\ \hfill & =E\left(s_e^2\right)+E\left\{{\left(\widehat{R}-R\right)}^2{s}_x^2\right\}-2E\left\{\left(\widehat{R}-R\right)s_{xe}\right\}\hfill \\ \hfill & =S_e^2+E\left\{{\left(\widehat{R}-R\right)}^2{\overline{q}}_s\right\}-2E\left\{\left(\widehat{R}-R\right){\overline{g}}_s\right\}+O\left(n^{-2}\right),(2.12)\hfill \end{array}$$

where ${\overline{q}}_s$ and ${\overline{g}}_s$ are sample means of $q_i$ and $g_i,$ respectively. In (2.12) we used

$$s_x^2=\left\{{\overline{q}}_s-{\left({\overline{x}}_s-\overline{X}\right)}^2\right\}\left(1+\frac{1}{n-1}\right),s_{xe}=\left\{{\overline{g}}_s-\left({\overline{x}}_s-\overline{X}\right){\overline{e}}_s\right\}\left(1+\frac{1}{n-1}\right)$$

and hence, using (2.1), (2.3) and (2.4), we get

$$\begin{array}{ll}E\left\{{\left(\widehat{R}-R\right)}^2\left({\overline{q}}_s-s_x^2\right)\right\}\hfill & =E\left\{{\overline{e}}_s^2{\left({\overline{x}}_s-\overline{X}\right)}^2/{\overline{X}}^2\right\}\left\{1+o\left(1\right)\right\}=O\left(n^{-2}\right)\hfill \\ E\left\{\left(\widehat{R}-R\right)({\overline{g}}_s-s_{xe}\right\}\hfill & =E\left\{{\overline{e}}_s\left({\overline{x}}_s-\overline{X}\right){\overline{e}}_s/\overline{X}\right\}\left\{1+o\left(1\right)\right\}=O\left(n^{-2}\right).\hfill \end{array}$$

From (2.1) and (2.12) it is seen that

$$\begin{array}{ll}\text{bias}\left(s_{\widehat{e}}^2\right)\hfill & =E\left\{{\left(\widehat{R}-R\right)}^2\left({\overline{q}}_s-\overline{Q}+\overline{Q}\right)\right\}-2E\left\{\left(\widehat{R}-R\right)\left({\overline{g}}_s-\overline{G}+\overline{G}\right)\right\}+O\left(n^{-2}\right)\hfill \\ \hfill & =\frac{1-f}{n{\overline{X}}^2}{S}_e^2\overline{Q}-2E\left[\left\{\frac{{\overline{e}}_s}{\overline{X}}-\frac{1}{{\overline{X}}^2}\left({\overline{x}}_s-\overline{X}\right){\overline{e}}_s\right\}\left({\overline{g}}_s-\overline{G}+\overline{G}\right)\right]+O\left(n^{-2}\right)\hfill \\ \hfill & =\frac{1-f}{n{\overline{X}}^2}{S}_e^2{S}_x^2-2\frac{1-f}{n}\left(\frac{S_{ge}}{\overline{X}}-\frac{S_{xe}^2}{{\overline{X}}^2}\right)+O\left(n^{-2}\right),(2.13)\hfill \end{array}$$

where we used $\overline{Q}=S_x^2\left(1-N^{-1}\right)$ and $\overline{G}=S_{xe}\left(1-N^{-1}\right).$ Note that it follows from (2.13) that for sufficiently large $n,$ the quantity $s_{\widehat{e}}^2$ leads to an overestimate of $S_e^2$ unless $S_{ge}$ is very positive. To our best knowledge, formula (2.13) is not mentioned elsewhere in the literature.

Based on (2.13), an alternative estimator of $\text{MSE}\text{(}{\widehat{\overline{Y}}}_R\text{)}$ that takes the bias of $s_{\widehat{e}}^2$ into account is

$${\widehat{\text{MSE}}}_2\left({\widehat{\overline{Y}}}_R\right)=\frac{1-f}{n}{\widehat{S}}_e^2\left\{1+3\left(\frac{1-f}{n}\right)C_x^2\left(1+2{\widehat{\rho }}_{x\widehat{e}}^2\right)-2\frac{\left(1-2f\right)}{n}{C}_x{\widehat{\rho }}_{\widehat{g}\widehat{e}}{s}_{\widehat{g}}/\left(s_x{s}_{\widehat{e}}\right)\right\},(2.14)$$

where ${\widehat{S}}_e^2$ is adjusted for the relative bias of $s_{\widehat{e}}^2$ that follows from (2.13). That is,

$${\widehat{S}}_e^2=s_{\widehat{e}}^2\left[1-\frac{1-f}{n}{C}_x^2\left(1+2{\widehat{\rho }}_{x\widehat{e}}^2\right)+2\frac{1-f}{n}{C}_x\frac{{\widehat{\rho }}_{\widehat{g}\widehat{e}}{s}_{\widehat{g}}}{s_x{s}_{\widehat{e}}}\right].$$

Note that we used here $S_{xe}^2/{\overline{X}}^2{S}_e^2={\rho }_{xe}^2{C}_x^2$ and $S_{ge}/\overline{X}{S}_e^2={\rho }_{ge}{S}_g{C}_x/S_e{S}_x.$ Finally, it should be noted that the other estimators ${\widehat{\rho }}_{x\widehat{e}}^2$ and ${\widehat{\rho }}_{\widehat{g}\widehat{e}}{s}_{\widehat{g}}/\left(s_x{s}_{\widehat{e}}\right)$ in (2.11) are also biased. However, it is less straightforward to derive that kind of bias. It is hoped that by taking all (co)variances from the sample, including $s_x^2,$ their bias is modest. In addition, in the simulations of Section 3, we found that replacing $C_x$ by $s_x/{\overline{x}}_s$ did not improve the results.

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