Bootstrap Confidence Intervals for the Population Mean Under Inverse Sampling Design

Abstract

Inverse sampling is commonly used for surveying rare (but not clustered) populations. However, when the sample size from the rare group is chosen too small, the customary unbiased estimator of the population mean appears to be highly skewed. In such a case, confidence intervals based on asymptotic normal theory have coverage rate smaller than the nominal level. As an approach to overcome this problem, we propose two resampling methods consisting of with-replacement bootstrap (BWR) and without replacement bootstrap (BWO) to construct confidence intervals for the population mean under simple inverse sampling without replacement. We carried out a simulation study to evaluate the behavior of suggested bootstrap methods, the normal approximation and the logarithmic transformation methods. Our simulation results suggest that the BWO method is preferable, since it provides intervals with coverage rate closer to the nominal level together with more balanced error rate.

Appendices

Appendix 1: Unbiasedness and the Variance of $\hat{\mu }^*_\mathrm{s}$ Under BWR Method

Let $os^*$ denote the bootstrap inverse sample selected with replacement from the original inverse sample s, and $os^*_C$ and $os^*_{\bar{C}}$ denote its parts in $s_C$ and $s_{\bar{C}}$, respectively. To prove unbiasedness of $\hat{\mu }^*_\mathrm{s}$, by using the conditional property of the mathematical expectation we have $E_*(\hat{\mu }^*_\mathrm{s})=E_*\big (E_*(\hat{\mu }^*_\mathrm{s}|~n_{bs})\big )$. Since $os^*$ given its size $n_{bs}$ is equivalent to a stratified random sampling with replacement of size $(r_b,n_{bs}-r_b)$ from $(s_C,s_{\bar{C}})$, we have

$$\begin{aligned} E_*(\hat{\mu }^*_\mathrm{s}|~n_{bs})=\frac{\hat{M}_b}{N}\bar{y}_{s_C}+\left( 1-\frac{\hat{M}_b}{N}\right) \bar{y}_{s_{\bar{C}}} \end{aligned}$$

(5)

Hence, $E_*(\hat{\mu }^*_\mathrm{s})=E_*\big (\frac{\hat{M}_b}{N}\bar{y}_{s_C}+(1-\frac{\hat{M}_b}{N})\bar{y}_{s_{\bar{C}}}\big )$. Now since $n_{bs}$ given s is distributed as a negative binomial with parameters $(r_b,\frac{r-1}{n_\mathrm{s}-1})$, it follows that $E_*(\frac{\hat{M}_b}{N})=\frac{r-1}{n_\mathrm{s}-1}=\frac{\hat{M}}{N}$. So, $E_*(\hat{\mu }^*_\mathrm{s})=\frac{\hat{M}}{N}\bar{y}_{s_C}+(1-\frac{\hat{M}}{N})\bar{y}_{s_{\bar{C}}}=\hat{\mu }_\mathrm{s}$.

To find the bootstrap variance of $\hat{\mu }^*_\mathrm{s}$ given s, it may use the conditional property of variance as:

$$\begin{aligned} v_b=\text {Var}_*(\hat{\mu }^*_\mathrm{s})=\text {Var}_*\big (E_*(\hat{\mu }^*_\mathrm{s}|~n_{bs})\big )+E_*\big (\text {Var}_*(\hat{\mu }^*_\mathrm{s}|~n_{bs})\big ). \end{aligned}$$

Using Eq. (5), the first term is $\text {Var}_*\big (E_*({\hat{\mu }}^*_\mathrm{s}|~n_{bs})\big )=\text {Var}_*(\frac{{\hat{M}_b}}{N})(\bar{y}_{s_C}-\bar{y}_{s_{\bar{C}}})^2.$ Also, for the interior variance of second term we have

$$\begin{aligned} \text {Var}_*(\hat{\mu }^*_\mathrm{s}|~n_{bs})=\bigg (\frac{\hat{M}_b}{N}\bigg )^2\text {Var}_*(\bar{y}^*_{os_C}|~n_{bs})+\bigg (1-\frac{\hat{M}_b}{N}\bigg )^2\text {Var}_*(\bar{y}^*_{os_{\bar{C}}}|~n_{bs}) \end{aligned}$$

Since $\bar{y}^*_{os_C}$ and $\bar{y}^*_{os_{\bar{C}}}$ given $n_{bs}$ are independent samples with replacement from $s_C$ and $s_{\bar{C}}$, we have $\text {Var}_*(\bar{y}^*_{os_C})=\frac{(r-1)s^2_C}{rr_b}$ and $\text {Var}_*(\bar{y}^*_{os_{\bar{C}}})=\frac{(n_\mathrm{s}-r-1)S^2_{\bar{C}}}{(n_\mathrm{s}-r)(n_{bs}-r)}$. Now giving expectation from the two last terms, we obtain

$$\begin{aligned} E_*\big (\text {Var}_*\big ({\hat{\mu }}^*_\mathrm{s}|~n_{bs}\big )\big )=E_*\big (\frac{{\hat{M}_b}}{N}\big )^2\frac{(r-1)s^2_C}{rr_b}+E_*\bigg (\frac{(1-\frac{\hat{M}_b}{N})^2}{{n_{bs}-r_b}}\bigg )\frac{(n_\mathrm{s}-r-1)s^2_{\bar{C}}}{(n_\mathrm{s}-r)}. \end{aligned}$$

Combining this with the previous result completes the proof.

Appendix 2: Unbiasedness of the Bootstrap Estimator Under BWO Method

Let $os^*$ denote the bootstrap inverse sample selected with replacement from the original inverse sample s, and $os^*_C$ and $os^*_{\bar{C}}$ denote its parts in $s_C$ and $s_{\bar{C}}$, respectively. To prove unbiasedness of $\hat{\mu }^*_\mathrm{s}$, by using the conditional property of the mathematical expectation we have let j denote the removed unit from $s_C$. Hence, by drawing an inverse sample without replacement from $U^*$ and by using the unbiasedness of (1), we have

$$\begin{aligned} E_*(\hat{\mu }^*_\mathrm{s})=E_*\big (E_*(\hat{\mu }^*_\mathrm{s}~|~j)\big ) \end{aligned}$$

$$\begin{aligned} =E_*\bigg (\frac{1}{N}\sum _{k\in {U^*}}y_k\bigg ) \end{aligned}$$

$$\begin{aligned} =\frac{1}{N}E_*\Bigg (\sum _{k\in {U^*_C}}y_k+\sum _{k\in {U^*_{\bar{C}}}}y_k\Bigg )~ \end{aligned}$$

$$\begin{aligned} =\frac{1}{N}E_*\Bigg (\frac{N}{n_\mathrm{s}-1}\sum _{k\in {s_{C(-j)}}}y_k+\frac{N}{n_\mathrm{s}-1}\sum _{k\in {s_{\bar{C}}}}y_k\Bigg ) \end{aligned}$$

$$\begin{aligned}=E_*\Bigg (\frac{r-1}{n_\mathrm{s}-1}\bar{y}_{s_{C(-j)}}+\frac{n_\mathrm{s}-r}{n_\mathrm{s}-1}\bar{y}_{s_{\bar{C}}}\Bigg ) \end{aligned}$$

$$\begin{aligned}=\bigg (\frac{r-1}{n_\mathrm{s}-1}\bigg )E_*(\bar{y}_{s_{C(-j)}})+\bigg (\frac{n_\mathrm{s}-r}{n_\mathrm{s}-1}\bigg )\bar{y}_{s_{\bar{C}}}. \end{aligned}$$

Now, since $E_*(\bar{y}_{s_{C(-j)}})=\bar{y}_{s_C}$, we have:

$$\begin{aligned} E_*(\hat{\mu }^*_\mathrm{s})=\hat{\mu }_\mathrm{s}, \end{aligned}$$

which guaranties the unbiasedness of the bootstrap estimator $\hat{\mu }^*_\mathrm{s}$.

Table 1 Standard deviation (top entry) and skewness (below that) of $\hat{\mu }_\mathrm{s}$ under five underlying models described in the text

Table 2 Coverage and standardized average length (in parenthesis) of the $90\%$ confidence intervals described in the text for $M=100$

Table 3 Coverage and standardized average length (in parenthesis) of the $90\%$ confidence intervals described in the text for $M=400$

Table 4 Lower and upper miscoverage rates of the $90\%$ confidence intervals described in the text for $M=100$

Table 5 Lower and upper miscoverage rates of the $90\%$ confidence intervals described in the text for $M=400$