An improved scrambled model for estimating the mean of a sensitive character

Abstract

This paper suggests a new randomized response model useful for gathering information on quantitative sensitive variable such as drug usage, tax evasion and induced abortions etc. The resultant estimator has been found to more efficient than the estimator of the Saha (2007) under some realistic conditions. We have illustrated results numerically.

Keywords

Bias variance scrambled response model sensitive variable estimation of mean

1. Introduction

In sociological, health, economic and psychological surveys, people often do not respond truthfully when asked personal or sensitive questions, or refuse to answer. So in such situations the procedures that protect anonymity are a solution. Randomized response (RR) technique and scrambled response procedure are the two extensively used ways to protect anonymity. The randomized response technique pioneered by Warner (1965) in an ingenious interviewing procedure for eliciting information on sensitive data while ensuring that respondents privacy is protected. However the scrambled response technique was initiated by Pollock and Bek (1976). Further some work have been carried out in this area by various authors including Himmelfarb and Edgell (1980), Eichhorn and Hayre (1983), Singh et al. (1998), Bar-Lev et al. (2004), Singh and Mathur (2005), Gupta et al. (2006), Saha (2007), Gupta and Shabbir (2008), Gjestvang and Singh (2009), Diana and Perri (2010, 2011), Gupta et al. (2012), Perri and Diana (2013), Hussain and Al-Zahrani (2016) and Kumari and Trisandhya (2017, 2019).

In this paper following the procedure adopted by Gjestvang and Singh (2009) and taking the clue from Saha (2007) we have developed a new randomized response model and the estimator for estimating the mean of a quantitative sensitive variable. Some analytical and numerical comparisons of efficiency are performed to set up the conditions under which improvements upon Saha’s model can be obtained.

2. Saha’s (2007) scrambled randomized response model and estimator

Let $\Omega=\{\Omega_{1},\Omega_{2},\ldots,\Omega_{N}\}$ be a finite population of $N$ units and $Y\geqslant 0$ be a quantitative sensitive variable under study with mean $E(Y)=\mu_{y}$ and $V(Y)=\sigma_{y}^{2}$ , assumed to be unknown. Let $W$ and $U$ be two positive independent random variables also independent of $Y$ whose distributions are known as well as $E(W)=\mu_{w}$ , $E(U)=\mu_{u}$ , $V(W)=\sigma_{W}^{2}$ and $V(U)=\sigma_{u}^{2}$ .

There are several devices available in the literature for estimating population mean $\mu_{y}$ of the variable $Y$ under investigation, for instance, see Odumade and Singh (2009). In most of these cases a coding mechanism of the response on $Y$ , i.e. the respondents are asked to algebraically perturb the true value of $Y$ through one or more random numbers generated from known scrambling distributions. Saha (2007) suggested to gather information on $Y$ by asking the interviewee to yield the scrambled randomized response

$\displaystyle Z=W(Y+U).$ (1)

Here the interviewer is completely unaware of the random numbers $W$ and $U$ used for scrambling true responses $Y$ . But the interviewer is having complete knowledge of the scrambling distributions. We also note that the respondent does not reveal to any one the scrambling numbers. This model combines multiplicative and additive models to induce larger confidence among the respondents about their privacy protection.

A simple random sample (SRS) of size $n$ is drawn with replacement ( $W R$ ) from the population $\Omega$ . Let $Z_{i}$ be the scrambled randomized response received from the $i^{\text{th}}$ selected individual $({i=1,2,\ldots,n})$ . Then we have

$\displaystyle Z_{i}=W({Y_{i}+U}),i=1,2,\ldots,n.$ (2)

Thus an unbiased estimator for the population mean $\mu_{y}$ is given by

$\displaystyle\hat{\mu}_{s}=\frac{\bar{Z}}{\mu_{w}}-\mu_{u},$ (3)

where $\bar{Z}=\frac{1}{n}\sum_{i=1}^{n}{Z_{i}}$ .

The variance of the estimator $\hat{\mu}_{s}$ is given by

$\displaystyle V({\hat{\mu}_{s}})=\frac{V(Z)}{n\mu_{w}^{2}}=\frac{\sigma_{z}^{2% }}{n\mu_{w}^{2}},$ (4)

where

$\displaystyle V(Z)=\sigma_{z}^{2}=[{\mu_{u}^{2}\sigma_{w}^{2}+\mu_{w}^{2}({% \sigma_{u}^{2}+\sigma_{y}^{2}})+\sigma_{w}^{2}({\sigma_{y}^{2}+\mu_{y}^{2}+% \sigma_{u}^{2}})+2\mu_{y}\mu_{u}\sigma_{w}^{2}}],=[{({\sigma_{w}^{2}+\mu_{w}^{% 2}})({\sigma_{u}^{2}+\mu_{u}^{2}})+\sigma_{w}^{2}({\sigma_{y}^{2}+\mu_{y}^{2}}% )+\mu_{w}^{2}({\sigma_{y}^{2}-\mu_{u}^{2}})+2\mu_{y}\mu_{u}\sigma_{w}^{2}}],=% \mu_{w}^{2}[{\mu_{u}^{2}({1+C_{u}^{2}})({1+C_{w}^{2}})+\mu_{y}^{2}C_{w}^{2}({1% +C_{y}^{2}})+\mu_{y}^{2}C_{y}^{2}+\mu_{u}^{2}({2RC_{w}^{2}-1})}],$ (5)

$C_{y}^{2}={\sigma_{y}^{2}}/{\mu_{y}^{2}}$ , $C_{u}^{2}={\sigma_{u}^{2}}/{\mu_{u}^{2}}$ and $C_{w}^{2}={\sigma_{w}^{2}}/\mu_{w}^{2}$ are the square of the coefficients variation ofthe variables $Y, U$ and $W$ respectively and $R=\mu_{y}/\mu_{u}$ .

3. Proposed randomized response model and estimator

Moving along the direction traced by Saha (2007), we consider a scrambled randomized response model in the way suggested by Gjestvang and Singh (2009). Let $\alpha$ and $\beta$ be two known positive real numbers. Consider a deck of cards in which $p$ is the proportion of cards bearing the statement: $W({Y_{i}+\alpha U})$ and $({1-p})$ be the proportion of cards bearing the statement: $W({Y_{i}-\beta U})$ . Let $p=\beta/{({\alpha+\beta})}$ be known. Each respondent is asked to draw one card secretly and report the scrambled response accordingly. Therefore, the response to the sensitive question is

$\displaystyle Z_{i}^{*}=\begin{cases}W({Y_{i}+\alpha U})\text{ with % probability},p=\beta/({\alpha+\beta}),\\ W({Y_{i}-\beta U})\text{ with probability},({1-p})=\alpha/({\alpha+\beta}).% \end{cases}$ (6)

Let $E_{d}$ be the expected value over all possible samples and $E_{R}$ be the expected value over the randomized device, then

$\displaystyle E({Z_{i}^{*}})=E_{d}E_{R}({Z_{i}^{*}}),=E_{d}E_{R}\{{pW({Y_{i}+% \alpha U})+({1-p})W({Y_{i}-\beta U})}\},=E_{d}[{p\{{Y_{i}+\alpha E_{R}(U)}\}E_% {R}(W)+({1-p})\{{Y_{i}-\beta E_{R}(U)}\}E_{R}(W)}],=E_{d}[{Y_{i}E_{R}(W)+\{{% \alpha p-\beta({1-p})}\}E_{R}(U)E_{R}(W)}],=E_{d}\left[\mu_{w}Y_{i}+\mu_{w}\mu% _{u}\left\{{\frac{\alpha\beta}{({\alpha+\beta})}-\frac{\alpha\beta}{({\alpha+% \beta})}}\right\}\right],=E_{d}({\mu_{w}Y_{i}}),=\mu_{w}E_{d}({Y_{i}}),=\mu_{w% }\mu_{y}\Rightarrow\mu_{y}=\frac{E({Z_{i}^{*}})}{\mu_{w}}.$ (7)

Thus an unbiased estimator of the population mean $\mu_{y}$ is given by

$\displaystyle\hat{\mu}_{y(P)}=\frac{\bar{Z}^{*}}{\mu_{w}},$ (8)

where $\bar{Z}^{*}=\frac{1}{n}\sum_{i=1}^{n}{Z_{i}^{*}}$ .

Let $V_{d}$ be the variance over all possible samples and $V_{R}$ be the variance over the randomization device; then we have

$\displaystyle V({\hat{\mu}_{y(P)}})=E_{d}V_{R}({\hat{\mu}_{y(P)}})+V_{d}E_{R}(% {\hat{\mu}_{y(P)}}),=E_{d}V_{R}\left({\frac{\bar{Z}^{*}}{\mu_{w}}}\right)+V_{d% }E_{R}\left({\frac{\bar{Z}^{*}}{\mu_{w}}}\right),=E_{d}\left\{{\frac{1}{\mu_{w% }^{2}}V_{R}({\bar{Z}^{*}})}\right\}+V_{d}\left\{{\frac{1}{\mu_{w}}E_{R}({\bar{% Z}^{*}})}\right\},=\frac{1}{\mu_{w}^{2}}\{{E_{d}V_{R}({\bar{Z}^{*}})}\}+\frac{% 1}{\mu_{w}^{2}}\{{V_{d}E_{R}({\bar{Z}^{*}})}\},=\frac{1}{\mu_{w}^{2}}[{E_{d}V_% {R}({\bar{Z}^{*}})+V_{d}E_{R}({\bar{Z}^{*}})}],=\frac{1}{\mu_{w}^{2}}\left[{E_% {d}V_{R}\left\{{\frac{1}{n}\sum_{i=1}^{n}{Z_{i}^{*}}}\right\}+V_{d}E_{R}\{{% \frac{1}{n}\sum_{i=1}^{n}{Z_{i}^{*}}}\}}\right],=\frac{1}{\mu_{w}^{2}}\left[{E% _{d}\left\{{\frac{1}{n^{2}}\sum_{i=1}^{n}{V_{R}({Z_{i}^{*}})}}\right\}+V_{d}% \left\{{\frac{1}{n}\sum_{i=1}^{n}{E_{R}({Z_{i}^{*}})}}\right\}}\right],=\frac{% 1}{\mu_{w}^{2}}\left[{\frac{1}{n^{2}}E_{d}\left\{{\sum_{i=1}^{n}{V_{R}({Z_{i}^% {*}})}}\right\}+\frac{1}{n^{2}}V_{d}\left\{{\sum_{i=1}^{n}{E_{R}({Z_{i}^{*}})}% }\right\}}\right],=\frac{1}{n^{2}\mu_{w}^{2}}\left[{\sum_{i=1}^{n}{E_{d}V_{R}(% {Z_{i}^{*}})}+\sum_{i=1}^{n}{V_{d}E_{R}({Z_{i}^{*}})}}\right],=\frac{1}{n^{2}% \mu_{w}^{2}}\left[{\sum_{i=1}^{n}{E_{d}V_{R}({Z_{i}^{*}})}+\sum_{i=1}^{n}{V_{d% }({\mu_{w}Y_{i}})}}\right],∼{}\because E_{R}({Z_{i}^{*}})=\mu_{w}Y_{i},=\frac{% 1}{n^{2}\mu_{w}^{2}}\left[{\sum_{i=1}^{n}{E_{d}V_{R}({Z_{i}^{*}})}+\sum_{i=1}^% {n}{\mu_{w}^{2}\sigma_{y}^{2}}}\right],=\frac{1}{n^{2}\mu_{w}^{2}}\left[{\sum_% {i=1}^{n}{E_{d}V_{R}({Z_{i}^{*}})}+n\mu_{w}^{2}\sigma_{y}^{2}}\right].$ (9)

Now,

$\displaystyle E_{d}V_{R}({Z_{i}^{*}})=E_{d}[{E_{R}({Z_{i}^{\ast 2}})-({E_{R}({% Z_{i}^{*}})})^{2}}],=E_{d}[{E_{R}({Z_{i}^{\ast 2}})-\mu_{w}^{2}Y_{i}^{2}}],∼{}% \because E_{R}({Z_{i}^{*}})=\mu_{w}Y_{i},=E_{d}[{E_{R}\{{pW^{2}({Y_{i}+\alpha U% })^{2}+({1-p})W^{2}({Y_{i}-\beta U})^{2}}\}-\mu_{w}^{2}Y_{i}^{2}}],=E_{d}[{E_{% R}\{{p({Y_{i}^{2}+\alpha^{2}U^{2}+2\alpha Y_{i}U})}}+{({1-p})({Y_{i}^{2}+\beta% ^{2}U^{2}-2\beta Y_{i}U})}\}W^{2}{-\mu_{w}^{2}Y_{i}^{2}}],=E_{d}[{E_{R}}\{{Y_{% i}^{2}W^{2}+2Y_{i}({\alpha p-\beta({1-p})})UW^{2}}+{({\alpha^{2}p+\beta^{2}({1% -p})})U^{2}W^{2}}\}{-\mu_{w}^{2}Y_{i}^{2}}].$

Since

$\displaystyle\{{\alpha p-\beta({1-p})}\}=\left\{{\frac{\alpha\beta}{({\alpha+% \beta})}-\frac{\alpha\beta}{({\alpha+\beta})}}\right\}=0$

and

$\displaystyle\{{\alpha^{2}p+\beta^{2}({1-p})}\}=\left\{{\frac{\alpha^{2}\beta}% {({\alpha+\beta})}+\frac{\alpha\beta^{2}}{({\alpha+\beta})}}\right\}=\frac{% \alpha\beta({\alpha+\beta})}{({\alpha+\beta})}=\alpha\beta,$

therefore,

$\displaystyle E_{d}V_{R}({Z_{i}^{*}})=E_{d}[{Y_{i}^{2}E_{R}({W^{2}})+\alpha% \beta E_{R}({U^{2}})E_{R}({W^{2}})-\mu_{w}^{2}Y_{i}^{2}}],=[{E_{d}({Y_{i}^{2}}% )E_{R}({W^{2}})+\alpha\beta E_{R}({U^{2}})E_{R}({W^{2}})-\mu_{w}^{2}E_{d}({Y_{% i}^{2}})}],=[{({\sigma_{y}^{2}+\mu_{y}^{2}})({\sigma_{w}^{2}+\mu_{w}^{2}})+% \alpha\beta({\sigma_{u}^{2}+\mu_{u}^{2}})({\sigma_{w}^{2}+\mu_{w}^{2}})-\mu_{w% }^{2}({\sigma_{y}^{2}+\mu_{y}^{2}})}],=[{\sigma_{w}^{2}({\sigma_{y}^{2}+\mu_{y% }^{2}})+\alpha\beta({\sigma_{u}^{2}+\mu_{u}^{2}})({\sigma_{w}^{2}+\mu_{w}^{2}}% )}].$ (10)

Putting Eq. (10) in Eq. (9) we get the variance of $\hat{\mu}_{y(P)}$ as

$\displaystyle V({\hat{\mu}_{y(P)}})=\frac{1}{n^{2}\mu_{w}^{2}}\left[{\sum_{i=1% }^{n}{\{{\sigma_{w}^{2}({\sigma_{y}^{2}+\mu_{y}^{2}})+\alpha\beta({\sigma_{u}^% {2}+\mu_{u}^{2}})({\sigma_{w}^{2}+\mu_{w}^{2}})}\}}+n\mu_{w}^{2}\sigma_{y}^{2}% }\right],=\frac{1}{n^{2}\mu_{w}^{2}}[{n\{{\sigma_{w}^{2}({\sigma_{y}^{2}+\mu_{% y}^{2}})+\alpha\beta({\sigma_{u}^{2}+\mu_{u}^{2}})({\sigma_{w}^{2}+\mu_{w}^{2}% })}\}+n\mu_{w}^{2}\sigma_{y}^{2}}],=\frac{1}{n\mu_{w}^{2}}[{\sigma_{w}^{2}({% \sigma_{y}^{2}+\mu_{y}^{2}})+\alpha\beta({\sigma_{u}^{2}+\mu_{u}^{2}})({\sigma% _{w}^{2}+\mu_{w}^{2}})+\mu_{w}^{2}\sigma_{y}^{2}}],=\frac{V({Z^{*}})}{n\mu_{w}% ^{2}}=\frac{\sigma_{z^{*}}^{2}}{n\mu_{w}^{2}},$ (11)

where

$\displaystyle V({Z^{*}})=\sigma_{z^{*}}^{2}=[{\sigma_{w}^{2}({\sigma_{y}^{2}+% \mu_{y}^{2}})+\alpha\beta({\sigma_{u}^{2}+\mu_{u}^{2}})({\sigma_{w}^{2}+\mu_{w% }^{2}})+\mu_{w}^{2}\sigma_{y}^{2}}],=[{\mu_{w}^{2}\mu_{y}^{2}C_{w}^{2}({1+C_{y% }^{2}})+\alpha\beta\mu_{u}^{2}\mu_{w}^{2}({1+C_{u}^{2}})({1+C_{w}^{2}})+\mu_{w% }^{2}\mu_{y}^{2}C_{y}^{2}}],=\mu_{w}^{2}[{\mu_{y}^{2}C_{w}^{2}({1+C_{y}^{2}})+% \alpha\beta\mu_{u}^{2}({1+C_{u}^{2}})({1+C_{w}^{2}})+\mu_{y}^{2}C_{y}^{2}}].$ (12)

An estimator of the variance of the suggested estimator $\hat{\mu}_{y(P)}$ of the population mean $\mu_{y}$ is given by

$\displaystyle\hat{V}({\hat{\mu}_{y(P)}})=\frac{\sum_{i=1}^{n}{({Z_{i}^{*}-\bar% {Z}^{*}})^{2}}}{n({n-1})\mu_{w}^{2}},$ (13)

where $\bar{Z}^{*}=\frac{1}{n}\sum_{i=1}^{n}{Z_{i}^{*}}$ .

4. Efficiency comparison

From Eqs (4) and (11) we have that

$\displaystyle V({\hat{\mu}_{y(P)}})<V({\hat{\mu}_{s}}),$ $\displaystyle\text{if }\frac{V({Z^{*}})}{n\mu_{w}^{2}}<\frac{V(Z)}{n\mu_{w}^{2% }},$ $\displaystyle\text{i.e. if }V({Z^{*}})<V(Z),$ $\displaystyle\text{i.e. if }\mu_{w}^{2}[{\mu_{y}^{2}C_{w}^{2}({1+C_{y}^{2}})+% \alpha\beta\mu_{u}^{2}({1+C_{u}^{2}})({1+C_{w}^{2}})+\mu_{y}^{2}C_{y}^{2}}]<% \mu_{w}^{2}[{\mu_{u}^{2}({1+C_{u}^{2}})({1+C_{w}^{2}})+\mu_{y}^{2}C_{w}^{2}({1% +C_{y}^{2}})+\mu_{y}^{2}C_{y}^{2}+\mu_{u}^{2}({2RC_{w}^{2}-1})}],$ $\displaystyle\text{i.e. if }\alpha\beta({1+C_{u}^{2}})({1+C_{w}^{2}})<[{({1+C_% {u}^{2}})({1+C_{w}^{2}})+({2RC_{w}^{2}-1})}],$ $\displaystyle\text{i.e. if }[{({1+C_{u}^{2}})({1+C_{w}^{2}})({1-\alpha\beta})+% ({2RC_{w}^{2}-1})}]>0$

which always holds if

$\displaystyle 0<\alpha\beta<1\text{ and }C_{w}^{2}>\frac{1}{2R},$ (14)

where $C_{y}^{2}={\sigma_{y}^{2}}/{\mu_{y}^{2}}$ , $C_{u}^{2}={\sigma_{u}^{2}}/{\mu_{u}^{2}}$ , $C_{w}^{2}={\sigma_{w}^{2}}/{\mu_{w}^{2}}$ and $R={\mu_{y}}/{\mu_{u}}$ .

Thus the proposed estimator $\hat{\mu}_{y(P)}$ is more efficient than the estimator $\hat{\mu}_{s}$ of Saha (2007) as long as the conditions in Eq. (14) are satisfied.

5. Numerical illustration

To cast light upon the performance of the proposed model over Saha’s model, we have computed the percent relative efficiency (PRE) of the proposed estimator $\hat{\mu}_{y(P)}$ with respect to the estimator $\hat{\mu}_{s}$ of Saha (2007) by using the following formula:

$\displaystyle\textit{PRE}({\hat{\mu}_{y(P)},\hat{\mu}_{s}})=\frac{V(Z)}{V({Z^{% *}})}\ast 100.$ (15)

From Eqs (5) and (12) we have

$\displaystyle\textit{PRE}({\hat{\mu}_{y(P)},\hat{\mu}_{s}})=\frac{[{\mu_{u}^{2% }({1+C_{u}^{2}})({1+C_{w}^{2}})+\mu_{y}^{2}C_{w}^{2}({1+C_{y}^{2}})+\mu_{y}^{2% }C_{y}^{2}+\mu_{u}^{2}({2RC_{w}^{2}-1})}]}{[{\mu_{y}^{2}C_{w}^{2}({1+C_{y}^{2}% })+\alpha\beta\mu_{u}^{2}({1+C_{u}^{2}})({1+C_{w}^{2}})+\mu_{y}^{2}C_{y}^{2}}]% }\ast 100.$ (16)

We consider a sensitive variable $Y$ with the mean $\mu_{y}=2$ and the coefficient of variation $C_{y}$ in the range of 0.1 to 2.0; and we assume that scrambling variable are Fisher distributed as in Eichhorn and Hayre (1983) and the Diana and Perri (2010).

We have computed the $\textit{PRE}({\hat{\mu}_{y(P)},\hat{\mu}_{s}})$ for following two cases:

(ii)

(i)

$W\sim F({10,5})$ and $U\sim F({5,5})$

$\mu_{w}=1.6667,$	$\mu_{u}=1.6667,$
$\sigma_{w}^{2}=7.2222,$	$\sigma_{u}^{2}=8.8889,$
$C_{w}^{2}=2.60,$	$C_{u}^{2}=3.20.$

(ii)

W\sim F({10,50})

and

U\sim F({1,5})

$\mu_{w}=1.0417,$	$\mu_{u}=1.6667,$
$\sigma_{w}^{2}=0.2736,$	$\sigma_{u}^{2}=22.2223,$
$C_{w}^{2}=0.255,$	$C_{u}^{2}=7.9997.$

Findings are shown in Tables 1 and 2.

Table 1

PREs of the suggested estimator $\hat{\mu}_{y(P)}$ with respect to the estimator $\hat{\mu}_{s}$ due to Saha (2007) in Case-I

CASE-I $W\sim F(10,5)$ , $U\sim F(5,5)$
$C_{y}=$ 0.1
$\beta$ $\alpha$	0.03	0.09	0.1	0.12	0.15	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
0.08	630.37	618.65	616.74	612.96	607.36	598.26	580.86	564.45	548.93	534.24	520.32	507.11	494.55
0.09	629.62	616.50	614.37	610.15	603.92	593.82	574.60	556.58	539.66	523.74	508.73	494.55	481.15
0.10	628.88	614.37	612.02	607.36	600.51	589.44	568.46	548.93	530.70	513.63	497.63	482.60	468.45
0.12	627.40	610.15	607.36	601.87	593.82	580.86	556.58	534.24	513.63	494.55	476.84	460.35	444.97
0.15	625.19	603.92	600.51	593.82	584.05	568.46	539.66	513.63	490.00	468.45	448.71	430.57	413.84
0.2	621.54	593.82	589.44	580.86	568.46	548.93	513.63	482.60	455.11	430.57	408.55	388.67	370.64
0.3	614.37	574.60	568.46	556.58	539.66	513.63	468.45	430.57	398.36	370.64	346.52	325.35	306.62
0.4	607.36	556.58	548.93	534.24	513.63	482.60	430.57	388.67	354.20	325.35	300.84	279.77	261.46
0.5	600.51	539.66	530.70	513.63	490.00	455.11	398.36	354.20	318.86	289.92	265.81	245.39	227.89
0.6	593.82	523.74	513.63	494.55	468.45	430.57	370.64	325.35	289.92	261.46	238.08	218.54	201.96
0.7	587.27	508.73	497.63	476.84	448.71	408.55	346.52	300.84	265.81	238.08	215.59	196.98	181.33
0.8	580.86	494.55	482.60	460.35	430.57	388.67	325.35	279.77	245.39	218.54	196.98	179.30	164.52
0.9	574.60	481.15	468.45	444.97	413.84	370.64	306.62	261.46	227.89	201.96	181.33	164.52	150.57
$C_{y}=$ 0.50
$\beta$ $\alpha$	0.03	0.09	0.1	0.12	0.15	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
0.08	500.38	493.33	492.17	489.87	486.47	480.90	470.13	459.84	449.98	440.54	431.49	422.80	414.46
0.09	499.93	492.03	490.73	488.17	484.36	478.16	466.22	454.86	444.04	433.72	423.87	414.46	405.45
0.10	499.49	490.73	489.30	486.47	482.28	475.45	462.37	449.98	438.24	427.10	416.51	406.43	396.83
0.12	498.60	488.17	486.47	483.11	478.16	470.13	454.86	440.54	427.10	414.46	402.54	391.29	380.65
0.15	497.27	484.36	482.28	478.16	472.11	462.37	444.04	427.10	411.41	396.83	383.25	370.57	358.70
0.2	495.07	478.16	475.45	470.13	462.37	449.98	427.10	406.43	387.68	370.57	354.91	340.52	327.26
0.3	490.73	466.22	462.37	454.86	444.04	427.10	396.83	370.57	347.57	327.26	309.19	293.01	278.44
0.4	486.47	454.86	449.98	440.54	427.10	406.43	370.57	340.52	314.98	293.01	273.90	257.13	242.29
0.5	482.28	444.04	438.24	427.10	411.41	387.68	347.57	314.98	287.99	265.25	245.84	229.08	214.46
0.6	478.16	433.72	427.10	414.46	396.83	370.57	327.26	293.01	265.25	242.29	223.00	206.55	192.36
0.7	474.11	423.87	416.51	402.54	383.25	354.91	309.19	273.90	245.84	223.00	204.04	188.05	174.38
0.8	470.13	414.46	406.43	391.29	370.57	340.52	293.01	257.13	229.08	206.55	188.05	172.59	159.48
0.9	466.22	405.45	396.83	380.65	358.70	327.26	278.44	242.29	214.46	192.36	174.38	159.48	146.93
$C_{y}=$ 1
$\beta$ $\alpha$	0.03	0.09	0.1	0.12	0.15	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
0.08	326.73	324.10	323.67	322.81	321.52	319.40	315.24	311.19	307.24	303.39	299.64	295.97	292.40
0.09	326.56	323.62	323.13	322.16	320.72	318.35	313.71	309.20	304.82	300.57	296.43	292.40	288.48
0.10	326.40	323.13	322.59	321.52	319.93	317.31	312.19	307.24	302.44	297.79	293.29	288.91	284.66
0.12	326.07	322.16	321.52	320.24	318.35	315.24	309.20	303.39	297.79	292.40	287.20	282.18	277.33
0.15	325.57	320.72	319.93	318.35	316.01	312.19	304.82	297.79	291.08	284.66	278.52	272.64	267.01
0.2	324.75	318.35	317.31	315.24	312.19	307.24	297.79	288.91	280.54	272.64	265.18	258.11	251.41
0.3	323.13	313.71	312.19	309.20	304.82	297.79	284.66	272.64	261.60	251.41	241.99	233.25	225.12
0.4	321.52	309.20	307.24	303.39	297.79	288.91	272.64	258.11	245.05	233.25	222.53	212.75	203.80
0.5	319.93	304.82	302.44	297.79	291.08	280.54	261.60	245.05	230.47	217.53	205.97	195.57	186.17
0.6	318.35	300.57	297.79	292.40	284.66	272.64	251.41	233.25	217.53	203.80	191.70	180.95	171.35
0.7	316.79	296.43	293.29	287.20	278.52	265.18	241.99	222.53	205.97	191.70	179.28	168.37	158.71
0.8	315.24	292.40	288.91	282.18	272.64	258.11	233.25	212.75	195.57	180.95	168.37	157.42	147.81
0.9	313.71	288.48	284.66	277.33	267.01	251.41	225.12	203.80	186.17	171.35	158.71	147.81	138.31
$C_{y}=$ 1.50
$\beta$ $\alpha$	0.03	0.09	0.1	0.12	0.15	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
0.08	231.60	230.52	230.34	229.98	229.44	228.56	226.80	225.08	223.38	221.70	220.05	218.43	216.82
0.09	231.53	230.31	230.11	229.71	229.11	228.11	226.15	224.22	222.33	220.46	218.63	216.82	215.05
0.1	231.46	230.11	229.89	229.44	228.78	227.68	225.51	223.38	221.29	219.24	217.22	215.25	213.30
0.10	231.33	229.71	229.44	228.91	228.11	226.80	224.22	221.70	219.24	216.82	214.47	212.16	209.90
0.12	231.12	229.11	228.78	228.11	227.13	225.51	222.33	219.24	216.23	213.30	210.46	207.69	204.99
0.2	230.79	228.11	227.68	226.80	225.51	223.38	219.24	215.25	211.40	207.69	204.10	200.64	197.29

Table 1, continued
CASE-I $W\sim F(10,5)$ , $U\sim F(5,5)$
$C_{y}=$ 1.50
$\beta$ $\alpha$	0.03	0.09	0.1	0.12	0.15	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
0.3	230.11	226.15	225.51	224.22	222.33	219.24	213.30	207.69	202.36	197.29	192.48	187.89	183.52
0.4	229.44	224.22	223.38	221.70	219.24	215.25	207.69	200.64	194.06	187.89	182.11	176.67	171.54
0.5	228.78	222.33	221.29	219.24	216.23	211.40	202.36	194.06	186.41	179.34	172.79	166.71	161.03
0.6	228.11	220.46	219.24	216.82	213.30	207.69	197.29	187.89	179.34	171.54	164.39	157.81	151.74
0.7	227.46	218.63	217.22	214.47	210.46	204.10	192.48	182.11	172.79	164.39	156.76	149.81	143.45
0.8	226.80	216.82	215.25	212.16	207.69	200.64	187.89	176.67	166.71	157.81	149.81	142.59	136.03
0.9	226.15	215.05	213.30	209.90	204.99	197.29	183.52	171.54	161.03	151.74	143.45	136.03	129.34
$C_{y}=$ 2
$\beta$ $\alpha$	0.03	0.09	0.1	0.12	0.15	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
0.08	182.90	182.36	182.27	182.09	181.83	181.38	180.50	179.62	178.76	177.90	177.05	176.21	175.37
0.09	182.87	182.26	182.16	181.96	181.66	181.16	180.17	179.19	178.22	177.26	176.31	175.37	174.45
0.10	182.83	182.16	182.05	181.83	181.49	180.94	179.84	178.76	177.69	176.63	175.58	174.55	173.53
0.12	182.83	182.16	182.05	181.83	181.49	180.94	179.84	178.76	177.69	176.63	175.58	174.55	173.53
0.15	182.77	181.96	181.83	181.56	181.16	180.50	179.19	177.90	176.63	175.37	174.14	172.92	171.72
0.2	182.50	181.16	180.94	180.50	179.84	178.76	176.63	174.55	172.52	170.53	168.59	166.70	164.85
0.3	182.16	180.17	179.84	179.19	178.22	176.63	173.53	170.53	167.64	164.85	162.14	159.53	156.99
0.4	181.83	179.19	178.76	177.90	176.63	174.55	170.53	166.70	163.03	159.53	156.16	152.94	149.85
0.5	181.49	178.22	177.69	176.63	175.06	172.52	167.64	163.03	158.67	154.54	150.61	146.88	143.33
0.6	181.16	177.26	176.63	175.37	173.53	170.53	164.85	159.53	154.54	149.85	145.44	141.28	137.36
0.7	180.83	176.31	175.58	174.14	172.02	168.59	162.14	156.16	150.61	145.44	140.62	136.10	131.86
0.8	180.50	175.37	174.55	172.92	170.53	166.70	159.53	152.94	146.88	141.28	136.10	131.28	126.79
0.9	180.17	174.45	173.53	171.72	169.07	164.85	156.99	149.85	143.33	137.36	131.86	126.79	122.09

Table 2

PREs of the suggested estimator $\hat{\mu}_{y(P)}$ with respect to the estimator $\hat{\mu}_{s}$ due to Saha (2007) in Case-II

CASE-II $W\sim F(10,50)$ , $U\sim F(1,5)$
$C_{y}=$ 0.1
$\beta$ $\alpha$	0.03	0.09	0.1	0.12	0.15	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
0.08	2758.00	2435.22	2388.63	2300.59	2180.07	2005.01	1727.57	1517.57	1353.09	1220.78	1112.04	1021.09	943.89
0.09	2735.34	2382.93	2332.83	2238.71	2110.96	1927.62	1642.34	1430.62	1267.25	1137.37	1031.64	943.89	869.90
0.10	2713.04	2332.83	2279.59	2180.07	2046.09	1855.98	1565.13	1353.09	1191.65	1064.63	962.07	877.54	806.66
0.12	2669.54	2238.71	2180.07	2071.55	1927.62	1727.57	1430.62	1220.78	1064.63	943.89	847.75	769.38	704.20
0.15	2606.83	2110.96	2046.09	1927.62	1773.58	1565.13	1267.25	1064.62	917.86	806.66	719.50	649.33	591.63
0.2	2508.61	1927.62	1855.98	1727.57	1565.13	1353.09	1064.63	877.54	746.38	649.33	574.61	515.32	467.11
0.3	2332.83	1642.34	1565.13	1430.62	1267.25	1064.63	806.66	649.33	543.35	467.11	409.64	364.76	328.74
0.4	2180.07	1430.62	1353.09	1220.78	1064.63	877.54	649.33	515.32	427.16	364.76	318.26	282.28	253.61
0.5	2046.08	1267.25	1191.65	1064.63	917.86	746.38	543.35	427.16	351.90	299.19	260.22	230.22	206.43
0.6	1927.61	1137.37	1064.63	943.89	806.66	649.33	467.11	364.76	299.19	253.61	220.08	194.38	174.05
0.7	1822.11	1031.64	962.07	847.75	719.50	574.61	409.64	318.26	260.22	220.08	190.67	168.19	150.45
0.8	1727.56	943.89	877.54	769.38	649.33	515.32	364.76	282.28	230.22	194.38	168.19	148.22	132.49
0.9	1642.34	869.90	806.66	704.28	591.63	467.11	328.74	253.61	206.43	174.05	150.45	132.49	118.36
$C_{y}=$ 0.5
$\beta$ $\alpha$	0.03	0.09	0.1	0.12	0.15	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
0.08	1390.07	1306.06	1293.03	1267.74	1231.61	1175.76	1078.00	995.24	924.29	862.77	808.94	761.43	719.19
0.09	1384.51	1291.42	1277.11	1249.42	1210.06	1149.70	1045.40	958.45	884.86	821.76	767.06	719.19	676.94
0.10	1378.98	1277.11	1261.57	1231.61	1189.25	1124.76	1014.72	924.29	848.66	784.47	729.30	681.39	639.38
0.12	1368.07	1249.42	1231.61	1197.48	1149.70	1078.00	958.45	862.77	784.47	719.19	663.94	616.58	575.52
0.15	1352.02	1210.06	1189.25	1149.70	1095.07	1014.72	884.86	784.47	704.53	639.38	585.26	539.59	500.53
0.2	1326.09	1149.70	1124.76	1078.00	1014.72	924.29	784.47	681.39	602.26	539.59	488.73	446.64	411.22
0.3	1277.10	1045.40	1014.72	958.45	884.86	784.47	639.38	539.59	466.74	411.22	367.51	332.19	303.07

Table 2, continued
CASE-II $W\sim F(10,50)$ , $U\sim F(1,5)$
$C_{y}=$ 0.5
$\beta$ $\alpha$	0.03	0.09	0.1	0.12	0.15	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
0.4	1231.61	958.45	924.29	862.77	784.47	681.394	539.59	446.64	381.01	332.19	294.47	264.43	239.96
0.5	1189.24	884.86	848.66	784.47	704.53	602.26	466.74	381.01	321.88	278.64	245.65	219.63	198.60
0.6	1149.69	821.76	784.47	719.19	639.38	539.59	411.22	332.19	278.64	239.96	210.71	187.82	169.41
0.7	1112.69	767.06	729.30	663.94	585.26	488.73	367.51	294.47	245.65	210.71	184.47	164.05	147.69
0.8	1077.99	719.19	681.39	616.58	539.59	446.64	332.19	264.43	219.63	187.82	164.05	145.62	130.92
0.9	1045.40	676.94	639.38	575.52	500.53	411.22	303.07	239.96	198.60	169.41	147.69	130.92	117.56
$C_{y}=$ 1
$\beta$ $\alpha$	0.03	0.09	0.1	0.12	0.15	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
0.08	594.61	580.30	577.98	573.40	566.66	555.77	535.21	516.11	498.33	481.73	466.21	451.65	437.97
0.09	593.70	577.69	575.11	570.01	562.53	550.49	527.88	507.07	487.83	469.99	453.42	437.97	423.55
0.10	592.78	575.11	572.26	566.66	558.46	545.30	520.76	498.33	477.76	458.81	441.31	425.10	410.04
0.12	590.97	570.01	566.66	560.08	550.49	535.21	507.07	481.73	458.81	437.97	418.95	401.50	385.45
0.15	588.26	562.53	558.46	550.49	538.95	520.76	487.83	458.81	433.06	410.04	389.34	370.64	353.65
0.2	583.81	550.49	545.30	535.21	520.76	498.33	458.81	425.10	396.01	370.64	348.32	328.55	310.89
0.3	575.11	527.88	520.76	507.07	487.83	458.81	410.04	370.64	338.15	310.89	287.70	267.73	250.36
0.4	566.66	507.07	498.33	481.73	458.81	425.10	370.64	328.55	295.04	267.73	245.05	225.92	209.55
0.5	558.45	487.83	477.76	458.81	433.06	396.01	338.15	295.04	261.68	235.10	213.42	195.40	180.18
0.6	550.48	469.99	458.81	437.97	410.04	370.64	310.89	267.73	235.10	209.55	189.01	172.14	158.04
0.7	542.74	453.42	441.31	418.95	389.34	348.32	287.70	245.05	213.42	189.01	169.62	153.84	140.74
0.8	535.21	437.97	425.10	401.50	370.64	328.55	267.73	225.92	195.40	172.14	153.84	139.05	126.85
0.9	527.88	423.55	410.04	385.45	353.65	310.89	250.36	209.55	180.18	158.04	140.74	126.85	115.46
$C_{y}=$ 1.50
$\beta$ $\alpha$	0.03	0.09	0.1	0.12	0.15	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
0.08	343.93	339.80	339.12	337.77	335.76	332.47	326.08	319.93	314.01	308.30	302.80	297.49	292.36
0.09	343.67	339.03	338.27	336.76	334.52	330.85	323.75	316.94	310.42	304.16	298.14	292.36	286.80
0.1	343.41	338.27	337.43	335.76	333.29	329.25	321.45	314.01	306.91	300.12	293.63	287.41	281.45
0.10	342.89	336.76	335.76	333.78	330.85	326.08	316.94	308.30	300.12	292.36	285.00	277.99	271.32
0.12	342.11	334.52	333.29	330.85	327.26	321.45	310.42	300.12	290.49	281.45	272.96	264.97	257.43
0.2	340.82	330.85	329.25	326.08	321.45	314.01	300.12	287.41	275.73	264.97	255.01	245.77	237.18
0.3	338.27	323.75	321.45	316.94	310.42	300.12	281.45	264.97	250.31	237.18	225.37	214.67	204.95
0.4	335.76	316.94	314.01	308.30	300.12	287.41	264.97	245.77	229.17	214.67	201.90	190.56	180.42
0.5	333.29	310.42	306.91	300.12	290.49	275.73	250.31	229.17	211.33	196.06	182.86	171.31	161.14
0.6	330.85	304.16	300.12	292.36	281.45	264.97	237.18	214.67	196.06	180.42	167.10	155.60	145.59
0.7	328.45	298.14	293.63	285.00	272.96	255.01	225.37	201.90	182.86	167.10	153.84	142.53	132.77
0.8	326.08	292.36	287.41	277.99	264.97	245.77	214.67	190.56	171.31	155.60	142.53	131.48	122.02
0.9	323.75	286.80	281.45	271.32	257.43	237.18	204.95	180.42	161.14	145.59	132.77	122.02	112.89
$C_{y}=$ 2
$\beta$ $\alpha$	0.03	0.09	0.1	0.12	0.15	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
0.08	242.69	240.97	240.69	240.12	239.28	237.89	235.16	232.49	229.87	227.32	224.82	222.38	219.99
0.09	242.58	240.65	240.34	239.70	238.76	237.20	234.15	231.17	228.27	225.44	222.68	219.99	217.36
0.10	242.47	240.34	239.98	239.28	238.24	236.52	233.15	229.87	226.69	223.60	220.58	217.65	214.79
0.12	242.47	240.34	239.98	239.28	238.24	236.52	233.15	229.87	226.69	223.60	220.58	217.65	214.79
0.15	242.26	239.70	239.28	238.45	237.20	235.16	231.17	227.32	223.60	219.99	216.50	213.12	209.84
0.2	241.40	237.20	236.52	235.16	233.15	229.87	223.60	217.65	212.01	206.66	201.57	196.73	192.11
0.3	240.34	234.15	233.15	231.17	228.27	223.60	214.79	206.66	199.12	192.11	185.58	179.47	173.76
0.4	239.28	231.17	229.87	227.32	223.60	217.65	206.66	196.73	187.70	179.47	171.93	165.00	158.61
0.5	238.24	228.27	226.69	223.60	219.11	212.01	199.12	187.70	177.53	168.40	160.16	152.69	145.89
0.6	237.20	225.44	223.60	219.99	214.79	206.66	192.11	179.47	168.40	158.61	149.90	142.09	135.06
0.7	236.18	222.68	220.58	216.50	210.65	201.57	185.58	171.93	160.16	149.90	140.87	132.86	125.72
0.8	235.16	219.99	217.65	213.12	206.66	196.73	179.47	165.00	152.69	142.09	132.86	124.76	117.59
0.9	234.15	217.36	214.79	209.84	202.82	192.11	173.76	158.61	145.89	135.06	125.72	117.59	110.45

Tables 1 and 2 exhibit that

(iii) (i)

In both the cases the $\textit{PRE}({\hat{\mu}_{y(P)},\hat{\mu}_{s}})$ is greater than 100%. Thus the proposed estimator $\hat{\mu}_{y(P)}$ is more efficient than the estimator $\hat{\mu}_{s}$ of Saha (2007) for the values of $({\alpha,\beta,C_{y}})$ closed in Tables 1 and 2;

(ii)

The $\textit{PRE}({\hat{\mu}_{y(P)},\hat{\mu}_{s}})$ decreases with increasing values of $({C_{y},\alpha,\beta})$ in both the cases;

(iii)

The gain in efficiency is substantial by using the proposed estimator $\hat{\mu}_{y(P)}$ over the estimator $\hat{\mu}_{s}$ of the Saha (2007) for smaller values of the coefficient of variation $C_{y}(\text{i.e.}∼{}0<C_{y}<1.0)$ and if ‘ $\alpha\beta$ ’ is closer to zero;

(iv)

The gain in efficiency is larger in Case-II than in Case-I except in few cases.

Thus for fixed value of $C_{y}$ , to obtain considerable gain in efficiency by using the proposed estimator $\hat{\mu}_{y(P)}$ over the estimator $\hat{\mu}_{s}$ due to Saha (2007), a suitable selection of $({\alpha,\beta})$ , based on practicable value of $p$ , should be made such that their product $\alpha\beta$ should remain near to zero. Gjestvang and Singh (2009) have mentioned that a practical choice of $({\alpha,\beta})$ , fixed by our experience from repeated surveys.

Footnotes

Acknowledgments

Authors are thankful to the learned referee for his valuable comments regarding improvement of the earlier draft of the paper.

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