On a generalized class of almost unbiased ratio estimators

Abstract

In sampling from finite populations to estimate the finite population mean/total of the study variable, one often observes available information on an associated auxiliary variable along with study variable to obtain an estimator, which is more efficient than the simple mean per unit estimator based on observations on study variable only. The classical ratio estimator is one such estimator, which is simple to compute and is more efficient than the simple mean per unit estimator under certain conditions. However, the ratio estimator in spite of its simplicity is a biased estimator having bias of $O(1/n)$ , $n$ being the sample size. The bias may be negligible when the sample size is very large. For small sample size the bias may be substantially large so as to make the estimate unreliable to be used in practice. Beale (1962) and Tin (1965) have suggested some modified forms of ratio estimator which remove the first order bias, thus reducing the biases to $O(1/n^{2})$ . Such modified ratio estimators are called Almost Unbiased Ratio estimators. This paper deals with construction of Generalized class of Almost unbiased ratio estimators using Srivastava’s (1971) generalized class of estimators and finds their expected values and variances in a generalized form. Further, as special cases Bahl-Tuteja’s (1991) ratio type exponential estimator and Swain’s (2014) square root transformation ratio type estimator are compared with regard to bias and efficiency along with numerical illustrations.

Keywords

Simple random sampling ratio estimator almost unbiased ratio estimator bias efficiency

1. Introduction

Large scale sample surveys are conducted in countries around the world, especially in developing countries to assess the present status of socio-economic scenario and also for future planning for development. In sample surveys auxiliary variables are often observed along with the main variable under study to estimate the parameters of the main variable with greater precision. In case of a single auxiliary variable positively correlated with the main variable the simplest method is the ratio method of estimation to estimate the population mean/population total/population ratio. The earliest use of ratio method of estimation in sample surveys is due to Cochran (1940) among many others. During last eight decades a large number of research papers on different aspects of ratio method of estimation and their modifications have been published. In spite of its simplicity the ratio estimate is a biased estimate, although bias decreases with increase in sample size. For small samples, the bias may be substantial, more so in stratification where the bias accumulates over strata to make the estimator sometimes unacceptable to be used in practice.This has made many research workers in sampling theory to devise techniques either changing the sampling scheme or at the estimation stage to reduce bias of ratio estimator completely or to almost unbiased.

Researches for reducing the bias of the ratio estimator date back to 1950’s through the illuminating works of Koop (1951), Lahiri (1951), Midzuno (1952), and Sen (1952), Hartley and Ross (1954), Goodman and Hartley (1958), Mickey (1959), Quenouille (1956), Durbin (1959), Pascual (1961), Williams (1961), Beale (1962), and Tin (1965) among many others. Some important discussions on bias reduction are found in Rao and Webster (1966), Chakrabarty (1968), and Mussa (1999). The ratio estimators whose biases of $O(1/n)$ are removed and thus reducing the biases to $O(1/n^{2})$ , $n$ being the sample size, are termed as almost unbiased ratio or ratio-type estimators.

Lahiri (1951), Midzuno (1952), and Sen (1952) proposed sampling schemes to make the ratio estimator unbiased. Murthy and Nanjamma (1959) and Murthy (1962) used interpenetrating sub-samples to derive almost unbiased estimates.

Hartley and Ross (1954) introduced an alternative estimator of the population ratio/mean using mean of ratios. de Pascual (1961) discussed unbiased ratio estimates in stratified random sampling. Rao and Swain (2014) derived a class of alternative Hartley-Ross unbiased estimators in simple random sampling. Williams (1961) proposed a method for deriving unbiased ratio and regression estimators of which Hartley-Ross estimator is a special case. Huchinson (1971) made a Monte Carlo study of Hartley-Ross estimator along with other choices.

Biradar and Singh (1992) considered a weighted combination of simple mean per unit estimator, ratio estimator, and mean of the ratios estimator to derive a class of unbiased estimators.

Al-Jaraha (2008) has made some illuminating discussions on unbiased ratio estimators and has generalized Hartley-Ross estimator by introducing inclusion probabilities.

Beale (1962) and Tin (1965) used the estimate of the first order bias of ratio estimator in different forms and suggested almost unbiased ratio-type estimators whose first order biases are removed and the ultimate biases reducing to $O(1/n^{2})$ . Durbin (1959) utilizing Quenouille’s (1956) technique proposed a method of bias reduction of ratio estimator by an adjustment after splitting the sample in two halves.

Chakrabarty (1968) and Mussa (1999) considered a bias reduction technique by taking a weighted combination of classical ratio estimator and almost unbiased estimators such as those due to Quenouille (1956) and Tin (1965). Such techniques remove the first order bias of the ratio estimator, thus reducing the bias to $O(1/n^{2})$ .

Srivastava (1971) proposed a generalized class of estimators which includes all ratio type estimators as special cases. Prabhu-Ajgaonkar (1993) proved the non-existence of an optimum estimator in Srivastava’s generalized class of estimators.

A new technique advocated by Singh and Singh (1991) is to make the linear variety of ratio-type estimators almost unbiased through ‘almost bias precipitate filtration’ (for details see Singh, 2003). In this paper a generalized class of Beale type and Tin type almost unbiased ratio estimators are constructed using Srivastava’s (1971) generalized class of estimators and some of its special cases are compared with regard to bias and efficiency along with numerical illustrations.

2. Ratio estimator its modifications

Let there be a finite population (S) of size $N$ distinguishable units $U_{1},U_{2},\ldots,U_{N}$ . Each unit is indexed by a pair of correlated real variables $(y,x)$ where $y$ is the study variable and $x$ is the auxiliary variable. The corresponding values of $(y,x)$ are $(Y_{1},X_{1}),(Y_{2},X_{2}),\ldots,(Y_{N},X_{N})$ .

A simple random sample $s$ of size $n(n<N)$ without replacement is selected from the finite population (S). The sample units are $u_{1},u_{2},\ldots,u_{n}$ with paired values $(y_{1},x_{1}),(y_{2},x_{2}),\ldots,(y_{n},x_{n})$ .

Define $\bar{Y}$ and $\bar{X}$ as the finite population means of $y$ and $x$ respectively.

The population ratio $R=\frac{\bar{Y}}{\bar{X}}$ .

The finite population variances of $y$ and $x$ , and covariance between $y$ and $x$ are respectively defined as

$\displaystyle\!\!\!\!\!\!\!\!S_{y}^{2}=(N\!-\!1)^{-1}\sum_{1}^{N}(Y_{i}\!-\!% \bar{Y})^{2},S_{x}^{2}=(N\!-\!1)^{-1}\sum_{1}^{N}(X_{i}\!-\!\bar{X})^{2},\text% { and }S_{xy}=(N\!-\!1)^{-1}\sum_{1}^{N}(X_{i}\!-\!\bar{X})(Y_{i}\!-\!\bar{Y}).$ $\displaystyle\!\!\!\!\!\!\!\!C_{y}^{2}=\frac{S_{y}^{2}}{\bar{Y}^{2}},C_{x}^{2}% =\frac{S_{x}^{2}}{\bar{X}^{2}}\text{ and }C_{xy}=\frac{S_{xy}}{\bar{Y}\bar{X}}% =\rho C_{x}C_{y}$

$C_{y}^{2}$ and $C_{x}^{2}$ are the squared coefficients of variations of $y$ and $x$ respectively, and $C_{xy}$ is the coefficient co-variation between $y$ and $x$ , and $\rho$ being the correlation coefficient between $x$ and $y$ .

The sample means of $y$ and $x$ are $\bar{y}$ and $\bar{x}$ respectively.

The sample variances of $y$ and $x$ are respectively $s_{y}^{2}=(n-1)^{-1}\sum_{1}^{n}(y_{i}-\bar{y})^{2}$ and $s_{x}^{2}=(n-1)^{-1}\sum_{1}^{n}(x_{i}-\bar{x})^{2}$ . The sample covariance between $y$ and $x$ is $s_{xy}=(n-1)^{-1}\sum_{1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})$ , $c_{x}^{2}=\frac{s_{x}^{2}}{\bar{x}^{2}}$ and $c_{yx}=\frac{s_{xy}}{\bar{y}\bar{x}}$ .

Sample ratio $r=\frac{\bar{y}}{\bar{x}}$ .

Write $\theta=\frac{1}{n}-\frac{1}{N}$ .

The ratio estimator of population mean $\bar{Y}$ , when $\bar{X}$ is known, is given by

$\displaystyle t_{r}=\frac{\bar{y}}{\bar{x}}\bar{X}.$ (1)

The ratio estimator of population ratio $R$ is

$\displaystyle r=\frac{\bar{y}}{\bar{x}}.$ (2)

The expected value of $t_{r}$ to first order of approximation, that is, to $O(1/n)$ is given by

$\displaystyle E(t_{r})=\bar{Y}[1+\theta(C_{x}^{2}-C_{xy})],\quad\text{(% Sukhatme et al., 1984)}$ (3)

The bias of $t_{r}$ to $O(1/n)$ is

$\displaystyle B(t_{r})=\theta\bar{Y}(C_{x}^{2}-C_{xy}),$ (4)

and the variance of $t_{r}$ to $O(1/n)$ is

$\displaystyle V(t_{r})=\theta\bar{Y}^{2}(C_{y}^{2}+C_{x}^{2}-2C_{xy}),\quad% \text{(Sukhatme et al., 1984)}$ (5)

Thus, if the sample size $n$ is sufficiently large such that the coefficient of variation of $\bar{x}$ less than say 0.1, the bias may be considered negligible compared to its standard deviation (Cochran, 1977). When the regression of $y$ on $x$ is linear and passes through the origin the classical ratio estimator attains the minimum possible variance. When this condition does not hold good, the ratio estimator is conditionally more efficient than the simple mean per unit estimator $\bar{y}$ if

$\displaystyle k>\frac{1}{2},\text{ where }k=\rho\frac{C_{y}}{C_{x}}.$ (6)

Hence, there exists other ratio-type of estimators which are less biased and more efficient than ratio estimator on certain zone of preference depending on the value of $k$ . Towards this end, some research workers have taken recourse to different transformation techniques on the auxiliary variable $x$ , for instance, power transformation, proposed by Srivastava (1967), exponential transformation suggested by Bahl and Tuteja (1992) and linear transformation suggested by Walsh (1970), Reddy (1974), Mohanty and Das (1971), Srivenkataramana (1978), Sisodia and Dwivedi (1981), Mohanty and Sahoo (1995), Upadhyaya and Singh (1984, 1999), Singh and Tailor (2003), Kadilar and Cingi (2004, 2006), Gupta and Shabbir (2007, 2008), Singh et al. (2008), Khoshnevisan et al. (2007) among others.

As the bias of ratio estimator $t_{r}$ to $O(1/n)$ becomes substantial for small sample sizes, research workers have constructd ratio-type estimators, whose biases to $O(1/n)$ are removed and the ultimate biases become of $O(1/n^{2})$ resulting in almost unbiased ratio estimators. In this paper discussions are restricted to Beale’s and Tin’s almost unbiased ratio estimators.

2.1 Almost unbiased ratio-type estimators

Beale (1962) proposed an almost unbiased ratio estimator given by

$\displaystyle t_{rB}=t_{r}\frac{1+\theta c_{xy}}{1+\theta c_{x}^{2}}=\bar{X}% \frac{\bar{y}\bar{x}+\theta s_{xy}}{\bar{x}^{2}+\theta s_{x}^{2}},$ (7)

where $c_{xy}$ and $c_{x}^{2}$ are consistent estimators of $C_{xy}$ and $C_{x}^{2}$ respectively.

Tin (1965) proposed another almost unbiased ratio estimators by subtracting the estimate of first order bias from the estimator itself given by

$\displaystyle t_{rT}=t_{r}[1+\theta(c_{xy}-c_{x}^{2})]=t_{r}[1+\theta(s_{xy}/% \bar{y}\bar{x})-\theta(s_{x}^{2}/\bar{x}^{2})]$ (8)

Both Beale’s and Tin’s estimators use same set of sample information $s_{xy}/\bar{x}\bar{y}$ and $s_{x}^{2}/\bar{x}^{2}$ to formulate their almost unbiased ratio-type estimators. Tin (1965) showed that in samping from finite populations and also under bivariate normality, the Beale’s estimator $t_{rB}$ is less biased than $t_{rT}$ and equally efficient with $t_{rT}$ to $O(1/n^{2})$ and empirically appears to be a better type of estimator as regards bias and efficiency. Beale’s estimator is in fractional form and has inbuilt property that it controls extreme values for the estimators. For instance, when $\bar{x}$ is small or even $\bar{x}=$ 0, Beale’s estimator is dominanted by $s_{xy}/s_{x}^{2}$ and hence will not give extreme values, where as Tin’s estimator is dominated by $s_{x}^{2}/\bar{x}^{2}$ and hence is extremely large when is $\bar{x}$ is small. Tin’s estimator is beset with a disvantage that it sometimes (on rare occasions) gives a negative estimate for a positive parameter (Tin, 1965). De-Graft Johnson (1969) and David (1971) have given illuminating discussions on almost unbiased ratio-type estimators. Srivastava et al. (1983) derived exact bias and mean square error of Beale’s estimator under bivariate normal model in the form an infinite series.

3. A generalized class of almost unbiased ratio estimators

Srivastava (1971) proposed a general class of estimators of the population mean $\bar{Y}$ of the study variable $y$ in the presence of a single auxiliary variable $x$ with known population mean $\bar{X}$ , is given by

$\displaystyle t_{g}=\bar{y}H(u),$ (9)

where (i) $\bar{y}$ is the sample mean of $y$ based on a simple random sample of size $n$ from the finite population consisting of $N$ units. (ii) $u=\frac{\bar{x}}{\bar{X}}$ and $H(\cdot)$ is a parametric function, such that it satisfies the following conditions (a) $H(1)=$ 1, (b) The first and second order derivatives of $H(u)$ with respect to $u$ exist and are known constants at a given point $u=$ 1.

In the present discourse it is further assumed that third and fourth order derivatives of $H(u)$ with respect $u=$ 1 exist and are known constants. Expanding $H(u)$ about the point $u=$ 1 in a fourth order Taylor’s series, we have

$\displaystyle H(u)=H(1+(u-1))=H(1)+(u-1)\left(\frac{\partial H}{\partial u}% \right)_{u=1}+\frac{(u-1)^{2}}{2}\left(\frac{\partial^{2}H}{\partial u^{2}}% \right)_{u=1}+(u-1)^{3}\frac{1}{6}\left(\frac{\partial^{3}H(u)}{\partial u^{3}% }\right)_{u=1}+(u-1)^{4}\frac{1}{24}\left(\frac{\partial^{4}H(u)}{\partial u^{% 4}}\right)_{u=1}+\ldots$

Write $\bar{y}=\bar{Y}(1+e_{o}),\bar{x}=\bar{X}(1+e_{1})$ with $E(e_{o})=E(e_{1})=0$ . Assuming that $|e_{1}|=|u-1|<$ 1, the higher degree terms greater than four may be assumed to be neglected. Thus we have

$\displaystyle t_{g}=\bar{y}H(u)=\bar{y}[(1+(u-1)H_{1}+(u-1)^{2}H_{2}+(u-1)^{3}% H_{3}+(u-1)^{4}H_{4}]$ (10)

where $H_{1}=\left(\frac{\partial H}{\partial u}\right)_{u=1}$ , $H_{2}=\frac{1}{2}\left(\frac{\partial^{2}H}{\partial u^{2}}\right)_{u=1}$ , $H_{3}=\frac{1}{6}\left(\frac{\partial^{3}H(u)}{\partial u^{3}}\right)_{u=1}$ , and $H_{4}=\frac{1}{24}\left(\frac{\partial^{4}H(u)}{\partial u^{4}}\right)_{u=1}$ where $H_{1}$ , $H_{2}$ $H_{3}$ , and $H_{4}$ are known constants.

Thus,

$\displaystyle t_{g}=\bar{Y}(1+e_{0})(1+e_{1}H_{1}+e_{1}^{2}H_{2}+e_{1}^{3}H_{3% }+e_{1}^{4}H_{4})=\bar{Y}(1+e_{1}H_{1}+e_{1}^{2}H_{2}+e_{1}^{3}H_{3}+e_{1}^{4}% H_{4}+e_{0}+e_{1}e_{0}H_{1}+e_{1}^{2}e_{0}H_{2}+e_{1}^{3}e_{0}H_{3}+e_{1}^{4}e% _{0}H_{4})$ (11)

Retaining first three terms in $t_{g}$ , we have to $O(1/n)$

$\displaystyle E(t_{g})=\bar{Y}[1+\theta(H_{2}C_{x}^{2}+H_{1}C_{xy})].$ (12)

where $C_{x}^{2}$ and $C_{xy}$ are coefficient of variation and coefficient of co-variation between $x$ and $y$ in the population.

To $O(1/n)$ ,

$\displaystyle\text{Bias}(t_{g})=B(t_{g})=\bar{Y}\theta[H_{2}C_{x}^{2}+H_{1}C_{% xy}]$ (13) $\displaystyle V(t_{g})=\bar{Y}^{2}\theta(H_{1}^{2}C^{2}_{x}+2H_{1}C_{xy}+C^{2}% _{y})$ (14)

Following Tin (1965), we subtract the estimate of the first order bias from the estimator $t_{g}$ itself to get an almost unbiased estimator in the form

$\displaystyle t_{gT}=t_{g}-\hat{B}(t_{g})=\bar{y}H(u)[1-\theta(H_{2}c_{x}^{2}+% H_{1}c_{xy})],$ (15)

where $c_{x}^{2}=\frac{s_{x}^{2}}{\bar{x}^{2}}$ and $c_{xy}=\frac{s_{xy}}{\bar{x}\bar{y}}$ are consistent estimators of $C_{x}^{2}$ and $C_{xy}$ respectively.

Earlier, Beale (1962) proposed an innovative technique, which is in ratio form, given in generalized form as

$\displaystyle t_{gB}=\bar{y}H(u)\left[\frac{1-H_{1}\theta\frac{s_{xy}}{\bar{x}% \bar{y}}}{1+H_{2}\theta\frac{s_{x}^{2}}{\bar{x}^{2}}}\right]$ (16)

Define the bivariate moments in finite population set up

$\displaystyle C_{ij}=\frac{\frac{1}{N}\sum\limits_{ij}(X_{i}-\bar{X})^{i}(Y_{i% }-\bar{Y})^{j}}{\bar{X}^{i}\bar{Y}^{j}}$

Thus, assuming $N-1\cong N$ for large $N$ , $C_{x}^{2}=C_{20}$ , $C_{y}^{2}=C_{02}$ , and $C_{xy}=C_{11}$ .

In the derivations that follow the results of bivariate moments given by Tin (1965) and De-Graft Johnson (1969) are used.

For large $N$ ,

$\displaystyle E(e_{1}^{2}e_{0}^{2})=\frac{1}{n^{2}}(2C_{11}^{2}+C_{20}C_{02}),% \quad E(e_{1}^{4})=\frac{3C_{20}^{2}}{n^{2}},\quad E(e_{1}^{3})=\frac{C_{30}}{% n^{2}},\quad E(e_{1}^{3}e_{0})=\frac{3C_{20}C_{11}}{n^{2}},$ $\displaystyle E(e_{1}e_{2})=\frac{1}{n}\frac{C_{21}}{C_{11}},\quad E(e_{1}e_{3% })=\frac{1}{n}\frac{C_{30}}{C_{20}},\quad E(e_{0}e_{2})=\frac{1}{n}\frac{C_{12% }}{C_{11}}.$

where $s_{xy}=S_{xy}(1+e_{2})$ and $s_{x}^{2}=S_{x}^{2}(1+e_{3})$ where $E(e_{2})=E(e_{3})=$ 0.

Assuming $|e_{0}|<$ 1, $|e_{1}|<$ 1, $|e_{2}|<$ 1, and $|e_{3}|<$ 1 for all possible samples and using traditional techniques (see Sukhatme et al., 1984) and keeping terms up to $O(1/n^{2})$ and assuming $(N-1)\cong N$ we find

$\displaystyle E(t_{g})=\bar{Y}+\bar{Y}\theta(H_{2}C_{20}+H_{1}C_{11})+\bar{Y}% \theta^{2}[H_{3}C_{30}+3H_{4}C_{20}^{2}+H_{2}C_{21}+3H_{3}C_{20}C_{11}]$ (17) $\displaystyle V(t_{g})=\bar{Y}^{2}\theta(H_{1}^{2}C_{20}+2H_{1}C_{11}+C_{02})+% \bar{Y}^{2}\theta^{2}\{C_{20}^{2}(2H_{2}^{2}+6H_{1}H_{3})+C_{11}^{2}(H_{1}^{2}% +4H_{2})+C_{20}C_{02}(H_{1}^{2}+2H_{2})+C_{30}(2H_{1}H_{2})\}+\bar{Y}^{2}% \theta^{2}\{C_{21}(2H_{1}^{2}+2H_{2})+C_{20}C_{11}(6H_{3}+10H_{1}H_{2})+2H_{1}% C_{12}\}$ (18)

Tin type generalized almost unbiased ratio estimator:

$\displaystyle\!\!\!\!\!\!\!\!\!\!E(t_{gT})=\bar{Y}\!+\!\bar{Y}\theta^{2}[C_{30% }(H_{3}\!+\!2H_{2}\!-\!H_{1}H_{2})\!+\!C_{21}(\!-\!H_{1}^{2}\!+\!H_{1})]\!+% \bar{Y}\theta^{2}[C_{20}^{2}(3H_{4}\!-\!4H_{2}\!+\!2H_{1}H_{2})\!+\!C_{20}C_{1% 1}(H_{3}\!-\!H_{1}H_{2}\!+\!2H_{2}\!-\!H_{1})].$ (19) $\displaystyle\!\!\!\!\!\!\!\!\!\!V(t_{gT})=\bar{Y}^{2}\theta(H_{1}^{2}C_{20}\!% +\!2H_{1}C_{11}\!+\!C_{02})\!+\!\bar{Y}^{2}\theta^{2}\{C_{11}^{2}(4H_{2}\!+\!2% H_{1}\!-\!H_{1}^{2})\!+\!H_{1}^{2}C_{20}C_{02}\}\!+\bar{Y}^{2}\theta^{2}\{C_{2% 0}^{2}(6H_{1}H_{3}\!+\!2H_{2}\!+\!4H_{1}H_{2}\!-\!2H_{1}^{2}H_{2})\!+\!C_{20}C% _{11}(\!-\!2H_{1}^{3}\!+\!4H_{1}^{2}\!+\!4H_{2}\!+\!4H_{1}H_{2}\!+\!10H_{3})\}$ (20)

Beale type generalized almost unbiased ratio estimator:

$\displaystyle\!\!\!\!\!\!\!\!\!\!E(t_{gB})=\bar{Y}\!+\!\bar{Y}\theta^{2}\{C_{3% 0}(3H_{2}-1)\!+\!C_{20}^{2}(3-6H_{2}\!+\!H_{2}^{2})\!+\!C_{20}C_{11}(3H_{2}\!+% \!H_{1}H_{2})\!+\!C_{21}(1\!+\!2H_{1}-H_{2})\}$ (21) $\displaystyle\!\!\!\!\!\!\!\!\!\!V(t_{gB})=\bar{Y}^{2}\theta(H_{1}^{2}C_{20}\!% +\!2H_{1}C_{11}\!+\!C_{02})\!+\!\bar{Y}^{2}\theta^{2}\{C_{11}^{2}(4H_{2}\!+\!2% H_{1}-H_{1}^{2})\!+\!H_{1}^{2}C_{20}C_{02}\}\!+\bar{Y}^{2}\theta^{2}\{C_{20}^{% 2}(6H_{4}\!+\!6H_{1}H_{3}\!+\!6H_{2}\!+\!8H_{1}H_{2}-2H_{1}^{2}H_{2}-6)\}\!+% \bar{Y}^{2}\theta^{2}\{C_{20}C_{11}(-2H_{1}^{3}\!+\!4H_{1}^{2}H_{2}-2H_{1}-2H_% {2}-2H_{1}H_{2}\!+\!12H_{3})$ (22)

Comparison of Beale type and Tin type Generalizes almost unbiased estimators:

$\displaystyle V(t_{gB})<V(t_{gT})\text{ if }\frac{\beta}{R}>\frac{6H_{4}+4H_{2% }+4H_{1}H_{2}-6}{6H_{2}-2H_{3}+6H_{1}H_{2}+2H_{1}+4H_{1}^{2}-4H_{1}^{2}H_{2}},$

provided, $(6H_{2}-2H_{3}+6H_{1}H_{2}+2H_{1}+4H_{1}^{2}-4H_{1}^{2}H_{2})\neq$ 0.

4. Some special classes of almost unbiased ratio-type estimators

Srivastava (1967) proposed a class of ratio-type estmators given by

$\displaystyle t_{s}=\bar{y}\left(\frac{\bar{X}}{\bar{x}}\right)^{\alpha},$

where $\alpha$ is a non-zero real number.

Here $H(u)=\left(\frac{\bar{X}}{\bar{x}}\right)^{\alpha}=u^{-\alpha}$ .

Thus, $H_{1}=-\alpha$ , $H_{2}=\frac{\alpha(\alpha+1)}{2}$ , $H_{3}=-\frac{\alpha(\alpha+1)(\alpha+2)}{6}$ and $H_{4}=\frac{\alpha(\alpha+1)(\alpha+2)(\alpha+3)}{24}$ .

Walsh (1970) suggested another class of estimators given by $t_{w}=\bar{y}\frac{\bar{X}}{\alpha\bar{x}+(1-\alpha)\bar{X}}$ , where $\alpha$ is a non-zero real number.

Here $H(u)=\frac{\bar{X}}{\alpha\bar{x}+(1-\alpha)\bar{X}}$ .

Thus, $H_{1}=-\alpha$ , $H_{2}=\alpha^{2}$ , $H_{3}=-\alpha^{3}$ and $H_{4}=\alpha^{4}$ .

Reddy (1974) suggested a class of estimators, given by

$\displaystyle t_{re}=\frac{\bar{y}\bar{X}}{\bar{x}+\alpha(\bar{x}-\bar{X})}.$

Here $H_{1}=-(\alpha+1)$ , $H_{2}=(\alpha+1)^{2}$ , $H_{3}=-(\alpha+1)^{3}$ , and $H_{4}=(\alpha+1)^{4}$ .

Khoshnevisan et al. (2007) following Srivastava (1967), Walsh (1970) and Reddy (1974) suggested a general family of estimators for estimating population mean of the study variable using some known functions of polulation parameter(s) of the auxiliary variable. This family of ratio-type estmators is given by

$\displaystyle t_{k}=\bar{y}\left(\frac{A\bar{X}+B}{\alpha(A\bar{x}+B)+(1-% \alpha)(A\bar{X}+B)}\right)^{g}.$

Here $H(u)=\left(\frac{A\bar{X}+B}{\alpha(A\bar{x}+B)+(1-\alpha)(A\bar{X}+B)}\right)% ^{g}$ .

Here $H_{1}=-\lambda\alpha g$ , $H_{2}=(\lambda\alpha)^{2}\frac{g(g+1)}{2}$ , $H_{3}=-\frac{(\lambda\alpha)^{3}g(g+1)(g+2)}{6}$ , and $H_{4}=\frac{(\lambda\alpha)^{4}g(g+1)(g+2)(g+3)}{24}$ , where $\lambda=\frac{A\bar{X}}{A\bar{X}+B}$ , $\alpha$ and $g$ are non-zero real numbers, and $A$ and $B$ are known constants or known values of population parameters of the auxiliary variable $x$ , such as standard deviation of $x$ , coefficient of variation of $x$ , correlation coefficient between $x$ and $y$ , coefficient of skewness of $x$ , coefficient of kurtosis of $x$ , minimum and maximum values of $x$ , etc.

Note: The expected values and variances of $t_{s}$ , $t_{w}$ , $t_{re}$ and $t_{k}$ to $O(1/n^{2})$ can be obtained by substituting corresponding $H_{1}$ , $H_{2}$ , $H_{3}$ , and $H_{4}$ in Eqs (17) and (18) respectively. The expected values of Tin-type and Beale-type almost unbiased ratio-type estimators to $O(1/n^{2})$ can be obtained by substituting $H_{1}$ , $H_{2}$ , $H_{3}$ , and $H_{4}$ in Eqs (19) and (21) respectively and for variances in Eqs (20) and (22) respectively.

5. Some special cases of almost unbiased ratio type estimators under simple random sampling

5.1 Almost unbiased classical ratio estimator

$\displaystyle t_{r}=\frac{\bar{y}\bar{X}}{\bar{x}},t_{rT}=\frac{\bar{y}\bar{X}% }{\bar{x}}\left[1-\theta\left(\frac{s_{x}^{2}}{\bar{x}^{2}}-\frac{s_{xy}}{\bar% {y}\bar{x}}\right)\right],t_{rB}=\frac{\bar{y}\bar{X}}{\bar{x}}\frac{\left(1+% \theta\frac{s_{xy}}{\bar{y}\bar{x}}\right)}{\left(1+\theta\frac{s_{x}^{2}}{% \bar{x}^{2}}\right)},H(u)=\frac{1}{u}=\frac{\bar{X}}{\bar{x}}.$

$H_{1}=$ $-$ 1, $H_{2}=$ 1, $H_{3}=$ $-$ 1, and $H_{4}=$ 1.

Substituting the values of $H_{1},H_{2},H_{3}$ , and $H_{4}$ in Eqs (17)–(22), we have to $O(1/n^{2})$

$\displaystyle\!\!\!\!\!\!\!\!E(t_{r})=\bar{Y}+\bar{Y}\theta(C_{20}-C_{11})+% \bar{Y}\theta^{2}[(C_{21}-C_{30})+3C_{20}(C_{20}-C_{11})]$ $\displaystyle\!\!\!\!\!\!\!\!V(t_{r})=\bar{Y}^{2}\theta^{2}(C_{20}-2C_{11}+C_{% 02})+\bar{Y}^{2}\theta^{2}(5C_{11}^{2}+3C_{20}C_{02}+4C_{21}-4C_{20}^{2}-2C_{3% 0}-16C_{20}C_{11}-2C_{12})$ $\displaystyle\!\!\!\!\!\!\!\!E(t_{rT})=\bar{Y}-\bar{Y}\theta^{2}[2(C_{21}-C_{3% 0})+3C_{20}(C_{20}-C_{11})]$ $\displaystyle\!\!\!\!\!\!\!\!V(t_{rT})=\bar{Y}^{2}\theta(C_{20}-2C_{11}+C_{02}% )+\bar{Y}^{2}\theta^{2}(C_{11}^{2}+C_{20}C_{02}+2C_{20}^{2}-4C_{20}C_{11})$ $\displaystyle\!\!\!\!\!\!\!\!E(t_{rB})=\bar{Y}-\bar{Y}\theta^{2}[2(C_{21}-C_{3% 0})+2C_{20}(C_{20}-C_{11})]$ $\displaystyle\!\!\!\!\!\!\!\!V(t_{rB})=V(t_{rT})$

These reslts are due to Tin (1965) and De-Graft Johnson (1969).

5.2 Almost unbiased ratio-type exponential estimator

Bahl-Tuteja (1991) proposed a ratio type exponential estimator as

$\displaystyle t_{\exp}=\bar{y}\exp\left(\frac{\bar{X}-\bar{x}}{\bar{X}+\bar{x}% }\right)$

Here, $H(u)=e^{\frac{1-u}{1+u}}$ , $H_{1}=-\frac{1}{2}$ , $H_{2}=\frac{3}{8}$ , $H_{3}=-\frac{13}{48}$ , and $H_{4}=\frac{73}{384}$ .

$\displaystyle t_{\exp T}=\bar{y}e^{\frac{\bar{X}-\bar{x}}{\bar{X}+\bar{x}}}% \left[1-\theta\left(\frac{3}{8}\frac{s_{x}^{2}}{\bar{x}^{2}}-\frac{1}{2}\frac{% s_{xy}}{\bar{y}\bar{x}}\right)\right],t_{\exp B}=\bar{y}e^{\frac{\bar{X}-\bar{% x}}{\bar{X}+\bar{x}}}\frac{\left(1+\frac{1}{2}\theta\frac{s_{xy}}{\bar{y}\bar{% x}}\right)}{\left(1+\frac{3}{8}\theta\frac{s_{x}^{2}}{\bar{x}^{2}}\right)}$

Substituting $H_{1},H_{2},H_{3},H_{4}$ in relevant expressions in Eqs (17)–(22), the following results are obtained.

The expected value and variance of Bahl-Tuteja ratio-type exponential estimator to $O(1/n^{2})$ as

$\displaystyle\!\!\!\!\!\!\!\!\!\!E(t_{\exp})=\bar{Y}+\bar{Y}\theta\left(\frac{% 3}{8}C_{30}\!-\!\frac{1}{2}C_{11}\right)+\bar{Y}\theta^{2}\left(\!-\!\frac{13}% {48}C_{30}+\frac{3}{8}C_{21}+\frac{73}{128}C_{20}^{2}\!-\!\frac{13}{16}C_{20}C% _{11}\right)$ $\displaystyle\!\!\!\!\!\!\!\!\!\!V(t_{\exp})=\bar{Y}^{2}\theta\left(\frac{1}{4% }C_{20}\!-\!C_{11}+C_{02}\right)+\bar{Y}^{2}\theta^{2}\left(\frac{35}{32}C_{20% }^{2}+\frac{7}{4}C_{11}^{2}+C_{20}C_{02}\!-\!\frac{3}{8}C_{30}+\frac{5}{2}C_{2% 1}\!-\!\frac{7}{2}C_{20}C_{11}\!-\!C_{12}\right)$

Tin type almost unbiased estimator for Bahl-Tuteja estimator

$\displaystyle E(t_{\exp T})=\bar{Y}+\bar{Y}\theta^{2}\left(\frac{2}{3}C_{30}-% \frac{3}{4}C_{21}+\frac{167}{128}C_{20}^{2}+\frac{7}{6}C_{20}C_{11}\right)$ $\displaystyle V(t_{\exp T})=\bar{Y}^{2}\theta\left(\frac{1}{4}C_{20}-C_{11}+C_% {02}\right)+\bar{Y}^{2}\theta^{2}\left(\frac{5}{8}C_{20}^{2}+\frac{1}{4}C_{11}% ^{2}+\frac{1}{4}C_{20}C_{02}-\frac{17}{24}C_{20}C_{11}\right)$

Beale type almost unbiased estimator for Bahl-Tuteja estimator

$\displaystyle E(t_{\exp B})=\bar{Y}+\bar{Y}\theta^{2}\left(\frac{1}{8}C_{30}-% \frac{3}{8}C_{21}+\frac{57}{64}C_{20}^{2}+\frac{15}{16}C_{20}C_{11}\right)$ $\displaystyle V(t_{\exp B})=\bar{Y}^{2}\theta\left(\frac{1}{4}C_{20}-C_{11}+C_% {02}\right)+\bar{Y}^{2}\theta^{2}\left(-\frac{223}{64}C_{20}^{2}+\frac{1}{4}C_% {11}^{2}+\frac{1}{4}C_{20}C_{02}-2C_{20}C_{11}\right)$

5.3 Almost unbiased square root transformation ratio estimators

Swain (2014) proposed a square root transformation estimator $t_{\textit{sqr}}$ given by

$\displaystyle t_{\textit{sqr}}=\bar{y}\left(\frac{\bar{X}}{\bar{x}}\right)^{1/% 2},$

which can be put in Srivstava’s generalized class, where $H(u)=\frac{1}{u^{1/2}}$ . We find $H_{1}=-\frac{1}{2}$ , $H_{2}=\frac{3}{8}$ , $H_{3}=-\frac{5}{16}$ , and $H_{4}=\frac{105}{128}$ .

The Tin type and Beale type almost unbiased square root transformation estimators are respectively

$\displaystyle t_{\textit{sqr}T}=\bar{y}\left(\frac{\bar{X}}{\bar{x}}\right)^{1% /2}\left[1-\theta\left(\frac{3}{8}\frac{s_{x}^{2}}{\bar{x}^{2}}-\frac{1}{2}% \frac{s_{xy}}{\bar{y}\bar{x}}\right)\right],t_{\textit{sqr}B}=\bar{y}\left(% \frac{\bar{X}}{\bar{x}}\right)^{1/2}\frac{\left(1+\frac{1}{2}\theta\frac{s_{xy% }}{\bar{y}\bar{x}}\right)}{\left(1+\frac{3}{8}\theta\frac{s_{x}^{2}}{\bar{x}^{% 2}}\right)}$

Substituting $H_{1},H_{2},H_{3},H_{4}$ in relevant expressions in Eqs (17)–(22), the following results are obtained.

The expressions for the expected value and variance of square root estimator to $O(1/n^{2})$ are given below

$\displaystyle\!\!\!\!\!\!\!\!\!\!E(t_{\textit{sqr}})=\bar{Y}+\bar{Y}\theta% \left(\frac{3}{8}C_{20}\!-\!\frac{1}{2}C_{11}\right)+\bar{Y}\theta^{2}\left(\!% -\!\frac{5}{16}C_{30}+\frac{3}{8}C_{21}+\frac{105}{128}C_{20}^{2}\!-\!\frac{15% }{16}C_{20}C_{11}\right)$ $\displaystyle\!\!\!\!\!\!\!\!\!\!V(t_{\textit{sqr}})=\bar{Y}^{2}\theta\left(% \frac{1}{4}C_{20}\!-\!C_{11}+C_{02}\right)+\bar{Y}^{2}\theta^{2}\left(\frac{39% }{32}C_{20}^{2}+\frac{7}{4}C_{11}^{2}+C_{20}C_{02}\!-\!\frac{3}{8}C_{30}+\frac% {5}{4}C_{21}\!-\!\frac{15}{4}C_{20}C_{11}\!-\!C_{12}\right)$

Tin type almost unbiased square root estimator

$\displaystyle E(t_{\textit{sqr}T})=\bar{Y}+\bar{Y}\theta^{2}\left[\left(\frac{% 5}{8}C_{30}-\frac{3}{4}C_{21}\right)+\left(-\frac{135}{128}C_{20}^{2}+\frac{9}% {8}C_{20}C_{11}\right)\right]$ $\displaystyle V(t_{\textit{sqr}T})=\bar{Y}^{2}\theta\left(\frac{1}{4}C_{20}-C_% {11}+C_{02}\right)+\bar{Y}^{2}\theta^{2}\left(\frac{3}{4}C_{20}^{2}+\frac{1}{4% }C_{11}^{2}+\frac{1}{4}C_{20}C_{02}-\frac{9}{8}C_{20}C_{11}\right)$

Beale type almost unbiased square root estimator

$\displaystyle E(t_{\textit{sqr}B})=\bar{Y}+\bar{Y}\theta^{2}\left[\left(\frac{% 1}{8}C_{30}-\frac{3}{8}C_{21}\right)+\left(\frac{57}{64}C_{20}^{2}+\frac{15}{1% 6}C_{20}C_{11}\right)\right]$ $\displaystyle V(t_{\textit{sqr}B})=\bar{Y}^{2}\theta\left(\frac{1}{4}C_{20}-C_% {11}+C_{02}\right)+\bar{Y}^{2}\theta^{2}\left(-\frac{183}{64}C_{20}^{2}+\frac{% 1}{4}C_{11}^{2}+\frac{1}{4}C_{20}C_{02}-\frac{33}{8}C_{20}C_{11}\right)$

Table 1
Comparison of almost unbiased estimators of ratio type exponential estimator

	B( $t_{\exp B}$ )	B( $t_{\exp T}$ )	V( $t_{\exp B}$ )	V( $t_{\exp T}$ )
Pop-1
$\theta=$ 1/10	0.000451522	$-$ 5.1199864E $-$ 05	0.145409	0.150557
$\theta=$ 1/20	0.00011288	$-$ 1.2799966E $-$ 05	0.073556	0.074843
$\theta=$ 1/50	1.80609E $-$ 05	$-$ 2.04799E $-$ 06	0.029627	0.029832
$\theta=$ 1/100	4.51522E $-$ 06	$-$ 5.11999E $-$ 07	0.014847	0.014899
Pop-2
$\theta=$ 1/10	0.00155	1.6076003E $-$ 03	0.15518	0.160936
$\theta=$ 1/20	0.000387	4.0190008E $-$ 04	0.078593	0.080032
$\theta=$ 1/50	6.2E $-$ 05	6.4304E $-$ 05	0.031678	0.031908
$\theta=$ 1/100	1.55E $-$ 05	1.6076E $-$ 05	0.015879	0.015937
Pop-3
$\theta=$ 1/10	0.018963	2.9385920E $-$ 02	0.014743	0.094776
$\theta=$ 1/20	0.004741	7.3464801E $-$ 03	0.02539	0.045398
$\theta=$ 1/50	0.000759	0.001175437	0.014481	0.017682
$\theta=$ 1/100	0.00019	0.000293859	0.007961	0.008761
Pop-5
$\theta=$ 1/10	0.000498	5.0625480E $-$ 04	0.166505	0.169192
$\theta=$ 1/20	0.000124	1.2656370E $-$ 04	0.08363	0.084302
$\theta=$ 1/50	1.99E $-$ 05	2.02502E $-$ 05	0.033543	0.03365
$\theta=$ 1/100	4.98E $-$ 06	5.06255E $-$ 06	0.016786	0.016813

Comments: In case of ratio type exponential estimator Beale type estimator is more efficient than Tin type estimator for all populations under consideration, Also, Beale type estimator is less biased than Tin type estimator, except for population 1 where Tin type estimator is marginally less biased than Beale type estimator.

Comparison of almost unbiased ratio-type exponential estimator and square root transformation ratio estimator in infinite populations when $x$ and y follow a bivariate normal distribution.

To $O(1/n^{2})$ ,

(1)

$t_{\textit{sqr}}$ will be less biased than $t_{\exp}$ if $\left|\frac{105}{128}-\frac{15}{16}\frac{\beta}{R}\right|<\left|\frac{73}{128}% -\frac{13}{16}\frac{\beta}{R}\right|$ .

(2)

$t_{\textit{sqr}}$ is more efficient than $t_{\exp}$ if $\frac{C_{11}}{C_{20}}>\frac{1}{2}$ .

That is, if $k>$ 1/2 or $\rho>\frac{1}{2}\sqrt{\frac{C_{20}}{C_{02}}}$ .

(3)

$t_{\exp B}$ will be less biased than $t_{\exp T}$ if $\left|\frac{57}{64}+\frac{15}{16}\frac{\beta}{R}\right|<\left|\frac{167}{128}+% \frac{7}{6}\frac{\beta}{R}\right|$ .

(4)

$t_{\exp B}$ is more efficient than $t_{\exp T}$ if $\frac{263}{64}+\frac{31}{24}\frac{\beta}{R}>$ 0, which is always true.

(5)

$t_{\textit{sqr}B}$ will be less biased than $t_{\textit{sqr}T}$ if $\left|\frac{57}{64}+\frac{15}{16}\frac{\beta}{R}\right|<\left|\frac{135}{128}+% \frac{11}{8}\frac{\beta}{R}\right|$ .

(6)

$t_{\textit{sqr}B}$ is more efficient than $t_{\textit{sqr}T}$ if $\frac{231}{64}+\frac{3}{4}\frac{\beta}{R}>$ 0, which is always true.

Note: $\frac{\beta}{R}$ is positive because $y$ and $x$ are positively correlated. Assume that $x$ and $y$ are positively measured.

5.3.1 Numerical illustrations

Consider four bivariate natural populations (1, 2, 3, and 5), each of size $N=$ 3164, with relevant parameters given by De-Graft Johnson (1969). The Beale type and Tin type estimators to $O(1/n^{2})$ are compared for Bahl-Tuteja’s estimator and Swain’s Square root transformation ratio estimator (Tables 1 and 2). Computations are carried out omitting the constant multipliers.

Table 2
Comparison of almost unbiased ratio type Square root transformation estimator

	B( $t_{\textit{sqr}B}$ )	B( $t_{\textit{sqr}T}$ )	V( $t_{\textit{sqr}B}$ )	V( $t_{\textit{sqr}T}$ )
Pop-1
$\theta=$ 1/10	0.00045152	$-$ 0.00179427	0.14606143	0.14988051
$\theta=$ 1/20	0.00011288	$-$ 0.00044857	0.07371873	0.07467350
$\theta=$ 1/50	0.00001806	$-$ 0.00007177	0.02965262	0.02980538
$\theta=$ 1/100	0.00000452	$-$ 0.00001794	0.01485383	0.01489202
Pop-2
$\theta=$ 1/10	0.00154964	$-$ 0.00048017	0.15592935	0.16030793
$\theta=$ 1/20	0.00038741	$-$ 0.00012004	0.07878053	0.07987517
$\theta=$ 1/50	0.00006199	$-$ 0.00001921	0.03170801	0.03188316
$\theta=$ 1/100	0.00001550	$-$ 0.00000480	0.01588664	0.01593043
Pop-3
$\theta=$ 1/10	0.01896277	$-$ 0.00493000	0.02561605	0.08892160
$\theta=$ 1/20	0.00474069	$-$ 0.00123250	0.02810839	0.04393478
$\theta=$ 1/50	0.00075851	$-$ 0.00019720	0.01491544	0.01744766
$\theta=$ 1/100	0.00018963	$-$ 0.00004930	0.00806974	0.00870279
Pop-5
$\theta=$ 1/10	0.00049767	$-$ 0.00041688	0.16684732	0.16884761
$\theta=$ 1/20	0.00012442	$-$ 0.00010422	0.08371552	0.08421559
$\theta=$ 1/50	0.00001991	$-$ 0.00001668	0.03355625	0.03363626
$\theta=$ 1/100	0.00000498	$-$ 0.00000417	0.01678980	0.01680980

Comments: In case of square root transformation estimator Beale type estimator is more efficient than Tin type estimator for all populations, but marginally more biased than Tin type estimator except for population 1.

6. Conclusions

Tin type and Beale type generalized class of almost unbiased ratio-type estimators have been derived using Srivastava’s (1971) generalized class of estimators. As special cases, Bahl-Tuteja’s and Swain’s square root transformation estimators are compared with regard to bias and efficiency. Under bivariate normality, $t_{\textit{sqr}}$ is more efficient than $t_{\exp}$ if $k>$ 1/2, but conditionally less biased than $t_{\exp}$ . In case of both these estimators Beale’s estimator is more efficient than Tin’s estimator to $O(1/n^{2})$ and conditionally less biased than Tin’s estimator. In contrast Beale’s estimator is less biased than Tin’s estimator to $O(1/n^{2})$ and of equal efficiency to that of Tin’s estimator to the same order in case of simple ratio estimator. Numerical illustrations show that Beal type estimator performs well compared to Tin type estimator for both Bahl-Tuteja and Swain’s square root transformation ratio estimators.

Footnotes

Acknowledgments

The author wishes to express sincere thanks to Professor Sarjinder Singh, Texas A & M University, USA for going through the manuscript and rendering considered opinion and comments.

References

Al-Jaraha

(2008). Unbiased Ratio Estimation for Finite Populations. Unpublished doctoral dissertation, Colorado State University, Fort Collins, CO, USA.

Bahl

, & Tuteja

R.K.

(1991). Ratio and product type exponential estimators. Jour Information and Optimization Sciences, 12, 159-164.

Beale

E.M.L.

(1962). Some uses of computers in operational research. Industrialle Organization, 31, 27-28.

Biradar

R.S.

, & Singh

H.P.

(1992). On a class of almost unbiased ratio estimators. Biom J, 34(8), 937-944.

Cochran

W.G.

(1940). The estimation of the yields of cereal experiments by sampling for the ratio of grains to total produce. Joun of Ag Sc, 30, 262-275.

Cochran

W.G.

(1977). Sampling Techniques. John Wiley and Sons (3rd ed), New York.

Chakrabarty

R.P.

(1968). Contribution to Theory of Ratio Type Estimators. Ph.D. thesis submitted to the Texas A and M University, USA.

David

I.P.

(1971). Contributions to Ratio Method of Estimation. Ph.D. dissertation submitted at Iowa University, Ames, IA, USA.

De-Graft Johnson

K.T.

(1969). Some Contributions to Theory of Two Phase Sampling. A dissertation submitted for the degree of Doctor of Philosophy, Iowa Stae University, IA, USA.

10.

Durbin

(1959). A note on the application Quennouille’s method of bias reduction in estimation of ratios. Biometrika, 46, 477-480.

11.

Goodman

L.A.

, & Hartley

H.O.

(1958). The precision of unbiased ratio estimators. Journ Amer Stat Assoc, 53, 491-508.

12.

Gupta

, & Shabbir

(2007). Use of transformed auxiliary variables in estimating population mean. Journ Statistical Planning and Inference, 137(5), 1606-1611.

13.

Gupta

, & Shabbir

(2008). On improvement in estimating population mean in simple random sampling. Journ Appl Stat, 35(5), 559-566.

14.

Hartley

H.O.

, & Ross

(1959). Unbiased ratio estimators. Nature, 174, 270-271.

15.

Huchinson

M.C.

(1971). A Monte Carlo comparison of some ratio estimates. Biometrika, 58(2), 313-321.

16.

Kadilar

, & Cingi

(2004). New ratio estimators using correlation coefficient. Intwer stat, 1-11.

17.

Kadilar

, & Cingi

(2006). Improvement in estimating population mean in simple random sampling. Applied Math Letters, 19(1), 75-79.

18.

Khoshnevisan

Singh

Chauhan

Swan

, & Smarandache

(2007). A general family estimators for estimating population mean using some known value of some population parameter(s). Far Eastern Journal of Theoretical Statistics, 22, 181-191.

19.

Koop

J.C.

(1951). A note on the bias of the ratio estimate. Bull Inter Stat Inst, 33, 141-146.

20.

Lahiri

D.B.

(1951). A method for sample selection providing unbiased ratio estimators. Bull Inter Stat Inst, 33, 133-140.

21.

Midzuno

(1952). On the sampling system with probability proportional to sum of sizes. Ann Inst Stat Math, 3, 99-108.

22.

Mohanty

, & Das

M.N.

(1971). Use of transformation in samping. Journ Ind Soc Agri Stat, XXIII(2), 83-87.

23.

Mohanty

, & Sahoo

(1995). A note on improving ratio method of estimation through linear transformation using certain known population parameters. Sankhya, Series B, 57, 93-102.

24.

Murthy

M.N.

, & Nanjamma

N.S.

(1959). Almost unbiased ratio esimates based on interpenetrating subsamples. Sankhya: The Indian Journal of Statistics, 21, 381-392.

25.

Murthy

M.N.

(1962). Almost unbiased estimators based on inter-penetrating sub-samples. Sankhya Series, 24, 303-314.

26.

Mussa

A.S.

(1999). A new bias-reducing modification of the finite population ratio estimators and a comparison among proposed alternatives. Journ Off Statistics, 15, 25-38.

27.

de Pascual

(1961). Unbiased ratio estimates in stratified random sampling. Journ Amer Statist Assoc, 56, 70-87.

28.

Prabhu-Ajgaonkar

S.G.

(1993). Non-existence of an optimum estimator in a class of ratio estimators. Statistical Papers, 34, 161-165.

29.

Quenouille

M.H.

(1956). Notes on bias in estimation. Biometrika, 43, 353-360.

30.

Rao

J.N.K.

, & Webster

J.T.

(1966). On two methods of Bias reduction in estimation of ratios. Biometrika, 53, 571-577.

31.

Rao

T.J.

, & Swain

A.K.P.C.

(2014). A note on the Hartley-Ross unbiased ratio estimator. Communications in Statistics-Theory and Methods, 43, 3162-3169.

32.

Reddy

V.N.

(1974). On a transformed ratio method of estimation. Sankhya C, 36, 59-70.

33.

Sen

A.R.

(1952). Present status of probability sampling and its use in estimation of farm characteristics (abstract). Econometrica, 27, 103.

34.

Singh

(2003). Advanced Sampling Theory with Applications. Kluwer Academic publishers.

35.

Singh

H.P.

Tailor

Singh

, & Kim

J.M.

(2008). A modified estimator of population mean. Statistical Papers, 49, 37-58.

36.

Singh

H.P.

, & Tailor

(2003). Use of correlation coefficient in estimating finite population mean. Statistics in Transition, 6(4) 555-560.

37.

Singh

, & Singh

(1991). Almost bias precipitate: A new technique. Aligarh Journ of Statistics, 11, 5-8.

38.

Sisodia

B.V.S.

, & Dwivedi

V.K.

(1981). A modified ratio estimator using coefficient of variation of auxiliary variable. Jour Ind Soc Agri Stat, 33, 13-18.

39.

Srivastava

V.K.

Dwivedi

T.D.

Chaubey

Y.P.

, & Bhatnagar

(1983). Finite sample properties of Beale’s ratio estimator. Commun Statist – Theory and Methods, 12(15), 1795-1805.

40.

Sukhatme

P.V.

Sukhatme

B.V.

, & Asok

(1984). Sampling Theory of Surveys with Applications, 3

{}^{\text{rd}}

edition, Iowa State University Press, Ames, IA, USA.

41.

Swain

A.K.P.C.

(2014). On an improved ratio type estimator of finite population mean in sample surveys. Revista Investigacion Operacional, 35, 49-57.

42.

Walsh

J.E.

(1970). Generalization of ratio estimator for population total. Sankhya A, 32, 99-106.

43.

Williams

W.H.

(1961). Generating unbiased ratio and regression estimators. Biometrics, 17(2), 267-274.

44.

Tin

(1965). Comparison of some ratio estimators. J Amer Stat Assoc, 60, 294-302.

45.

Upadhyaya

L.N.

, & Singh

H.P.

(1984). On the estimation of of population mean with known coefficient variation. Biom Journ, 26(8), 915-922.

46.

Upadhyaya

L.N.

, & Singh

H.P.

(1999). Use of transformed auxiliary variable in estimating finite population mean. Biom Journ, 41(5), 627-636.

On a generalized class of almost unbiased ratio estimators

Abstract

Keywords

1. Introduction

2. Ratio estimator its modifications

5. Some special cases of almost unbiased ratio type estimators under simple random sampling

5.1 Almost unbiased classical ratio estimator

5.2 Almost unbiased ratio-type exponential estimator

5.3 Almost unbiased square root transformation ratio estimators

Table 1 Comparison of almost unbiased estimators of ratio type exponential estimator

Table 2 Comparison of almost unbiased ratio type Square root transformation estimator

Footnotes

Acknowledgments

References

Table 1
Comparison of almost unbiased estimators of ratio type exponential estimator

Table 2
Comparison of almost unbiased ratio type Square root transformation estimator