Corrigendum

This article refers to:

Simultaneous estimation of means of two sensitive variables

Article title: Simultaneous estimation of means of two sensitive variables

Authors: Zawar Hussain and Maryam Murtaza

Journal: Communications in Statistics—Theory and Methods

DOI: 10.1080/03610926.2018.1481981

Volume issue: 47 (2):324–343

There are two purposes of this corrigendum on the above mentioned article; to provide corrected expressions of ${\sigma }_{z_2}^2$, ${\sigma }_{Z_1{Z}_2}$, $E(S_1^3),$ and $E(S_2^3)$ in Section 2, Theorem 2.3, at page 328, and to provide corrected numerical results of relative efficiency given in Table 2.2.1 (see Table 1 in supplementary material) at page 331 of the of the above mentioned article.

On page 328, in the proof of Theorem 2.3, authors used the following expressions. (1) $$E\left(S_1^3\right)={\gamma }_{30}-3{\theta }_1{\gamma }_{20}+{\theta }_1^3$$(1) (2) $$E\left(S_2^3\right)={\gamma }_{03}-3{\theta }_2{\gamma }_{02}+{\theta }_2^3$$(2)

Though, it is straightforward to derive the above expectations, Ahmed, Sedory, and Singh (2018), perhaps, committed this error by chance. These expressions in Equations (1)(1) $$E\left(S_1^3\right)={\gamma }_{30}-3{\theta }_1{\gamma }_{20}+{\theta }_1^3$$(1) and (2)(2) $$E\left(S_2^3\right)={\gamma }_{03}-3{\theta }_2{\gamma }_{02}+{\theta }_2^3$$(2) should be corrected as (3) $$E\left(S_1^3\right)={\gamma }_{30}+3{\theta }_1{\gamma }_{20}+{\theta }_1^3$$(3) (4) $$E\left(S_2^3\right)={\gamma }_{03}+3{\theta }_2{\gamma }_{02}+{\theta }_2^3$$(4)

But again, in Section 2, Theorem 2.3, at page 328, using Equations (1)(1) $$E\left(S_1^3\right)={\gamma }_{30}-3{\theta }_1{\gamma }_{20}+{\theta }_1^3$$(1) and (2)(2) $$E\left(S_2^3\right)={\gamma }_{03}-3{\theta }_2{\gamma }_{02}+{\theta }_2^3$$(2) , the authors claimed that

${\sigma }_{z_2}^2=({\sigma }_{y_1}^2+{\mu }_{y_1}^2)[P({\gamma }_{40}+4{\gamma }_{30}{\theta }_1+6{\gamma }_{20}{\theta }_2^2+{\theta }_1^4)+(1-P)({\gamma }_{20}+{\theta }_1^2)({\gamma }_{02}+{\theta }_2^2)]$$+({\sigma }_{y_2}^2+{\mu }_{y_2}^2)[(1-P)({\gamma }_{04}+4{\gamma }_{03}{\theta }_2+6{\gamma }_{02}{\theta }_2^2+{\theta }_2^4)+P({\gamma }_{20}+{\theta }_1^2)({\gamma }_{02}+{\theta }_2^2)]+2({\sigma }_{y_1{y}_2}+{\mu }_{y_1}{\mu }_{y_2})[P{\theta }_2({\gamma }_{30}-3{\theta }_1{\gamma }_{20}+{\theta }_1^3)+(1-P){\theta }_1({\gamma }_{03}-3{\theta }_2{\gamma }_{02}+{\theta }_2^3)]-{[{\mu }_{y_1}\{P({\gamma }_{20}+{\theta }_1^2)+(1-P){\theta }_1{\theta }_2\}+{\mu }_{y_2}\{P{\theta }_1{\theta }_2+(1-P)({\gamma }_{02}+{\theta }_2^2)\}]}^2\text{(5)}$

and (6) $$\begin{array}{c}{\sigma }_{z_1{z}_2}=({\sigma }_{y_1}^2+{\mu }_{y_1}^2)\left\{P\right({\gamma }_{30}-3{\theta }_2{\gamma }_{02}+{\theta }_2^3)+(1-P\left){\theta }_2\right({\gamma }_{20}+{\theta }_1^2\left)\right\}\\ +({\sigma }_{y_2}^2+{\mu }_{y_2}^2)\left\{P{\theta }_1\right({\gamma }_{02}+{\theta }_2^2)+(1-P\left)\right({\gamma }_{03}-3{\theta }_2{\gamma }_{02}+{\theta }_2^3\left)\right\}\\ +2({\sigma }_{y_1{y}_2}+{\mu }_{y_1}{\mu }_{y_2})\left\{P{\theta }_2\right({\gamma }_{20}+{\theta }_1^2)+(1-P\left){\theta }_1\right({\gamma }_{02}+{\theta }_2^2\left)\right\}\\ -({\theta }_1{\mu }_{y_1}+{\theta }_2{\mu }_{y_2})[{\mu }_{y_1}\{P({\gamma }_{20}+{\theta }_1^2)+(1-P){\theta }_1{\theta }_2\}+{\mu }_{y_2}\{P{\theta }_1{\theta }_2+(1-P)({\gamma }_{02}+{\theta }_2^2)\}]\end{array}$$(6)

Again, the above expressions Equations (5)(7) $$\begin{array}{c}{\sigma }_{z_2}^2=({\sigma }_{y_1}^2+{\mu }_{y_1}^2)\left[P\right({\gamma }_{40}+4{\gamma }_{30}{\theta }_1+6{\gamma }_{20}{\theta }_2^2+{\theta }_1^4)+(1-P\left)\right({\gamma }_{20}+{\theta }_1^2\left)\right({\gamma }_{02}+{\theta }_2^2\left)\right]\\ +({\sigma }_{y_2}^2+{\mu }_{y_2}^2)\left[\right(1-P\left)\right({\gamma }_{04}+4{\gamma }_{03}{\theta }_2+6{\gamma }_{02}{\theta }_2^2+{\theta }_2^4)+P({\gamma }_{20}+{\theta }_1^2\left)\right({\gamma }_{02}+{\theta }_2^2\left)\right]\\ +2({\sigma }_{y_1{y}_2}+{\mu }_{y_1}{\mu }_{y_2})\left[P{\theta }_2\right({\gamma }_{30}+3{\theta }_1{\gamma }_{20}+{\theta }_1^3)+(1-P\left){\theta }_1\right({\gamma }_{03}+3{\theta }_2{\gamma }_{02}+{\theta }_2^3\left)\right]\\ -{\left[{\mu }_{y_1}\right\{P({\gamma }_{20}+{\theta }_1^2)+(1-P){\theta }_1{\theta }_2\}+{\mu }_{y_2}\{P{\theta }_1{\theta }_2+(1-P)({\gamma }_{02}+{\theta }_2^2)\left\}\right]}^2\end{array}$$(7) and (6)(7) $$\begin{array}{c}{\sigma }_{z_2}^2=({\sigma }_{y_1}^2+{\mu }_{y_1}^2)\left[P\right({\gamma }_{40}+4{\gamma }_{30}{\theta }_1+6{\gamma }_{20}{\theta }_2^2+{\theta }_1^4)+(1-P\left)\right({\gamma }_{20}+{\theta }_1^2\left)\right({\gamma }_{02}+{\theta }_2^2\left)\right]\\ +({\sigma }_{y_2}^2+{\mu }_{y_2}^2)\left[\right(1-P\left)\right({\gamma }_{04}+4{\gamma }_{03}{\theta }_2+6{\gamma }_{02}{\theta }_2^2+{\theta }_2^4)+P({\gamma }_{20}+{\theta }_1^2\left)\right({\gamma }_{02}+{\theta }_2^2\left)\right]\\ +2({\sigma }_{y_1{y}_2}+{\mu }_{y_1}{\mu }_{y_2})\left[P{\theta }_2\right({\gamma }_{30}+3{\theta }_1{\gamma }_{20}+{\theta }_1^3)+(1-P\left){\theta }_1\right({\gamma }_{03}+3{\theta }_2{\gamma }_{02}+{\theta }_2^3\left)\right]\\ -{\left[{\mu }_{y_1}\right\{P({\gamma }_{20}+{\theta }_1^2)+(1-P){\theta }_1{\theta }_2\}+{\mu }_{y_2}\{P{\theta }_1{\theta }_2+(1-P)({\gamma }_{02}+{\theta }_2^2)\left\}\right]}^2\end{array}$$(7) are incorrect and should be corrected as: (7) $$\begin{array}{c}{\sigma }_{z_2}^2=({\sigma }_{y_1}^2+{\mu }_{y_1}^2)\left[P\right({\gamma }_{40}+4{\gamma }_{30}{\theta }_1+6{\gamma }_{20}{\theta }_2^2+{\theta }_1^4)+(1-P\left)\right({\gamma }_{20}+{\theta }_1^2\left)\right({\gamma }_{02}+{\theta }_2^2\left)\right]\\ +({\sigma }_{y_2}^2+{\mu }_{y_2}^2)\left[\right(1-P\left)\right({\gamma }_{04}+4{\gamma }_{03}{\theta }_2+6{\gamma }_{02}{\theta }_2^2+{\theta }_2^4)+P({\gamma }_{20}+{\theta }_1^2\left)\right({\gamma }_{02}+{\theta }_2^2\left)\right]\\ +2({\sigma }_{y_1{y}_2}+{\mu }_{y_1}{\mu }_{y_2})\left[P{\theta }_2\right({\gamma }_{30}+3{\theta }_1{\gamma }_{20}+{\theta }_1^3)+(1-P\left){\theta }_1\right({\gamma }_{03}+3{\theta }_2{\gamma }_{02}+{\theta }_2^3\left)\right]\\ -{\left[{\mu }_{y_1}\right\{P({\gamma }_{20}+{\theta }_1^2)+(1-P){\theta }_1{\theta }_2\}+{\mu }_{y_2}\{P{\theta }_1{\theta }_2+(1-P)({\gamma }_{02}+{\theta }_2^2)\left\}\right]}^2\end{array}$$(7) (8) $$\begin{array}{c}{\sigma }_{z_1{z}_2}=({\sigma }_{y_1}^2+{\mu }_{y_1}^2)\left\{P\right({\gamma }_{30}+3{\theta }_2{\gamma }_{02}+{\theta }_2^3)+(1-P\left){\theta }_2\right({\gamma }_{20}+{\theta }_1^2\left)\right\}\\ +({\sigma }_{y_2}^2+{\mu }_{y_2}^2)\left\{P{\theta }_1\right({\gamma }_{02}+{\theta }_2^2)+(1-P\left)\right({\gamma }_{03}+3{\theta }_2{\gamma }_{02}+{\theta }_2^3\left)\right\}\\ +2({\sigma }_{y_1{y}_2}+{\mu }_{y_1}{\mu }_{y_2})\left\{P{\theta }_2\right({\gamma }_{20}+{\theta }_1^2)+(1-P\left){\theta }_1\right({\gamma }_{02}+{\theta }_2^2\left)\right\}\\ -({\theta }_1{\mu }_{y_1}+{\theta }_2{\mu }_{y_2})[{\mu }_{y_1}\{P({\gamma }_{20}+{\theta }_1^2)+(1-P){\theta }_1{\theta }_2\}+{\mu }_{y_2}\{P{\theta }_1{\theta }_2+(1-P)({\gamma }_{02}+{\theta }_2^2)\}]\end{array}$$(8)

The simple error could have been ignored by taking it a typographical mistake but at page 331, using Equations (1)(1) $$E\left(S_1^3\right)={\gamma }_{30}-3{\theta }_1{\gamma }_{20}+{\theta }_1^3$$(1) , (2)(2) $$E\left(S_2^3\right)={\gamma }_{03}-3{\theta }_2{\gamma }_{02}+{\theta }_2^3$$(2) , (5)(6) $$\begin{array}{c}{\sigma }_{z_1{z}_2}=({\sigma }_{y_1}^2+{\mu }_{y_1}^2)\left\{P\right({\gamma }_{30}-3{\theta }_2{\gamma }_{02}+{\theta }_2^3)+(1-P\left){\theta }_2\right({\gamma }_{20}+{\theta }_1^2\left)\right\}\\ +({\sigma }_{y_2}^2+{\mu }_{y_2}^2)\left\{P{\theta }_1\right({\gamma }_{02}+{\theta }_2^2)+(1-P\left)\right({\gamma }_{03}-3{\theta }_2{\gamma }_{02}+{\theta }_2^3\left)\right\}\\ +2({\sigma }_{y_1{y}_2}+{\mu }_{y_1}{\mu }_{y_2})\left\{P{\theta }_2\right({\gamma }_{20}+{\theta }_1^2)+(1-P\left){\theta }_1\right({\gamma }_{02}+{\theta }_2^2\left)\right\}\\ -({\theta }_1{\mu }_{y_1}+{\theta }_2{\mu }_{y_2})[{\mu }_{y_1}\{P({\gamma }_{20}+{\theta }_1^2)+(1-P){\theta }_1{\theta }_2\}+{\mu }_{y_2}\{P{\theta }_1{\theta }_2+(1-P)({\gamma }_{02}+{\theta }_2^2)\}]\end{array}$$(6) , and (6)(6) $$\begin{array}{c}{\sigma }_{z_1{z}_2}=({\sigma }_{y_1}^2+{\mu }_{y_1}^2)\left\{P\right({\gamma }_{30}-3{\theta }_2{\gamma }_{02}+{\theta }_2^3)+(1-P\left){\theta }_2\right({\gamma }_{20}+{\theta }_1^2\left)\right\}\\ +({\sigma }_{y_2}^2+{\mu }_{y_2}^2)\left\{P{\theta }_1\right({\gamma }_{02}+{\theta }_2^2)+(1-P\left)\right({\gamma }_{03}-3{\theta }_2{\gamma }_{02}+{\theta }_2^3\left)\right\}\\ +2({\sigma }_{y_1{y}_2}+{\mu }_{y_1}{\mu }_{y_2})\left\{P{\theta }_2\right({\gamma }_{20}+{\theta }_1^2)+(1-P\left){\theta }_1\right({\gamma }_{02}+{\theta }_2^2\left)\right\}\\ -({\theta }_1{\mu }_{y_1}+{\theta }_2{\mu }_{y_2})[{\mu }_{y_1}\{P({\gamma }_{20}+{\theta }_1^2)+(1-P){\theta }_1{\theta }_2\}+{\mu }_{y_2}\{P{\theta }_1{\theta }_2+(1-P)({\gamma }_{02}+{\theta }_2^2)\}]\end{array}$$(6) , relative efficiency results for different choices of parameters are provided in the form of Table 2.2.1 (see Table 1 in supplementary files) using the SAS codes given at pages 338–343, by committing the same error. This needs correction. The corrected results on relative efficiency, using Equations (3)(3) $$E\left(S_1^3\right)={\gamma }_{30}+3{\theta }_1{\gamma }_{20}+{\theta }_1^3$$(3) , (4)(4) $$E\left(S_2^3\right)={\gamma }_{03}+3{\theta }_2{\gamma }_{02}+{\theta }_2^3$$(4) , (7)(7) $$\begin{array}{c}{\sigma }_{z_2}^2=({\sigma }_{y_1}^2+{\mu }_{y_1}^2)\left[P\right({\gamma }_{40}+4{\gamma }_{30}{\theta }_1+6{\gamma }_{20}{\theta }_2^2+{\theta }_1^4)+(1-P\left)\right({\gamma }_{20}+{\theta }_1^2\left)\right({\gamma }_{02}+{\theta }_2^2\left)\right]\\ +({\sigma }_{y_2}^2+{\mu }_{y_2}^2)\left[\right(1-P\left)\right({\gamma }_{04}+4{\gamma }_{03}{\theta }_2+6{\gamma }_{02}{\theta }_2^2+{\theta }_2^4)+P({\gamma }_{20}+{\theta }_1^2\left)\right({\gamma }_{02}+{\theta }_2^2\left)\right]\\ +2({\sigma }_{y_1{y}_2}+{\mu }_{y_1}{\mu }_{y_2})\left[P{\theta }_2\right({\gamma }_{30}+3{\theta }_1{\gamma }_{20}+{\theta }_1^3)+(1-P\left){\theta }_1\right({\gamma }_{03}+3{\theta }_2{\gamma }_{02}+{\theta }_2^3\left)\right]\\ -{\left[{\mu }_{y_1}\right\{P({\gamma }_{20}+{\theta }_1^2)+(1-P){\theta }_1{\theta }_2\}+{\mu }_{y_2}\{P{\theta }_1{\theta }_2+(1-P)({\gamma }_{02}+{\theta }_2^2)\left\}\right]}^2\end{array}$$(7) , and (8)(8) $$\begin{array}{c}{\sigma }_{z_1{z}_2}=({\sigma }_{y_1}^2+{\mu }_{y_1}^2)\left\{P\right({\gamma }_{30}+3{\theta }_2{\gamma }_{02}+{\theta }_2^3)+(1-P\left){\theta }_2\right({\gamma }_{20}+{\theta }_1^2\left)\right\}\\ +({\sigma }_{y_2}^2+{\mu }_{y_2}^2)\left\{P{\theta }_1\right({\gamma }_{02}+{\theta }_2^2)+(1-P\left)\right({\gamma }_{03}+3{\theta }_2{\gamma }_{02}+{\theta }_2^3\left)\right\}\\ +2({\sigma }_{y_1{y}_2}+{\mu }_{y_1}{\mu }_{y_2})\left\{P{\theta }_2\right({\gamma }_{20}+{\theta }_1^2)+(1-P\left){\theta }_1\right({\gamma }_{02}+{\theta }_2^2\left)\right\}\\ -({\theta }_1{\mu }_{y_1}+{\theta }_2{\mu }_{y_2})[{\mu }_{y_1}\{P({\gamma }_{20}+{\theta }_1^2)+(1-P){\theta }_1{\theta }_2\}+{\mu }_{y_2}\{P{\theta }_1{\theta }_2+(1-P)({\gamma }_{02}+{\theta }_2^2)\}]\end{array}$$(8) are available in table provided as supplementary file.

Same errors are committed in Section 3 of the abovementioned article when dealing with stratified random sampling. These errors must be removed by following the corrections as made in Equations (3)(3) $$E\left(S_1^3\right)={\gamma }_{30}+3{\theta }_1{\gamma }_{20}+{\theta }_1^3$$(3) , (4)(4) $$E\left(S_2^3\right)={\gamma }_{03}+3{\theta }_2{\gamma }_{02}+{\theta }_2^3$$(4) , (7)(7) $$\begin{array}{c}{\sigma }_{z_2}^2=({\sigma }_{y_1}^2+{\mu }_{y_1}^2)\left[P\right({\gamma }_{40}+4{\gamma }_{30}{\theta }_1+6{\gamma }_{20}{\theta }_2^2+{\theta }_1^4)+(1-P\left)\right({\gamma }_{20}+{\theta }_1^2\left)\right({\gamma }_{02}+{\theta }_2^2\left)\right]\\ +({\sigma }_{y_2}^2+{\mu }_{y_2}^2)\left[\right(1-P\left)\right({\gamma }_{04}+4{\gamma }_{03}{\theta }_2+6{\gamma }_{02}{\theta }_2^2+{\theta }_2^4)+P({\gamma }_{20}+{\theta }_1^2\left)\right({\gamma }_{02}+{\theta }_2^2\left)\right]\\ +2({\sigma }_{y_1{y}_2}+{\mu }_{y_1}{\mu }_{y_2})\left[P{\theta }_2\right({\gamma }_{30}+3{\theta }_1{\gamma }_{20}+{\theta }_1^3)+(1-P\left){\theta }_1\right({\gamma }_{03}+3{\theta }_2{\gamma }_{02}+{\theta }_2^3\left)\right]\\ -{\left[{\mu }_{y_1}\right\{P({\gamma }_{20}+{\theta }_1^2)+(1-P){\theta }_1{\theta }_2\}+{\mu }_{y_2}\{P{\theta }_1{\theta }_2+(1-P)({\gamma }_{02}+{\theta }_2^2)\left\}\right]}^2\end{array}$$(7) , and (8)(8) $$\begin{array}{c}{\sigma }_{z_1{z}_2}=({\sigma }_{y_1}^2+{\mu }_{y_1}^2)\left\{P\right({\gamma }_{30}+3{\theta }_2{\gamma }_{02}+{\theta }_2^3)+(1-P\left){\theta }_2\right({\gamma }_{20}+{\theta }_1^2\left)\right\}\\ +({\sigma }_{y_2}^2+{\mu }_{y_2}^2)\left\{P{\theta }_1\right({\gamma }_{02}+{\theta }_2^2)+(1-P\left)\right({\gamma }_{03}+3{\theta }_2{\gamma }_{02}+{\theta }_2^3\left)\right\}\\ +2({\sigma }_{y_1{y}_2}+{\mu }_{y_1}{\mu }_{y_2})\left\{P{\theta }_2\right({\gamma }_{20}+{\theta }_1^2)+(1-P\left){\theta }_1\right({\gamma }_{02}+{\theta }_2^2\left)\right\}\\ -({\theta }_1{\mu }_{y_1}+{\theta }_2{\mu }_{y_2})[{\mu }_{y_1}\{P({\gamma }_{20}+{\theta }_1^2)+(1-P){\theta }_1{\theta }_2\}+{\mu }_{y_2}\{P{\theta }_1{\theta }_2+(1-P)({\gamma }_{02}+{\theta }_2^2)\}]\end{array}$$(8) .