Sensitivity analysis of censoring schemes in progressively type-II right censored order statistics

Abstract

We study the sensitivity of some optimality criteria based on progressively type-II right censored order statistics scheme changes and explain how the sensitivity analysis helps to find the optimal censoring schemes. We find that determining an optimal censoring plan among a class of one-step censoring schemes is not always recommended. We consider optimality criteria as the model output of a sensitivity analysis problem and quantify how this model depends on its input factor and censoring scheme, using local and global sensitivity methods. Finally, we propose a simple method to find the optimal scheme among all possible censoring schemes.

Appendix

For the purpose of providing explicit estimators, we use the method purposed by Balakrishnan [1]. We present an approximated maximum likelihood estimation. The likelihood function based on the progressively censored sample upon denoting $Z^{{\mathcal {R}}}_{i:m:n}={\left( X^{{\mathcal {R}}}_{i:m:n}-\mu \right) }/{\sigma }$, can be written down as

$$\begin{aligned} \displaystyle L={\sigma }^{-m}C\prod ^m_{i=1} {f\left( z_{i:m:n}\right) {\left\{ 1 - F \left( z_{i:m:n}\right) \right\} }^{R_i}} \end{aligned}$$

for $z_{1:m:n}<\cdots <z_{m:m:n}$, where f(z) and F(z) denote the density and distribution functions of a standard normal random variable. Now by realizing that $f^{'}(z)=-zf(z)$ we obtain the likelihood equations for $\mu$ and $\sigma$ as

$$\begin{aligned} \displaystyle \frac{\partial {\ln (L)}}{\partial \mu }= \frac{1}{\sigma }\left( \sum ^m_{i=1}{z_{i:m:n}}+ \sum ^m_{i=1}{R_i\frac{f\left( z_{i:m:n}\right) }{1-F \left( z_{i:m:n}\right) }} \right) =0 \end{aligned}$$

(9)

and

$$\begin{aligned} \displaystyle \frac{\partial {\ln (L)}}{\partial \sigma }= -\frac{1}{\sigma }\left( m-\sum ^m_{i=1}{z^2_{i:m:n}}- \sum ^m_{i=1}{R_iz_{i:m:n} \frac{f\left( z_{i:m:n}\right) }{1-F\left( z_{i:m:n}\right) }} \right) =0. \end{aligned}$$

(10)

The likelihood equations (9) and (10) do not admit explicit solutions except in the complete sample case. But by expanding ${f\left( z_{i:m:n}\right) }/{\left\{ 1 - F\left( z_{i:m:n}\right) \right\} }$ in a Taylor series around the points ${\zeta }_{i:m:n}=F^{-1}\left( p_{i:m:n} \right)$, we can approximate

$$\begin{aligned} \displaystyle \frac{f\left( z_{i:m:n}\right) }{1-F\left( z_{i:m:n}\right) } \simeq {\gamma }_i+{\delta }_iz_{i:m:n}, \end{aligned}$$

(11)

where

$$\begin{aligned}&\displaystyle p_{r:m:n}=1-\prod ^m_{i=m-r+1}{\frac{i+R_{m-i+1}+ \cdots +R_m}{i+1+R_{m-i+1}+\cdots +R_m}}, \\&\quad \displaystyle \gamma _i=\frac{f\left( {\zeta }_{i:m:n}\right) }{1-p_{i:m:n}} \left\{ 1-\frac{{\zeta }_{i:m:n}f\left( {\zeta }_{i:m:n}\right) }{1-p_{i:m:n}}+ {\zeta }^2_{i:m:n}\right\} \end{aligned}$$

and

$$\begin{aligned} \displaystyle \delta _i=\frac{f\left( {\zeta }_{i:m:n}\right) }{{\left( 1-p_{i:m:n}\right) }^2} \left\{ f\left( {\zeta }_{i:m:n}\right) -{\zeta }_{i:m:n}\left( 1-p_{i:m:n}\right) \right\} \end{aligned}$$

for $i=1,2,\ldots , m$.

It is easy to see ${\delta }_i>0$ whenever ${\zeta }_{i:m:n}\le 0$, $i=1,2, \ldots , m$. Also when $0<{\zeta }_{i:m:n}<h_2$

$$\begin{aligned}&\displaystyle \left| {\zeta }_{i:m:n}\left( 1-p_{i:m:n}\right) \right| = {\zeta }_{i:m:n}\left\{ 1-F\left( {\zeta }_{i:m:n}\right) \right\} = {\zeta }_{i:m:n}\int ^{h_2}_{{\zeta }_{i:m:n}}{f(t)}dt\\&\quad \le \int ^{h_2}_{{\zeta }_{i:m:n}}{xf(t)}dt= f\left( {\zeta }_{i:m:n}\right) - f\left( h_2\right) \le f\left( {\zeta }_{i:m:n}\right) \end{aligned}$$

and consequently ${\delta }_i>0$, $i=1,2,\ldots , m$.

By using (11), (9) and (10), we obtain

$$\begin{aligned} \displaystyle \frac{\partial {\ln (L)}}{\partial \mu }\simeq \frac{\partial {\ln \left( L^*\right) }}{\partial \mu }= \frac{1}{\sigma }\left\{ \sum ^m_{i=1}{R_i{\gamma }_i}+ \sum ^m_{i=1}{\left( 1+{R_i\delta }_i\right) z_{i:m:n}}\right\} =0 \end{aligned}$$

(12)

and

$$\begin{aligned} \displaystyle \frac{\partial {\ln (L)}}{\partial \sigma }\simeq \frac{\partial {\ln \left( L^*\right) }}{\partial \sigma }= -\frac{1}{\sigma } \left( m-\sum ^m_{i=1}{R_i{\gamma }_iz_{i:m:n}}- \sum ^m_{i=1}{\left( 1+{R_i\delta }_i\right) z^2_{i:m:n}}\right) =0. \end{aligned}$$

(13)

Upon solving (12) and (13), we derive the approximate maximum likelihood estimators of $\mu$ and $\sigma$ as

$$\begin{aligned} \displaystyle {{\widehat{\mu }}}^*=\frac{{{\widehat{\sigma }}}^* \sum ^m_{i=1}{R_i{\gamma }_i}}{\sum ^m_{i=1}{\left( 1+{R_i\delta }_i\right) }}+ \frac{\sum ^m_{i=1}{\left( 1+{R_i\delta }_i\right) X^{{\mathcal {R}}}_{i:m:n}}}{\sum ^m_{i=1}{\left( 1+{R_i\delta }_i\right) }} \end{aligned}$$

(14)

and

$$\begin{aligned} \displaystyle {{\widehat{\sigma }}}^*=\frac{-B+\sqrt{B^2-4AC}}{2A}, \end{aligned}$$

where

$$\begin{aligned}&\displaystyle a = m\sum ^m_{i=1}{\left( 1+{R_i\delta }_i\right) }, \\&\displaystyle B=\left\{ \sum ^m_{i=1}{R_i{\gamma }_i}\right\} \left\{ \sum ^m_{i=1}{\left( 1+{R_i\delta }_i\right) X^{{\mathcal {R}}}_{i:m:n}}\right\} - \left\{ \sum ^m_{i=1}{R_i{\gamma }_i X^{{\mathcal {R}}}_{i:m:n}}\right\} \left\{ \sum ^m_{i=1}{\left( 1+{R_i\delta }_i\right) }\right\} \end{aligned}$$

and

$$\begin{aligned} \displaystyle C=\left\{ \sum ^m_{i=1}{\left( 1+{R_i\delta }_i\right) X^{{\mathcal {R}}}_{i:m:n}}\right\} ^2 - \left\{ \sum ^m_{i=1}{\left( 1+{R_i\delta }_i\right) }\right\} \left\{ \sum ^m_{i=1}{\left( 1+{R_i\delta }_i\right) X_{i:m:n}^{2{\mathcal {R}}}}\right\} . \end{aligned}$$

It is important to mention here that upon solving (13) we obtain a quadratic equation in $\sigma$ which has two roots. However, one of them drops out.

The conditional bias of ${{\widehat{\mu }}}^*$ can be computed exactly from (14). But it is difficult to determine the conditional bias of ${{\widehat{\sigma }}}^*$ exactly. However, the conditional bias of ${{\widehat{\sigma }}}^*$ may be evaluated approximately by

$$\begin{aligned} \displaystyle {E\left\{ \frac{\partial {\ln \left( L^*\right) }}{\partial \sigma }\right\} }/ {E\left\{ -\frac{{\partial }^2{\ln \left( L^*\right) }}{\partial {\sigma }^2}\right\} }; \end{aligned}$$

see Balakrishnan [1].

In addition, we obtain the approximated Fisher information from (12) and (13) as

$$\begin{aligned} \displaystyle I=\left[ \begin{array}{ll} \displaystyle I_{11}&{}I_{12} \\ \displaystyle I_{12}&{}I_{22} \end{array} \right] , \end{aligned}$$

where

$$\begin{aligned} \displaystyle I_{12}=E\left\{ -\frac{{\partial }^2{\ln \left( L^*\right) }}{\partial \mu \partial \sigma }\right\} = V_1\left\{ \frac{\sum ^m_{i=1}{\left( 1+{R_i\delta }_i\right) }}{{\sigma }^2}\right\} = V_1E\left\{ -\frac{{\partial }^2{\ln \left( L^*\right) }}{\partial {\mu }^2}\right\} = V_1 I_{11} \end{aligned}$$

and

$$\begin{aligned} \displaystyle I_{22} = E\left\{ -\frac{{\partial }^2{\ln \left( L^*\right) }}{\partial {\sigma }^2}\right\} = V_2\left\{ \frac{\sum ^m_{i=1}{\left( 1+{R_i\delta }_i\right) }}{{\sigma }^2}\right\} = V_2E\left\{ -\frac{{\partial }^2{\ln \left( L^*\right) }}{\partial {\mu }^2}\right\} = V_2 I_{11}, \end{aligned}$$

where

$$\begin{aligned} \displaystyle V_1=\frac{\sum ^m_{i=1}{R_i{\gamma }_i}}{\sum ^m_{i=1}{ \left( 1+{R_i\delta }_i\right) }}+ 2\left[ \frac{\sum ^m_{i=1}{\left( 1+{R_i\delta }_i\right) E\left( Z_{i:m:n}\right) }}{\sum ^m_{i=1}{\left( 1+{R_i\delta }_i\right) }}\right] \end{aligned}$$

and

$$\begin{aligned} \displaystyle V_2=3\left[ \frac{\sum ^m_{i=1}{\left( 1+{R_i\delta }_i\right) E\left( Z^2_{i:m:n}\right) }}{\sum ^m_{i=1}{\left( 1+{R_i\delta }_i\right) }}\right] + 2\left[ \frac{\sum ^m_{i=1}{R_i{\gamma }_iE \left( Z_{i:m:n}\right) }}{\sum ^m_{i=1}{\left( 1+{R_i\delta }_i \right) }}\right] - \frac{m}{\sum ^m_{i=1}{ \left( 1+{R_i\delta }_i\right) }}. \end{aligned}$$

It follows

$$\begin{aligned} \displaystyle Var \left( {{\widehat{\mu }}}^*\right)&= {} \frac{{\sigma }^2}{\sum ^m_{i=1}{\left( 1+{R_i\delta }_i\right) }} \left\{ \frac{V_2}{V_2-V^2_1} \right\} , \\ \displaystyle Var \left( {{\widehat{\sigma }}}^*\right)&= {} \frac{{\sigma }^2}{\sum ^m_{i=1}{\left( 1+{R_i\delta }_i\right) }} \left\{ \frac{1}{V_2-V^2_1} \right\} \end{aligned}$$

and

$$\begin{aligned} \displaystyle Cov \left( {{\widehat{\mu }}}^*, {{\widehat{\sigma }}}^*\right) = -\frac{{\sigma }^2}{\sum ^m_{i=1} {\left( 1+{R_i\delta }_i \right) }} \left\{ \frac{V_1}{V_2-V^2_1}\right\} . \end{aligned}$$