Type II Censored Samples in Experimental Design Under Jones and Faddy’s Skew t Distribution

Abstract

In this study, parameters of the one-way analysis of variance (ANOVA) model based on the type II censored samples are estimated when the distribution of the error terms is Jones and Faddy’s skew t. In the estimation procedure, maximum likelihood (ML) methodology is used. Solutions of the likelihood equations are obtained using an iterative method because of the nonlinear terms existing in the likelihood equations. A modified version of the ML methodology is also used. It gives the explicit estimators of the model parameters and has no computational difficulties. A new test statistic is proposed for testing the equality of the treatment effects. An extensive Monte Carlo simulation study is carried out to compare the powers of the test statistics based on the ML and the modified ML estimators of the parameters. Real data are analyzed at the end of the study to illustrate the implementation of the proposed methodologies.

Appendices

Appendix A

1.1 A1: Elements of the Hessian Matrix H

$$\begin{aligned} \dfrac{\partial ^2 \ln L}{\partial \mu ^2} & =-\dfrac{(a+0.5)}{\sigma ^2}\sum _{i=1}^{c}\sum _{j=r_{1i}+1}^{n-r_{2i}}\dfrac{v[3z_{ij}(v+z_{ij}^2)^{1/2}+(v+3z_{ij}^2)]}{[(v+z_{ij}^2)^{3/2}+z_{ij}(v+z_{ij}^2)]^2} \\& \quad +\dfrac{(b+0.5)}{\sigma ^2}\sum _{i=1}^{c}\sum _{j=r_{1i}+1}^{n-r_{2i}}\dfrac{v[3z_{ij}(v+z_{ij}^2)^{1/2}-(v+3z_{ij}^2)]}{[(v+z_{ij}^2)^{3/2}-z_{ij}(v+z_{ij}^2)]^2} \\& \quad + \sum _{i=1}^{c}\dfrac{r_{1i}}{\sigma ^2}\dfrac{f^{'}(z_{ir_{1i}+1})F(z_{ir_{1i}+1})-f(z_{ir_{1i}+1})^2}{F(z_{ir_{1i}+1})^2} \\& \quad -\sum _{i=1}^{c}\dfrac{r_{2i}}{\sigma ^2}\dfrac{f^{'}(z_{in_{i}-r_{2i}})(1-F(z_{in_{i}-r_{2i}}))-f(z_{in_{i}-r_{2i}})^2}{(1-F(z_{in_{i}-r_{2i}}))^2}, \end{aligned}$$

$$\begin{aligned} \dfrac{\partial ^2 \ln L}{\partial \tau _{i}^2} & =-\dfrac{(a+0.5)}{\sigma ^2}\sum _{j=r_{1i}+1}^{n-r_{2i}}\dfrac{v[3z_{ij}(v+z_{ij}^2)^{1/2}+(v+3z_{ij}^2)]}{[(v+z_{ij}^2)^{3/2}+z_{ij}(v+z_{ij}^2)]^2} \\& \quad +\dfrac{(b+0.5)}{\sigma ^2}\sum _{j=r_{1i}+1}^{n-r_{2i}}\dfrac{v[3z_{ij}(v+z_{ij}^2)^{1/2}-(v+3z_{ij}^2)]}{[(v+z_{ij}^2)^{3/2}-z_{ij}(v+z_{ij}^2)]^2}\\& \quad +\dfrac{r_{1i}}{\sigma ^2}\dfrac{f^{'}(z_{ir_{1i}+1})F(z_{ir_{1i}+1})-f(z_{ir_{1i}+1})^2}{F(z_{ir_{1i}+1})^2} \\& \quad - \dfrac{r_{2i}}{\sigma ^2}\dfrac{f^{'}(z_{in_{i}-r_{2i}})(1-F(z_{in_{i}-r_{2i}}))-f(z_{in_{i}-r_{2i}})^2}{(1-F(z_{in_{i}-r_{2i}}))^2}, \end{aligned}$$

$$\begin{aligned} \dfrac{\partial ^2 \ln L}{\partial \sigma ^2} & =\sum _{i=1}^{c}\dfrac{n_{i}-r_{1i}-r_{2i}}{\sigma ^2}+\dfrac{a+0.5}{\sigma ^2}\sum _{i=1}^{c}\sum _{i=r_{1i}+1}^{n_{i}-r_{2i}}z_{ij}\dfrac{v}{(v+z_{ij}^2)^{3/2}+z_{ij}(v+z_{ij}^2)} \\& \quad -\dfrac{(a+0.5)}{\sigma ^2}\sum _{i=1}^{c}\sum _{j=r_{1i}+1}^{n-r_{2i}}z_{ij}^2\dfrac{v[3z_{ij}(v+z_{ij}^2)^{1/2}+(v+3z_{ij}^2)]}{[(v+z_{ij}^2)^{3/2}+z_{ij}(v+z_{ij}^2)]^2} \\& \quad -\dfrac{b+0.5}{\sigma ^2}\sum _{i=1}^{c}\sum _{i=r_{1i}+1}^{n_{i}-r_{2i}}z_{ij}\dfrac{v}{(v+z_{ij}^2)^{3/2}-z_{ij}(v+z_{ij}^2)} \\& \quad +\dfrac{(b+0.5)}{\sigma ^2}\sum _{i=1}^{c}\sum _{j=r_{1i}+1}^{n-r_{2i}}z_{ij}^2\dfrac{v[3z_{ij}(v+z_{ij}^2)^{1/2}-(v+3z_{ij}^2)]}{[(v+z_{ij}^2)^{3/2}-z_{ij}(v+z_{ij}^2)]^2} \\& \quad + \sum _{i=1}^{c}\dfrac{r_{1i}}{\sigma ^2}z_{ir_{1i}+1}\left[ \dfrac{{f(z_{ir_{1i}+1}})}{F(z_{ir_{1i}+1})}+\dfrac{f^{'}(z_{ir_{1i}+1})F(z_{ir_{1i}+1})-f(z_{ir_{1i}+1})^2}{F(z_{ir_{1i}+1})^2}\right] \\& \quad - \sum _{i=1}^{c}\dfrac{r_{2i}}{\sigma ^2}z_{in_{i}-r_{2i}}\left[ \dfrac{f(z_{in_{i}-r_{2i})}}{1-F(z_{in_{i}-r_{2i}})}+\dfrac{f^{'}(z_{in_{i}-r_{2i}})(1-F(z_{in_{i}-r_{2i}}))-f(z_{in_{i}-r_{2i}})^2}{(1-F(z_{in_{i}-r_{2i}}))^2}\right] , \end{aligned}$$

$$\begin{aligned} \dfrac{\partial ^2 \ln L}{\partial \mu \partial \tau _{i}} & =-\dfrac{(a+0.5)}{\sigma ^2}\sum _{j=r_{1i}+1}^{n-r_{2i}}z_{ij}\dfrac{v[3z_{ij}(v+z_{ij}^2)^{1/2}+(v+3z_{ij}^2)]}{[(v+z_{ij}^2)^{3/2}+z_{ij}(v+z_{ij}^2)]^2} \\& \quad +\dfrac{(b+0.5)}{\sigma ^2}\sum _{j=r_{1i}+1}^{n-r_{2i}}z_{ij}\dfrac{v[3z_{ij}(v+z_{ij}^2)^{1/2}-(v+3z_{ij}^2)]}{[(v+z_{ij}^2)^{3/2}-z_{ij}(v+z_{ij}^2)]^2} \\& \quad + \dfrac{r_{1i}}{\sigma ^2}\left[ \dfrac{{f(z_{ir_{1i}+1}})}{F(z_{ir_{1i}+1})}+\dfrac{f^{'}(z_{ir_{1i}+1})F(z_{ir_{1i}+1})-f(z_{ir_{1i}+1})^2}{F(z_{ir_{1i}+1})^2}\right] \\& \quad - \dfrac{r_{2i}}{\sigma ^2}\left[ \dfrac{f(z_{in_{i}-r_{2i})}}{1-F(z_{in_{i}-r_{2i}})}+\dfrac{f^{'}(z_{in_{i}-r_{2i}})(1-F(z_{in_{i}-r_{2i}}))-f(z_{in_{i}-r_{2i}})^2}{(1-F(z_{in_{i}-r_{2i}}))^2}\right] , \end{aligned}$$

$$\begin{aligned} \dfrac{\partial ^2 \ln L}{\partial \mu \partial \sigma } & =-\dfrac{(a+0.5)}{\sigma ^2}\sum _{i=1}^{c}\sum _{j=r_{1i}+1}^{n-r_{2i}}z_{ij}\dfrac{v[3z_{ij}(v+z_{ij}^2)^{1/2}+(v+3z_{ij}^2)]}{[(v+z_{ij}^2)^{3/2}+z_{ij}(v+z_{ij}^2)]^2} \\& \quad +\dfrac{(b+0.5)}{\sigma ^2}\sum _{i=1}^{c}\sum _{j=r_{1i}+1}^{n-r_{2i}}z_{ij}\dfrac{v[3z_{ij}(v+z_{ij}^2)^{1/2}-(v+3z_{ij}^2)]}{[(v+z_{ij}^2)^{3/2}-z_{ij}(v+z_{ij}^2)]^2} \\& \quad + \sum _{i=1}^{c}\dfrac{r_{1i}}{\sigma ^2}\left[ \dfrac{{f(z_{ir_{1i}+1}})}{F(z_{ir_{1i}+1})}+\dfrac{f^{'}(z_{ir_{1i}+1})F(z_{ir_{1i}+1})-f(z_{ir_{1i}+1})^2}{F(z_{ir_{1i}+1})^2}\right] \\& \quad - \sum _{i=1}^{c}\dfrac{r_{2i}}{\sigma ^2}\left[ \dfrac{f(z_{in_{i}-r_{2i})}}{1-F(z_{in_{i}-r_{2i}})}+\dfrac{f^{'}(z_{in_{i}-r_{2i}})(1-F(z_{in_{i}-r_{2i}}))-f(z_{in_{i}-r_{2i}})^2}{(1-F(z_{in_{i}-r_{2i}}))^2}\right] \end{aligned}$$

and

$$\begin{aligned} \dfrac{\partial ^2 \ln L}{\partial \tau _{i}\partial \sigma } & =-\dfrac{(a+0.5)}{\sigma ^2}\sum _{j=r_{1i}+1}^{n-r_{2i}}z_{ij}\dfrac{v[3z_{ij}(v+z_{ij}^2)^{1/2}+(v+3z_{ij}^2)]}{[(v+z_{ij}^2)^{3/2}+z_{ij}(v+z_{ij}^2)]^2} \\& \quad + \dfrac{(b+0.5)}{\sigma ^2}\sum _{j=r_{1i}+1}^{n-r_{2i}}z_{ij}\dfrac{v[3z_{ij}(v+z_{ij}^2)^{1/2}-(v+3z_{ij}^2)]}{[(v+z_{ij}^2)^{3/2}-z_{ij}(v+z_{ij}^2)]^2} \\& \quad + \dfrac{r_{1i}}{\sigma ^2}\left[ \dfrac{{f(z_{ir_{1i}+1}})}{F(z_{ir_{1i}+1})}+\dfrac{f^{'}(z_{ir_{1i}+1})F(z_{ir_{1i}+1})-f(z_{ir_{1i}+1})^2}{F(z_{ir_{1i}+1})^2}\right] \\& \quad - \dfrac{r_{2i}}{\sigma ^2}\left[ \dfrac{f(z_{in_{i}-r_{2i})}}{1-F(z_{in_{i}-r_{2i}})}+\dfrac{f^{'}(z_{in_{i}-r_{2i}})(1-F(z_{in_{i}-r_{2i}}))-f(z_{in_{i}-r_{2i}})^2}{(1-F(z_{in_{i}-r_{2i}}))^2}\right] . \end{aligned}$$

1.2 A2: Elements of the Hessian Matrix $H^{\star }$ for the Reparametrized Model

$$\begin{aligned} \dfrac{\partial ^2 \ln L}{\partial \mu _{i}^2} & =-\dfrac{(a+0.5)}{\sigma ^2}\sum _{j=r_{1i}+1}^{n-r_{2i}}\dfrac{v[3z_{ij}(v+z_{ij}^2)^{1/2}+(v+3z_{ij}^2)]}{[(v+z_{ij}^2)^{3/2}+z_{ij}(v+z_{ij}^2)]^2} \\& \quad + \dfrac{(b+0.5)}{\sigma ^2}\sum _{j=r_{1i}+1}^{n-r_{2i}}\dfrac{v[3z_{ij}(v+z_{ij}^2)^{1/2}-(v+3z_{ij}^2)]}{[(v+z_{ij}^2)^{3/2}-z_{ij}(v+z_{ij}^2)]^2} \\& \quad + \dfrac{r_{1i}}{\sigma ^2}\dfrac{f^{'}(z_{ir_{1i}+1})F(z_{ir_{1i}+1})-f(z_{ir_{1i}+1})^2}{F(z_{ir_{1i}+1})^2} \\& \quad - \dfrac{r_{2i}}{\sigma ^2}\dfrac{f^{'}(z_{in_{i}-r_{2i}})(1-F(z_{in_{i}-r_{2i}}))-f(z_{in_{i}-r_{2i}})^2}{(1-F(z_{in_{i}-r_{2i}}))^2}, \end{aligned}$$

$$\begin{aligned} \dfrac{\partial ^2 \ln L}{\partial \mu _{i}\partial \mu _{l}}=0 \quad {\text{for all}} \quad i \ne l \ (i=1, 2, \ldots ,c ; \ l=1, 2, \ldots ,c), \end{aligned}$$

and

$$\begin{aligned} \dfrac{\partial ^2 \ln L}{\partial \mu _{i}\partial \sigma } & =-\dfrac{(a+0.5)}{\sigma ^2}\sum _{j=r_{1i}+1}^{n-r_{2i}}z_{ij}\dfrac{v[3z_{ij}(v+z_{ij}^2)^{1/2}+(v+3z_{ij}^2)]}{[(v+z_{ij}^2)^{3/2}+z_{ij}(v+z_{ij}^2)]^2} \\& \quad + \dfrac{(b+0.5)}{\sigma ^2}\sum _{j=r_{1i}+1}^{n-r_{2i}}z_{ij}\dfrac{v[3z_{ij}(v+z_{ij}^2)^{1/2}-(v+3z_{ij}^2)]}{[(v+z_{ij}^2)^{3/2}-z_{ij}(v+z_{ij}^2)]^2} \\& \quad + \dfrac{r_{1i}}{\sigma ^2}\left[ \dfrac{{f(z_{ir_{1i}+1}})}{F(z_{ir_{1i}+1})}+\dfrac{f^{'}(z_{ir_{1i}+1})F(z_{ir_{1i}+1})-f(z_{ir_{1i}+1})^2}{F(z_{ir_{1i}+1})^2}\right] \\& \quad -\dfrac{r_{2i}}{\sigma ^2}\left[ \dfrac{f(z_{in_{i}-r_{2i})}}{1-F(z_{in_{i}-r_{2i}})}+\dfrac{f^{'}(z_{in_{i}-r_{2i}})(1-F(z_{in_{i}-r_{2i}}))-f(z_{in_{i}-r_{2i}})^2}{(1-F(z_{in_{i}-r_{2i}}))^2}\right] \end{aligned}$$

Appendix B

MML estimators have the following properties:

Lemma 1

For n tends to infinity, $\hat{\mu }$ is normally distributed with mean $\mu$ and variance $\sigma ^2/m$.

Proof

This follows from the fact that the likelihood equation given in (7) and the modified likelihood equation given in (17) are asymptotically equivalent and $\partial \ln L^{*}/\partial \mu$ assumes the form, see Kendall and Stuart (1961):

$$\begin{aligned} \dfrac{\partial \ln L}{\partial \mu }\cong \dfrac{\partial \ln L^*}{\partial \mu } & =\dfrac{m}{\sigma^2}(K+D\hat{\sigma}-\mu )\\ & =\dfrac{m}{\sigma^2} (\hat{\mu}-\mu ). \end{aligned}$$

The normality of $\hat{\mu}$ follows from the fact that $E \left( \frac{\partial ^{r} \ln L^*}{\partial \mu ^{r}}\right) =0$ for all $r \ge 3$, see Bartlett (1953).

Lemma 2

For n tends to infinity, $\hat{\tau}_{i}$ is normally distributed with mean $\tau _{i}$ and variance $\sigma^2/m_{i}$.

Proof

This follows from the fact that the likelihood equation given in (8) and the modified likelihood equation given in (18) are asymptotically equivalent and $\partial \ln L^{*}/\partial \tau _{i}$ assumes the form, see Kendall and Stuart (1961):

$$\begin{aligned} \dfrac{\partial \ln L}{\partial \tau_{i}}\cong \dfrac{\partial \ln L^*}{\partial \tau _{i}} & =\dfrac{m_{i}}{\sigma^2}\{[(K_{i}-K)+(D_{i}-D)\hat{\sigma}]-\tau _{i}\} \\ & =\dfrac{m_{i}}{\sigma ^2} (\hat{\tau}_{i}-\tau _{i} ). \end{aligned}$$

The normality of $\hat{\tau }_{i}$ follows from the fact that $E \left( \frac{\partial ^{r} \ln L^*}{\partial \tau _{i}^{r}}\right) =0$ for all $r \ge 3$, see Bartlett (1953).

Lemma 3

Conditional on $\mu +\tau _{i}$ being known, the distribution of $A\hat{\sigma}^{2}/\sigma ^2$ is asymptotically Chi square with A degrees of freedom.

Proof

This follows from the fact that $B_0/\sqrt{AC_0}\cong 0$ and $\partial \ln L^{*}/\partial \sigma$ assumes the form, see Kendall and Stuart (1961):

$$\begin{aligned} \dfrac{\partial \ln L}{\partial \sigma }\cong \dfrac{\partial \ln L^*}{\partial \sigma }=\dfrac{A}{\sigma ^3}\left( \dfrac{C_0}{A}-\sigma ^2\right) , \end{aligned}$$

where $B_0$ and $C_0$ are the same as B and C, respectively, with $K_i$ is replaced by $\mu +\tau _{i}$, see Tiku (1982) and Senoglu (2007).