Discriminating between log-normal and log-logistic distributions in the presence of type-II censoring

Abstract

The log-normal and the log-logistic distributions are two of the most commonly used distributions for studying positively skewed lifetime data. Both the distributions share number of interesting properties, and for a certain range of parameters their cumulative and hazard functions can also be similar in nature. However, selecting a more appropriate distribution and discriminating among them for a given data to best fit is an important issue. Further, when the data are observed in the presence of some censoring scheme the problem becomes more challenging. In this paper, we address the problem of selecting a more appropriate distribution by discriminating based on the random samples drawn in the presence of type-II censoring. We consider the difference of the maximized log-likelihood functions, and compute the asymptotic distribution of the discrimination statistic. We further propose a modified discriminating approach, and compute the probabilities of correct selection to check the performance of the discrimination procedure. Finally, simulation study is conducted, and two real data sets are analysed for the illustration purpose.

Appendices

Appendix A

Proof of Lemma 1

The use of the Theorems 1 and 2 of Bhattacharyya (1985) will easily establish the proof of parts (i) and (ii) of Lemma 1. Further, it can be observed that log-normal distribution satisfies the regularity conditions of Theorem 1 and 2. Now, we proceed in the following way to prove part (iii) of Lemma 1. Observe that, from (6) and part (iii) of Lemma 1, we have

$$\begin{aligned} \frac{T_r}{n}= & {} \frac{1}{n}\left( L_{LN}({{\hat{\sigma }}},{{\hat{\mu }}})-L_{LL}({{\hat{\gamma }}}, {{\hat{\xi }}}) \right) =\frac{1}{n}\left( \sum _{i=1}^{r}g(x_{i},{{\hat{\beta }}})+(n-r)h(x_{r},{{\hat{\beta }}})\right) \end{aligned}$$

(14)

$$\begin{aligned} \text{ and } \frac{T_*}{n}= & {} \frac{1}{n}\left( \sum _{i=1}^{r}g(x_{i},{{\tilde{\beta }}})+(n-r)h(x_{r},\tilde{\beta })\right) , \end{aligned}$$

(15)

where $\beta =(\sigma ,\mu ,\gamma ,\xi )^T, {\hat{\beta }}=({\hat{\sigma }},{\hat{\mu }},{\hat{\gamma }},{\hat{\xi }})^T, g(x_{i}, \beta )=\ln \left( \frac{f_{LN}(x_{i},\sigma ,\mu )}{f_{LL}(x_{i}, \gamma ,\xi )}\right) , h(x_{r},\beta )=\ln \left( \frac{1- F_{LN}(x_{r},\sigma ,\mu )}{1-F_{LL}(x_{r},\gamma ,\xi )}\right)$, and ${{\tilde{\beta }}}=(\sigma ,\mu ,{{\tilde{\gamma }}},{{\tilde{\xi }}})^T$. Now, our main objective is to show

$$\begin{aligned} \frac{1}{\sqrt{n}}\left( T_r-E_{LN}(T_r)\right) -\frac{1}{\sqrt{n}}\left( T_*-E_{LN}(T_*)\right)\overset{P}{\rightarrow } & {} 0 \end{aligned}$$

(16)

Here, $\overset{P}{\rightarrow }$ stands for convergence in probability. Now, to prove equation (16), it will be enough to prove

$$\begin{aligned} \frac{1}{\sqrt{n}}\left( T_r-T_*\right)\overset{P}{\rightarrow } & {} 0, \end{aligned}$$

(17)

$$\begin{aligned} \text{ and } E_{LN}\left( \frac{1}{\sqrt{n}}(T_r - T_*) \right)\overset{P}{\rightarrow } & {} 0 \end{aligned}$$

(18)

Now, using (14) and (15), $(1/\sqrt{n})\left( T_r-T_*\right)$ can be written as

$$\begin{aligned}{} & {} \sqrt{n}\left( \frac{1}{n}\sum _{i=1}^{r}\left( g(x_{i};{{\hat{\beta }}})-g(x_{i};{{\tilde{\beta }}}) \right) +\left( \frac{n-r}{n} \right) \left( h(x_{r};{{\hat{\beta }}})-h(x_{r};{{\tilde{\beta }}}) \right) \right) \nonumber \\{} & {} \quad = \sqrt{n}\left( \frac{1}{n}\sum _{i=1}^{n}\left( g(x_i;{{\hat{\beta }}})-g(x_i;{{\tilde{\beta }}}) \right) \cdot 1_{x_i\le x_{r}} +\left( \frac{n-r}{n} \right) \left( h(x_{r};{{\hat{\beta }}})-h(x_{r};{{\tilde{\beta }}}) \right) \right) \nonumber \\{} & {} \qquad \overset{a.e.}{=}\ \sqrt{n}\left( \frac{1}{n}\sum _{i=1}^{n}\left( g(x_i;{{\hat{\beta }}})-g(x_i;{{\tilde{\beta }}}) \right) \cdot 1_{x_i\le \alpha } +\left( \frac{n-r}{n} \right) \left( h(x_{r};{{\hat{\beta }}})-h(x_{r};{{\tilde{\beta }}}) \right) \right) \nonumber \\{} & {} \quad =\sum _{j=1}^{4}\left( \frac{1}{n}\sum _{i=1}^{n}g'_j(x_i;\tilde{\beta }_j) \cdot 1_{x_i\le \alpha }+\left( \frac{n-r}{n} \right) h'_j(x_{r};{{\tilde{\beta }}}_j)\right) \sqrt{n}({\hat{\beta _j}}-{{\tilde{\beta }}}_j)+o_p(1). \end{aligned}$$

(19)

Here $\overset{a.e.}{=}$ and $o_p(1)$, stands for asymptotically equivalent and convergence in probability to zero respectively, where $g'_j=\left( \frac{\partial }{\partial \beta _j}\right) g(.), h'_j=\left( \frac{\partial }{\partial \beta _j}\right) h(.),{\hat{\beta _j}}$ and ${{\tilde{\beta }}}_j$ are the $j^{th}$ components of ${{\hat{\beta }}}$ and ${{\tilde{\beta }}}$ respectively for $j = 1, 2, 3, 4$. It should be noted that the second equality is produced using the Theorem 2 of Bhattacharyya (1985), as $\sum _{i=1}^{n}g(x_i;.)\cdot 1_{x_i\le \alpha }$ and $\sum _{i=1}^{n}g(x_i;.)\cdot 1_{x_i\le x_{r}}$ are asymptotically equivalent. Now, with the implementation of the weak law of large numbers and the fact that $h'(.)$ is a continuous function, we have,

$$\begin{aligned}{} & {} \left( \frac{1}{n}\sum _{i=1}^{n} g'_j(x_i;\tilde{\beta }_j)\cdot 1_{x_i\le \alpha }+\frac{n-r}{n} h'_j(x_{r};\tilde{\beta }_j) \right) \\{} & {} \qquad \overset{P}{\rightarrow }\ E_{LN}\left( g'_j(x_i;\beta _j) \cdot 1_{x_i\le \alpha }+(1-p) h'_j(\alpha ; \beta _j)\right) \Big |_{\beta ={{\tilde{\beta }}}}. \end{aligned}$$

Now, following the same line as Fearn and Nebenzahl (1991), we have

$$\begin{aligned} \left| \frac{1}{n}\sum _{i=1}^{n} g'_j(x_i;\tilde{\beta }_j)\cdot 1_{x_i\le \alpha }+\frac{n-r}{n} h'_j(x_{r};\tilde{\beta }_j) \right| \text { is asymptotically bounded}. \end{aligned}$$

(20)

Therefore, using dominated convergence theorem, we can write

$$\begin{aligned}{} & {} E_{LN}\left( g'_j(x_i;\beta _j) \cdot 1_{x_i\le \alpha }+(1-p) h'_j(\alpha ; \beta _j)\right) \Big |_{\beta ={{\tilde{\beta }}}}\nonumber \\{} & {} \quad = \frac{\partial }{\partial \beta _j}\left( \int _{0}^{\alpha }\ln \left( \frac{f_{LN}(x;\sigma ,\mu )}{f_{LL}(x;\gamma ,\xi )}\right) f_{LN}(x;\sigma ,\mu ) dx\right. \nonumber \\{} & {} \qquad \left. +(1-p) \ln \left( \frac{1- F_{LN}(\alpha ; \sigma , \mu )}{1- F_{LL}(\alpha ; \gamma ,\xi )}\right) \right) \Bigg |_{\beta ={{\tilde{\beta }}}}\nonumber \\{} & {} \quad = 0. \end{aligned}$$

(21)

Now, equation (17) follows from the use of equation (21), and $\sqrt{n}({\hat{\beta _j}}-{{\tilde{\beta }}}_j)$ converges to a normal distribution with mean zero and finite variance. Next, to prove (18), by making use of (19), observe that

$$\begin{aligned}{} & {} E_{LN}\left| \frac{1}{\sqrt{n}}\sum _{i=1}^{r}\left( g(x_{i};{{\hat{\beta }}})-g(x_{i};{{\tilde{\beta }}}) \right) +\sqrt{n}\left( \frac{n-r}{n} \right) \left( h(x_{r};{{\hat{\beta }}})-h(x_{r};{{\tilde{\beta }}}) \right) \right| \\{} & {} \quad \le \sum _{j=1}^{4}E_{LN}\left| \left( \frac{1}{n}\sum _{i=1}^{n}g'_j(x_i;{{\tilde{\beta }}}_j) \cdot 1_{x_i\le \alpha } + \left( \frac{n-r}{n} \right) h'_j(x_{r};{{\tilde{\beta }}}_j)\right) \sqrt{n}({\hat{\beta _j}}-{{\tilde{\beta }}}_j) \right| +o(1) \\{} & {} \quad \le \sum _{j=1}^{4}\left( E_{LN}\left( \sqrt{n}({\hat{\beta _j}}-{{\tilde{\beta }}}_j) \right) ^2E_{LN}\right. \\{} & {} \qquad \left. \left( \frac{1}{n}\sum _{i=1}^{n}g'_j(x_i;\tilde{\beta }_j) \cdot 1_{x_i\le \alpha } + \left( \frac{n-r}{n} \right) h'_j(x_{r};{{\tilde{\beta }}}_j) \right) ^2\right) ^{1/2}+o(1). \end{aligned}$$

Therefore, by using equations (20) and (21), it is observed that

$$\begin{aligned} E_{LN}\left| \frac{1}{n}\sum _{i=1}^{n}g'_j(x_i;{{\tilde{\beta }}}_j) \cdot 1_{x_i\le \alpha }+\left( \frac{n-r}{n} \right) h'_j(x_{r};{{\tilde{\beta }}}_j) \right| ^2\overset{P}{\rightarrow }\ 0. \end{aligned}$$

Since, $\lim _{n\rightarrow \infty }E_{LN}\left( \sqrt{n}({\hat{\beta _j}}-{{\tilde{\beta }}}_j) \right) ^2<\infty ~\forall ~ j = 1, 2, 3, 4$, and therefore equation (18) follows. $\square$

Appendix B

$T_*$ is independent of the parameters of the parent distributions

Here, we show numerically that the values of $T_*$ are not dependent on the parameters but only depends on the censoring proportion p. We, consider the case of the log-normal distribution. We used the results of Lemma 1 for calculating the values of $T_*$ for different values of $\sigma$ and $\mu$, the shape and the scale parameters of the log-normal distribution, for different censoring proportions, p= 0.9, 0.8, 0.7, 0.6, 0.5, 0.4 and 0.3. The tabulated values of $T_*$ are presented in Table 11, and from the reported values, it can be observed that the values of $T_*$ remains more or less same for different values of $\sigma$ and $\mu$ for the particular value of p. But it changes as we change the value of p. Hence, we can conclude based on the numerical observations that $T_*$ values only depend on p and are independent of the values of the parent parameters. This also holds true when the log-logistic distribution is considered as parent distribution.

Table 11 Values of $T_*$ when the parent distribution is log-normal