12.2 Estimation of Shape or Power Parameter

Let us now consider the class of the lifetime distributions, whose distribution functions are expressed by

$$F(t;\alpha,\gamma,\sigma)= G\left(\left(\frac{t-\gamma}{\sigma}\right)^{\alpha}\right)\,,$$ (12.11)

where $G(\cdot)$ is also a distribution function. For the Weibull model, $G(t)=1-\exp(-t)$ is an exponential distribution. Nagatsuka and Kamakura ([24], [25]) proposed a new method using the location-scale-free transformation of data set to estimate the power parameter in the Castillo-Hadi model ([2]). That is, let $T_1, \ldots,T_n$ be independently distributed according to the distribution function (12.11). Consider the $W$-transformation to be defined as

$$W_i = \frac{T_i - T_{(1)}}{T_{(n)}- T_{(1)}}, \ \ \ (i=2,\ldots,n-1),$$ (12.12)

where $T_{(k)}$ is the $k$-th order statistic of $T_i$'s. The new random variables $W_i$'s derived by this $W$-transformation are then free from location and scale parameter. The arithmetic mean of $W_i$'s gives the approximation to the original distribution of $T$. Let $V_i, i=1,\ldots,n$ be i.i.d. distributed with common distribution function $F_V(v)$, and let the $i$-th order statistic $V_{(i)}$ have the marginal distribution function $F_{V_{(i)}}(v)$. Then

$$F_{v}(v)=\frac{1}{n}\sum _{i=1}^n F_{V_{(i)}}(v)\,.$$ (12.13)

This equation indicates that the arithmetic mean of the marginal distributions of $n$ order statistics is exactly the original distribution. In the case of the Castillo-Hadi Model, [25] provided a theorem regarding this approximation, i.e.,

Theorem 1 ([25])  
The mixture of the marginal distributions of $W_{(i)}$, $i=2,\ldots,n-1$:

$$F^{(n)}(w)=\frac{1}{n-2}\sum _{i=2}^{n-1} F_{W_{(i)}}(w)$$ (12.14)

is the approximate distribution of $W_i$'s and the limiting distribution (12.14) is the power function distribution with parameter $1/\alpha$. That is

$$\notag \lim _{n\rightarrow\infty}\frac{1}{n-2}\sum _{i=2}^{n-1} F_{W_{(i)}}(w) =w^{\frac{1}{\alpha}}, \ \ \ 0<w<1.$$

In the case of the Weibull distribution, the marginal distribution of $W_{(i)}$ is calculated as

$$F_{W_{(i)}}(w) = \Pr\left(W_{(i)}\le w\right)$$

$$= \Pr\left(\frac{T_{(i)} - T_{(1)}}{T_{(n)}-T_{(1)}}\le w\right)$$

$$= \int_0^{\infty}\int_u^{\infty} n(n-1)f(u)f(v)\Bigg[ \sum _{k=i-1}^{n-2}\binom{n-2}{k}%\right.$$

$$\times\left\{F((1-w)u+wv)-F(u)\right\}^k$$

$$\times \left\{F(v)-F((1-w)u+wv)\right\}^{n-k-2} \Bigg]{\text{d}}v{\text{d}}u$$

$$= \int _0^1\int _u^1 n(n-1)\sum _{k=i-1}^{n-2}\binom{n-2}{k} \times\left[1-\exp\left\{-\alpha(w,m,u,v)\right\}-u\right]^k$$

$$\times \left[v-(1-\exp\left\{-\alpha(w,m,u,v)\right\})\right]^{n-k-2}\,,$$ (12.15)

where

$$\notag \alpha(w,m,u,v)=\left[(1-w)\left\{-\log(1-u)\right\}^{\frac{1}{m}}+w\left\{-\log(1-v)\right\}^{\frac{1}{m}}\right]^m\,.$$

Calculations show that $F^{(n)}(w)$ has a first moment of

$$\mu _n(m) = \int _0 ^{\infty}\left\{1-F^{(n)}(w)\right\}{\text{d}}w$$

$$= -\frac{1}{n-2} + \frac{n(n-1)}{m}\int _0^1\int _u^1 (v-u)^{n-3}$$

$$\times \frac{\Gamma\left(\frac{1}{m},-\log(1-u),-\log(1-v)\right)... ...{\frac{1}{m}}-\left\{-\log(1-u)\right\}^{\frac{1}{m}}}{\text{d}}v{\text{d}}u\,.$$ (12.16)

where $\Gamma(\cdot,\cdot,\cdot)$ is the incomplete generalized gamma function defined by

$$\notag \Gamma(a,z_0,z_1)=\int _{z_0}^{z_1}t^{a-1}e^{-t}{\text{d}}t\,.$$

Now, an estimating of the shape parameter $m$ is obtained by equating the theoretical population mean with sample mean of $W$-transformed $W$'s. [24] provided a table for obtaining estimates and concluded based on simulation studies that the robust estimate of $m$ is possible without using any existing threshold parameter.