On the exact distribution of the likelihood ratio test for testing the homogeneity of scale parameters of several two-parameter exponential distributions: complete and censored samples

Abstract

We consider the problem of providing the exact distribution of the likelihood ratio test (LRT) statistic for testing the homogeneity of scale parameters of $k \ge 2$ two-parameter exponential distributions. To this end, we apply the Millen inverse transform and the Jacobi polynomial expansion to the moments of LRT statistic. We consider the problem when the data are either complete or censored under the different kinds of Type II censoring, such as the Type II right, progressively Type II right, and double Type II censoring schemes. We also discuss the exact null distribution of the LRT when the data are censored under the Type I censoring scheme.

Appendix

Proof of Lemma 1

Note that $\varvec{Z} = \left( {Z_{1} , \ldots ,Z_{k} } \right) = \left( {\frac{{Y_{1} }}{{\mathop \sum \nolimits_{i = 1}^{k} Y_{i} }}, \ldots ,\frac{{Y_{k} }}{{\mathop \sum \nolimits_{i = 1}^{k} Y_{i} }}} \right)$ has a Dirichlet distribution with parameter $(\theta_{1} - 1, \ldots ,\theta_{k} - 1)$. Therefore, the $h$th moment of $L$ under $H_{0}$ is

$$ \begin{aligned} E_{{H_{0} }} \left( {L^{h} } \right) & = \frac{{{{\Theta }}^{h} }}{{\mathop \prod \nolimits_{i = 1}^{k} \theta_{i}^{{\frac{{\theta_{i} h}}{{{\Theta }}}}} }}\mathop \int \nolimits_{0}^{1} \ldots \mathop \int \nolimits_{0}^{1} \mathop \prod \limits_{i = 1}^{k} z_{i}^{{\theta_{i} h/{{\Theta }}}} \frac{{{{\Gamma }}\left( {{{\Theta }} - k} \right)}}{{\mathop \prod \nolimits_{i = 1}^{k} {{\Gamma }}\left( {\theta_{i} - 1} \right)}}\mathop \prod \limits_{i = 1}^{k} z_{i}^{{\theta_{i} - 2}} dz_{1} \ldots dz_{k} \\ & = \frac{{{{\Theta }}^{h} }}{{\mathop \prod \nolimits_{i = 1}^{k} \theta_{i}^{{\theta_{i} h/{{\Theta }}}} }}\frac{{{{\Gamma }}\left( {{{\Theta }} - k} \right)}}{{{{\Gamma }}\left( {{{\Theta }} - k + h} \right)}}\mathop \prod \limits_{i = 1}^{k} \frac{{{{\Gamma }}\left( {\theta_{i} - 1 + \theta_{i} h/{{\Theta }}} \right)}}{{{{\Gamma }}\left( {\theta_{i} - 1} \right)}}. \\ \end{aligned} $$

Now, let $V_{i} = \sigma_{i} Y_{i} /\mathop \sum \nolimits_{i = 1}^{k} \sigma_{i} Y_{i}$, $i = 1, \ldots ,k$. It is easily verified that $Y_{i} = (V_{i} /\sigma_{i} )/\mathop \sum \nolimits_{i = 1}^{k} (V_{i} /\sigma_{i} )$ and thus the joint density function of $\varvec{V} = (V_{1} , \ldots ,V_{k} )$ is

$$ f_{\varvec{V}} \left( \varvec{v} \right) = \frac{{{{\Gamma }}\left( {{{\Theta }} - k} \right)}}{{\mathop \prod \nolimits_{i = 1}^{k} {{\Gamma }}\left( {\theta_{i} - 1} \right)}}\frac{{\mathop \prod \nolimits_{i = 1}^{k} \left( {1/\sigma_{i} } \right)^{{\theta_{i} - 1}} }}{{\left( {\mathop \sum \nolimits_{i = 1}^{k} \frac{{v_{i} }}{\sigma }_{i} } \right)^{{{{\Theta }} - k}} }}\mathop \prod \limits_{i = 1}^{k} v_{i}^{{\theta_{i} - 2}} . $$

Therefore, the $h$th moment of $L$ is

$$ \begin{aligned} E\left( {L^{h} } \right) & = \frac{{{{\Theta }}^{h} }}{{\mathop \prod \nolimits_{i = 1}^{k} \theta_{i}^{{\theta_{i} h/{{\Theta }}}} }}E_{\varvec{V}} \left( {\mathop \prod \limits_{i = 1}^{k} V_{i}^{{\frac{{\theta_{i} h}}{{{\Theta }}}}} } \right) = \frac{{{{\Theta }}^{h} }}{{\mathop \prod \nolimits_{i = 1}^{k} \theta_{i}^{{\theta_{i} h/{{\Theta }}}} }}\mathop \int \nolimits_{0}^{1} \ldots \mathop \int \nolimits_{0}^{1} \mathop \prod \limits_{i = 1}^{k} v_{i}^{{\theta_{i} h/{{\Theta }}}} f_{\varvec{V}} \left( \varvec{v} \right)dv_{1} \ldots dv_{k} \\ & = \frac{{{{\Theta }}^{h} {{\Gamma }}\left( {{{\Theta }} - k} \right)}}{{\mathop \prod \nolimits_{i = 1}^{k} \sigma_{i}^{{\theta_{i} - 1}} \theta_{i}^{{\theta_{i} h/{{\Theta }}}} {{\Gamma }}\left( {\theta_{i} - 1} \right)}}\mathop \int \nolimits_{0}^{1} \ldots \mathop \int \nolimits_{0}^{1} \left( {\mathop \sum \limits_{i = 1}^{k} \frac{{v_{i} }}{\sigma }_{i} } \right)^{{k - {{\Theta }}}} \mathop \prod \limits_{i = 1}^{k} v_{i}^{{\theta_{i} + \frac{{\theta_{i} h}}{{{\Theta }}} - 2}} dv_{1} \ldots dv_{k} \\ & = \frac{{{{\Theta }}^{h} {{\Gamma }}\left( {{{\Theta }} - k} \right)}}{{{{\Gamma }}\left( {{{\Theta }} - k + h} \right)}}\mathop \prod \limits_{i = 1}^{k} \frac{{{{\Gamma }}(\theta_{i} + \theta_{i} h/{{\Theta }} - 1)}}{{\sigma_{i}^{{\theta_{i} - 1}} {{\Gamma }}\left( {\theta_{i} - 1} \right)\theta_{i}^{{\theta_{i} h/{{\Theta }}}} }}E_{\varvec{D}} \left( {\left( {\mathop \sum \limits_{i = 1}^{k} D_{i} /\sigma_{i} } \right)^{{k - {{\Theta }}}} } \right). \\ \end{aligned} $$

This completes the proof.□

Proof of Theorem 1

Here, we present a sketch of proof. For more detail, one can refer to Nagarsenker (1976). Let $f_{L} (.)$ denote the density function of $L$. If $f_{L} (.)$ can be obtained in a closed form, then $F_{L} (.)$ is obtained directly. By applying the Mellin inverse transform to $E_{{H_{0} }} (L^{h} )$ and changing variable $h + {{\Theta }} = t$, we have

$$ \begin{aligned} f_{L} \left( l \right) & = \frac{1}{2\pi i}\mathop \int \nolimits_{ - i\infty }^{ + i\infty } l^{ - h - 1} E_{{H_{0} }} \left( {L^{h} } \right) dh \\ & = \frac{{{{\Gamma }}\left( {{{\Theta }} - k} \right)}}{{\mathop \prod \nolimits_{i = 1}^{k} {{\Gamma }}\left( {\theta_{i} - 1} \right)}} l^{{{{\Theta }} - 1}} \mathop \prod \limits_{i = 1}^{k} \left( {\frac{{\theta_{i} }}{{{\Theta }}}} \right)^{{\theta_{i} }} \frac{1}{2\pi i}\mathop \int \nolimits_{{{{\Theta }} - i\infty }}^{{{{\Theta }} + i\infty }} l^{ - t} \phi \left( t \right) dt, \\ \end{aligned} $$

where $\phi \left( t \right) = (\mathop \prod \nolimits_{i = 1}^{k} {{\Gamma }}\left( {\theta_{i} t/{{\Theta }} - 1} \right)\left( {{{\Theta }}/\theta_{i} } \right)^{{\theta_{i} t/{{\Theta }}}} )/{{\Gamma }}\left( {t - k} \right)$. We also generally have$\log {{\Gamma }}(x + a) = \frac{1}{2}\log \left( {2\pi } \right) + \left( {x + a - \frac{1}{2}} \right)\log \left( x \right) - x - \mathop \sum \nolimits_{r = 1}^{m} \frac{{\left( { - 1} \right)^{r} B_{r + 1} \left( a \right)}}{{r\left( {r + 1} \right)x^{r} }} + R_{m + 1} \left( x \right),$where $R_{m} (x)$ is the remainder such that $\left| {R_{m} \left( x \right)} \right| \le C/|x^{m} |$, $C$ is a constant independent of $x$ and $B_{r} (a)$ is the Bernoulli polynomial of degree $r$ and order one; see Nagarsenker (1976). Therefore, the expansion of $\log (\phi \left( t \right))$ is

$$ \begin{aligned} & \log \left( {\phi \left( t \right)} \right) = v\log \left( {2\pi } \right) - 1.5\mathop \sum \limits_{i = 1}^{k} \log \left( {\frac{{\theta_{i} }}{{{\Theta }}}} \right) - v\log \left( t \right) \\ & \quad + \mathop \sum \limits_{r = 1}^{m} \left( {\frac{1}{t}} \right)^{r} \left\{ {\frac{{\left( { - 1} \right)^{r} }}{{r\left( {r + 1} \right)}}\left( {B_{r + 1} \left( { - k} \right) - \mathop \sum \limits_{i = 1}^{k} B_{r + 1} \left( { - 1} \right)\left( {\frac{{{\Theta }}}{{\theta_{i} }}} \right)^{r} } \right)} \right\} + \mathop \sum \limits_{i = 1}^{k} R_{m + 1} \left( {\frac{{\theta_{i} t}}{{{\Theta }}}} \right) - R_{m + 1} \left( t \right), \\ \end{aligned} $$

where $v = (k - 1)/2$. By letting $m \to \infty$,e have

$$ \phi \left( t \right) = \left( {2\pi } \right)^{v} t^{ - v} \mathop \prod \limits_{i = 1}^{k} \left( {\frac{{\theta_{i} }}{{{\Theta }}}} \right)^{ - 1.5} \left\{ {1 + \mathop \sum \limits_{j = 1}^{\infty } \beta_{j} /t^{j} } \right\}, $$

(A.1)

where $\beta_{j}$s are obtained recursively as $\beta_{j} = \frac{1}{j}\mathop \sum \nolimits_{s = 1}^{j} s \alpha_{s} \beta_{j - s}$ with $\beta_{0} = 1$ in which

$$ \alpha_{s} = \frac{{\left( { - 1} \right)^{s} }}{{s\left( {s + 1} \right)}}\left\{ {B_{s + 1} \left( { - k} \right) - \mathop \sum \limits_{i = 1}^{k} B_{s + 1} \left( { - 1} \right)\left( {\frac{{{\Theta }}}{{\theta_{i} }}} \right)^{s} } \right\}, $$

see Kalinin and Shalaevskii (1971). Note that $\left( {2\pi } \right)^{ - v} \mathop \prod \nolimits_{i = 1}^{k} \left( {\theta_{i} /{{\Theta }}} \right)^{1.5} \phi \left( t \right) = O\left( {t^{ - v} } \right)$. Therefore, $\left( {2\pi } \right)^{ - v} \mathop \prod \nolimits_{i = 1}^{k} \left( {\theta_{i} /{{\Theta }}} \right)^{1.5} \phi \left( t \right)$ can be expanded as a factorial series of the form

$$ \left( {2\pi } \right)^{ - v} \mathop \prod \limits_{i = 1}^{k} \left( {\theta_{i} /{{\Theta }}} \right)^{1.5} \phi \left( t \right) = \mathop \sum \limits_{r = 1}^{\infty } R_{r} {{\Gamma }}(t)/{{\Gamma }}(t + r + v), $$

(A.2)

for any $a \ge 0$; see Nair (1940). The coefficient $R_{r}$ can be obtained by equating the coefficients of the two series (A.1) and (A.2). In this regard, note that $\log ({{\Gamma }}(t)/{{\Gamma }}(t + r + v))$ can be expanded as $- \left( {v + n} \right)\log \left( t \right) + \sum\nolimits_{j = 1}^{\infty } {A_{r,j} } /t^{j}$ where

$$ A_{r,j} = \frac{1}{{j\left( {j + 1} \right)}}\left( { - 1} \right)^{j - 1} \left\{ {B_{j + 1} \left( 0 \right) - B_{j + 1} \left( {v + r} \right)} \right\}. $$

Therefore, we can write $\frac{\varGamma \left( t \right)}{{\varGamma \left( {t + r + v} \right)}} = t^{{ - \left( {v + r} \right)}} \left\{ {1 + \sum\nolimits_{j = 1}^{\infty } {\frac{{C_{r,j} }}{{t^{j} }}} } \right\}$ where $C_{r,j} = \frac{1}{j}\sum\nolimits_{q = 1}^{j} q A_{r,q} C_{r,j - q}$ with $C_{r,0} = 1$. Finally, the coefficients $R_{r}$ are obtained recursively by $\sum\nolimits_{j = 0}^{i} {R_{i - j} } C_{i - j,j} = \beta_{i}$ with $R_{0} = 1$. Now, $f_{L} (l)$ can be obtained by integrating term by term of (A.2) since a factorial series is uniformly convergent in a half-plane; see Nagarsenker (1976). Therefore, we have

$$ \begin{aligned} f_{L} \left( l \right) & = \left( {2\pi } \right)^{v} l^{{{{\Theta }} - 1}} \frac{{{{\Gamma }}\left( {{{\Theta }} - k} \right)}}{{\mathop \prod \nolimits_{i = 1}^{k} {{\Gamma }}\left( {\theta_{i} - 1} \right)}} \mathop \prod \limits_{i = 1}^{k} \left( {\frac{{\theta_{i} }}{{{\Theta }}}} \right)^{{\theta_{i} - 1.5}} \frac{1}{2\pi i}\mathop \int \nolimits_{{{{\Theta }} - i\infty }}^{{{{\Theta }} + i\infty }} l^{ - t} \mathop \sum \limits_{r = 0}^{\infty } R_{r} \frac{{{{\Gamma }}\left( t \right)}}{{{{\Gamma }}\left( {t + r + v} \right)}}dt \\ & = \left( {2\pi } \right)^{v} l^{{{{\Theta }} - 1}} \frac{{{{\Gamma }}\left( {{{\Theta }} - k} \right)}}{{\mathop \prod \nolimits_{i = 1}^{k} {{\Gamma }}\left( {\theta_{i} - 1} \right)}} \mathop \prod \limits_{i = 1}^{k} \left( {\frac{{\theta_{i} }}{{{\Theta }}}} \right)^{{\theta_{i} - 1.5}} \mathop \sum \limits_{r = 0}^{\infty } R_{r} \frac{1}{2\pi i}\mathop \int \nolimits_{{{{\Theta }} - i\infty }}^{{{{\Theta }} + i\infty }} l^{ - t} \frac{{{{\Gamma }}\left( t \right)}}{{{{\Gamma }}\left( {t + r + v} \right)}} dt. \\ \end{aligned} $$

However, $\frac{1}{2\pi i}\mathop \int \nolimits_{{{{\Theta }} - i\infty }}^{{{{\Theta }} + i\infty }} l^{ - t} {{\Gamma }}(t)/{{\Gamma }}\left( {t + r + v} \right) dt = l\left( {1 - l} \right)^{v + r - 1} /{{\Gamma }}(v + r)$; see Nagarsenker (1976). This completes the proof.□

Proof of Theorem 2

Note that the support of $L$ is $(0, 1)$ since ${{\Lambda }}$ is the LRT statistic. By matching the first two moments of Beta $(p,q)$ random variable and $L$, we have

$$ p = \frac{{E_{{H_{0} }} \left( L \right)\left( {E_{{H_{0} }} \left( L \right) - E_{{H_{0} }} \left( {L^{2} } \right)} \right)}}{{E_{{H_{0} }} \left( {L^{2} } \right) - E_{{H_{0} }}^{2} \left( L \right)}} \;\;{\text{and}}\;\; q = \frac{{\left( {E_{{H_{0} }} \left( L \right) - E_{{H_{0} }} \left( {L^{2} } \right)} \right)\left( {1 - E_{{H_{0} }} \left( L \right)} \right)}}{{E_{{H_{0} }} \left( {L^{2} } \right) - E_{{H_{0} }}^{2} \left( L \right)}}. $$

In this case, it can be easily verified that $c_{1} = c_{2} = 0$. It completes the proof.□