On the exact distribution of the likelihood ratio test statistic for testing the homogeneity of the scale parameters of several inverse Gaussian distributions

Abstract

Several researchers have addressed the problem of testing the homogeneity of the scale parameters of several independent inverse Gaussian distributions based on the likelihood ratio test. However, only approximations of the distribution function of the test statistic are available in the literature. In this note, we present the exact distribution of the likelihood ratio test statistic for testing the equality of the scale parameters of several independent inverse Gaussian populations in a closed form. To this end, we apply the Mellin inverse transform and the Jacobi polynomial expansion to the moments of the likelihood ratio test statistic. We also propose an approximate method based on the Jacobi polynomial expansion. Finally, we apply an accurate numerical method, which is based on the inverse of characteristic function, to obtain a near-exact approximation of the likelihood ratio test statistic distribution. The proposed methods are illustrated via numerical and real data examples.

Appendix: proofs

Proof of Lemma 1

It is well known that $W_{i} = \lambda_{i} V_{i}$ has a Chi square distribution with $n_{i} - 1$ degrees of freedom. Note that the $LR$ can be rewritten as

$$ LR = \mathop \prod \limits_{i = 1}^{k} \left( {\left( {\frac{{W_{i} /\lambda_{i} }}{{\mathop \sum \nolimits_{j = 1}^{k} W_{j} /\lambda_{j} }}} \right)^{{n_{i} /2}} f_{i}^{{ - n_{i} /2}} } \right) . $$

If $\varvec{Z} = \left( {Z_{1} , \ldots ,Z_{k} } \right) = \left( {\frac{{W_{1} }}{{\mathop \sum \nolimits_{i = 1}^{k} w_{i} }}, \ldots ,\frac{{W_{k} }}{{\mathop \sum \nolimits_{i = 1}^{k} W_{i} }}} \right)$, it is known that $\varvec{Z}$ has a Dirichlet distribution with parameter $\left( {n_{1}^{*} , \ldots ,n_{k}^{*} } \right)$. It can be verified that

$$ \varvec{Y} = \left( {Y_{1} , \ldots ,Y_{k} } \right) = \left( {\frac{{Z_{1} /\lambda_{1} }}{{\mathop \sum \nolimits_{i = 1}^{k} Z_{i} /\lambda_{i} }}, \ldots ,\frac{{Z_{k} /\lambda_{k} }}{{\mathop \sum \nolimits_{i = 1}^{k} Z_{i} /\lambda_{i} }}} \right) = \left( {\frac{{W_{1} /\lambda_{1} }}{{\mathop \sum \nolimits_{j = 1}^{k} W_{j} /\lambda_{j} }}, \ldots ,\frac{{W_{k} /\lambda_{k} }}{{\mathop \sum \nolimits_{j = 1}^{k} W_{j} /\lambda_{j} }}} \right), $$

has the following density function

$$ f_{\varvec{Y}} \left( {y_{1} , \ldots ,y_{k} } \right) = \frac{{{{\varGamma }}\left( {N^{*} } \right)}}{{\mathop \prod \nolimits_{i = 1}^{k} \varGamma \left( {n_{i}^{*} } \right)}}\frac{{\mathop \prod \nolimits_{i = 1}^{k} \lambda_{i}^{{n_{i}^{*} }} }}{{\left( {\mathop \sum \nolimits_{i = 1}^{k} \lambda_{i} y_{i} } \right)^{{N^{*} }} }}\mathop \prod \limits_{i = 1}^{k} y_{i}^{{n_{i}^{*} - 1}} , $$

see Kharrati-Kopaei and Malekzadeh (2019). Therefore, the $h$th moment of $L$ under $H_{1}$ is

$$ \begin{aligned} E_{{H_{1} }} \left( {L^{h} } \right) & = \mathop \prod \limits_{i = 1}^{k} f_{i}^{{ - f_{i} h}} E_{{H_{1} }} \left( {\mathop \prod \limits_{i = 1}^{k} \left( {\frac{{V_{i} }}{V}} \right)^{{f_{i} h}} } \right) = \mathop \prod \limits_{i = 1}^{k} f_{i}^{{ - f_{i} h}} E_{{H_{1} }} \left( {\mathop \prod \limits_{i = 1}^{k} \left( {\frac{{W_{i} /\lambda_{i} }}{{\mathop \sum \nolimits_{j = 1}^{k} W_{j} /\lambda_{j} }}} \right)^{{f_{i} h}} } \right) \\ & = \frac{{\varGamma \left( {N^{*} } \right)}}{{\varGamma \left( {N^{*} + h} \right)}}\mathop \prod \limits_{i = 1}^{k} \left( {f_{i}^{{ - f_{i} h}} \lambda_{i}^{{n_{i}^{*} }} \frac{{\varGamma \left( {f_{i} h + n_{i}^{*} } \right)}}{{\varGamma \left( {n_{i}^{*} } \right)}}} \right)E_{\varvec{D}} \left( {\left( {\mathop \sum \limits_{i = 1}^{k} \lambda_{i} D_{i} } \right)^{{ - N^{*} }} } \right), \\ \end{aligned} $$

for detail, see Kharrati-Kopaei and Malekzadeh (2019). Note that $E_{{H_{1} }} \left( {L^{h} } \right)$ reduces to $E_{{H_{0} }} \left( {L^{h} } \right)$ under $H_{0}$ since $\mathop \sum \nolimits_{i = 1}^{k} D_{i} = 1$.

Proof of Theorem 1

Let $f_{L} \left( . \right)$ denote the PDF of $L$. If one can obtain $f_{L} \left( . \right)$ in a closed form, then $F_{L} \left( . \right)$ is obtained directly. The proof is similar to that of Kharrati-Kopaei and Malekzadeh (2019), and hence we present a sketch of the proof. By applying the MIT to $E_{{H_{0} }} \left( {L^{h} } \right)$ and changing variable $h + N/2 = t$, we have

$$ \begin{aligned} f_{L} \left( x \right) & = \frac{1}{2\pi i}\mathop \int \limits_{ - i\infty }^{ + i\infty } x^{ - h - 1} E_{{H_{0} }} \left( {L^{h} } \right) dh \\ & = \mathop \prod \limits_{i = 1}^{k} f_{i}^{{n_{i} /2}} \frac{{\varGamma \left( {N^{*} } \right)}}{{\mathop \prod \nolimits_{i = 1}^{k} \varGamma \left( {n_{i}^{*} } \right)}} x^{{\frac{N}{2} - 1}} \frac{1}{2\pi i}\mathop \int \limits_{{\frac{N}{2} - i\infty }}^{{\frac{N}{2} + i\infty }} x^{ - t} \phi \left( t \right) dt, \\ \end{aligned} $$

where $\phi \left( t \right) = \left( {\mathop \prod \nolimits_{i = 1}^{k} \varGamma \left( {f_{i} t - 1/2} \right)f_{i}^{{ - f_{i} t}} } \right)/\varGamma \left( {t - k/2} \right)$. One can expand $\phi \left( t \right)$ as

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

(A.1)

where $v = \left( {k - 1} \right)/2$, and $\beta_{j}$ 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 $\alpha_{s} = \frac{{\left( { - 1} \right)^{s} }}{{s\left( {s + 1} \right)}}\left\{ {B_{s + 1} \left( { - k} \right) - B_{s + 1} \left( { - 1} \right)\mathop \sum \limits_{i = 1}^{k} f_{i}^{ - s} } \right\}$ where $B_{r} \left( a \right)$ is the Bernoulli polynomial of degree $r$ and order one; see Kalinin (1971), Nagarsenker (1976). Note that $\left( {2\pi } \right)^{ - v} \left( {\mathop \prod \nolimits_{i = 1}^{k} f_{i} } \right)\phi \left( t \right)$ can be expanded as a factorial series of the form

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

(A.2)

see Nair (1940). The coefficient $R_{r}$ can be obtained by equating the coefficients of the two series in (A.1) and (A.2). In this regard, note that $\log \left( {\varGamma \left( t \right)/\varGamma \left( {t + r + v} \right)} \right)$ can be expanded as $- \left( {v + n} \right)\log \left( t \right) + \mathop \sum \nolimits_{j = 1}^{\infty } A_{r,j} /t^{j}$ where $A_{r,j} = \frac{{\left( { - 1} \right)^{j - 1} }}{{j\left( {j + 1} \right)}}\left\{ {B_{j + 1} \left( 0 \right) - B_{j + 1} \left( {v + r} \right)} \right\}.$ Therefore, we can write $\frac{\varGamma \left( t \right)}{{\varGamma \left( {t + r + v} \right)}} = t^{{ - \left( {v + r} \right)}} \left\{ {1 + \mathop \sum \nolimits_{j = 1}^{\infty } \frac{{C_{r,j} }}{{t^{j} }}} \right\}$ where $C_{r,j} = \frac{1}{j}\mathop \sum \nolimits_{q = 1}^{j} q A_{r,q} C_{r,j - q}$ with $C_{r,0} = 1$. Consequently, the coefficients $R_{r}$ are obtained recursively by $\mathop \sum \nolimits_{j = 0}^{i} R_{i - j} C_{i - j,j} = \beta_{i}$ with $R_{0} = 1$. Now, $f_{L} \left( x \right)$ 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

$$ \begin{aligned} f_{L} \left( x \right) & = \left( {2\pi } \right)^{v} x^{{\frac{N}{2} - 1}} \frac{{\varGamma \left( {N^{*} } \right)}}{{\mathop \prod \nolimits_{i = 1}^{k} \varGamma \left( {n_{i}^{*} } \right)}} \mathop \prod \limits_{i = 1}^{k} f_{i}^{{\frac{{n_{i} }}{2} - 1}} \frac{1}{2\pi i}\mathop \int \limits_{{\frac{N}{2} - i\infty }}^{{\frac{N}{2} + i\infty }} x^{ - t} \mathop \sum \limits_{r = 0}^{\infty } R_{r} \frac{\varGamma \left( t \right)}{{\varGamma \left( {t + r + v} \right)}}dt \\ & = \left( {2\pi } \right)^{v} x^{{\frac{N}{2} - 1}} \frac{{\varGamma \left( {N^{*} } \right)}}{{\mathop \prod \nolimits_{i = 1}^{k} \varGamma \left( {n_{i}^{*} } \right)}} \mathop \prod \limits_{i = 1}^{k} f_{i}^{{\frac{{n_{i} }}{2} - 1}} \mathop \sum \limits_{r = 0}^{\infty } R_{r} \frac{1}{2\pi i}\mathop \int \limits_{{\frac{N}{2} - i\infty }}^{{\frac{N}{2} + i\infty }} x^{ - t} \frac{\varGamma \left( t \right)}{{\varGamma \left( {t + r + v} \right)}} dt. \\ \end{aligned} $$

However, the Mellin transform of $\varGamma \left( t \right)/\varGamma \left( {t + r + v} \right)$ is $\left( {1 - x} \right)^{v + r - 1} /\varGamma \left( {v + r} \right)$; see Nagarsenker (1976). This completes the proof.

Proof of Theorem 2

Note that the support of $L$ is $\left( {0, 1} \right)$ since $LR$ is the LRT statistic. Therefore, the Jacobi polynomial expansion of $F_{L} \left( x \right)$ can be expressed as

$$ F_{L} \left( x \right) = \mathop \sum \limits_{n = 0}^{\infty } c_{n} \frac{{\varGamma \left( {n + q} \right)\varGamma \left( {p + q} \right)}}{{n!\varGamma \left( q \right)\varGamma \left( {n + p + q - 1} \right)}}\mathop \sum \limits_{s = 0}^{n} \left( {\begin{array}{*{20}c} n \\ s \\ \end{array} } \right)\left( { - 1} \right)^{s} \frac{{\varGamma \left( {n + p + q + s - 1} \right)}}{{\varGamma \left( {p + q + s} \right)}}F_{{Beta\left( {p,q + s} \right)}} \left( x \right) $$

where $\varGamma \left( . \right)$ is the gamma function and $F_{{{\text{Beta}}\left( {\nu_{1} ,\nu_{2} } \right)}} \left( . \right)$ denotes the distribution function of a Beta variable with parameters $\nu_{1}$ and $\nu_{2}$ and

$$ c_{n} = \frac{{\varGamma \left( p \right)\varGamma \left( q \right)\left( {2n + p + q - 1} \right)}}{{\varGamma \left( {p + q} \right)\varGamma \left( {p + n} \right)}}\mathop \sum \limits_{s = 0}^{n} \left( {\begin{array}{*{20}c} n \\ s \\ \end{array} } \right)\left( { - 1} \right)^{s} \frac{{\varGamma \left( {n + p + q + s - 1} \right)}}{{\varGamma \left( {q + s} \right)}}E_{{H_{0} }} \left( {1 - L} \right)^{s} , $$

see Derksen and Sullivan (1990), Kharrati-Kopaei and Malekzadeh (2019), Luke (1969), Provost (2005), Reinking (2002). For the convergence of the JPE, see Alexits (1961). The parameters $p$ and q are usually determined by matching the moments of the Beta $\left( {p,q} \right)$ random variable and the statistic $L$; see Boik (1993), Bolkhovskaya et al. (2002). By matching the first two moments of Beta $\left( {p,q} \right)$ 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 $c_{0} = 1$ and $c_{1} = c_{2} = 0$. It completes the proof.

Proof of Lemma 2

Note that $Y = - 2\log \left( {LR} \right) = \mathop \sum \nolimits_{j = 1}^{k} n_{j} \log \left( {f_{j} } \right) - \mathop \sum \nolimits_{j = 1}^{k} n_{j} { \log }\left( {V_{j} /V} \right)$ where $\left( {V_{1} /V, \ldots ,V_{k} /V} \right)$ has a Dirichlet distribution with parameter $\left( {n_{1}^{*} , \ldots ,n_{k}^{*} } \right)$ under $H_{0}$. Therefore,

$$ \begin{aligned} {\text{CF}}_{Y} \left( t \right) & = E\left( {\exp \left\{ {itY} \right\}} \right) = \mathop \prod \limits_{j = 1}^{k} f_{j}^{{itn_{j} }} E\left( {\exp \left\{ { - it\mathop \sum \limits_{j = 1}^{k} n_{j} { \log }\left( {V_{j} /V} \right)} \right\}} \right) \\ & = \mathop \prod \limits_{j = 1}^{k} f_{j}^{{itn_{j} }} \mathop \int \limits_{0}^{1} \ldots \mathop \int \limits_{0}^{1} \frac{{\varGamma \left( {N^{*} } \right)}}{{\mathop \prod \nolimits_{j = 1}^{k} \varGamma \left( {n_{j}^{*} } \right)}}\mathop \prod \limits_{j = 1}^{k} x_{i}^{{ - itn_{j} }} \mathop \prod \limits_{j = 1}^{k} x_{i}^{{n_{j}^{*} - 1}} dx_{1} \ldots dx_{k} \\ & = \frac{{\varGamma \left( {N^{*} } \right)}}{{\mathop \prod \nolimits_{j = 1}^{k} \varGamma \left( {n_{j}^{*} } \right)}}\mathop \prod \limits_{j = 1}^{k} f_{j}^{{itn_{j} }} \mathop \int \limits_{0}^{1} \ldots \mathop \int \limits_{0}^{1} \mathop \prod \limits_{j = 1}^{k} x_{i}^{{n_{j}^{*} - itn_{j} - 1}} dx_{1} \ldots dx_{k} \\ & = \frac{{\varGamma \left( {N^{*} } \right)}}{{\mathop \prod \nolimits_{j = 1}^{k} \varGamma \left( {n_{j}^{*} } \right)}}\frac{{\mathop \prod \nolimits_{j = 1}^{k} \varGamma \left( {n_{j}^{*} - itn_{j} } \right)}}{{\varGamma \left( {N^{*} - itN} \right)}}\mathop \prod \limits_{j = 1}^{k} f_{j}^{{itn_{j} }} . \\ \end{aligned} $$

This completes the proof.