The Likelihood Ratio Test of Equality of Mean Vectors with a Doubly Exchangeable Covariance Matrix

Abstract

The authors derive the LRT statistic for the test of equality of mean vectors when the covariance matrix has what is called a double exchangeable structure. A second expression for this statistic, based on determinants of Wishart matrices with a block-diagonal parameter matrix, allowed for the expression of the distribution of this statistic as that of a product of independent Beta random variables. Moreover, the split of the LRT statistic into three independent components, induced by this second representation, will then allow for the expression of the exact distribution of the LRT statistic in a very manageable finite closed form for most cases and the obtention of very sharp near-exact distributions for the other cases. Numerical studies show that, as expected, due to the way they are built, these near-exact distributions are indeed asymptotic not only for increasing sample sizes but also for increasing values of all other parameters in the distribution, besides lying very close to the exact distribution even for extremely small samples.

Appendices

Appendix A: Proof of Expression (16)

For ease of exposition let us split the proof of (16) in the proof of three consecutive results. First let us prove the following result.

Result 1: The MLE of $\Sigma$ under $H_1$, which is $A^*$ into (8), may be equivalently written as

$$\begin{aligned} A^*=P_{vm}\otimes \widehat{\Psi }_{1|H_1}+Q_{vm}\otimes \widehat{\Psi }_{2|H_1}+R_{vm}\otimes \widehat{\Psi }_{3|H_1}, \end{aligned}$$

(A.1)

where, from (7),

$$\begin{aligned} \begin{array}{l} \widehat{\Psi }_{1|H_1}=\widehat{U}_{|H_1}+(v-1)\widehat{V}_{|H_1}+v(m-1)\widehat{M}_{|H_1}\\ \widehat{\Psi }_{2|H_1}=\widehat{U}_{|H_1}+(v-1)\widehat{V}_{H_1}-v\widehat{M}_{|H_1}\\ \widehat{\Psi }_{3|H_1}=\widehat{U}_{|H_1}-\widehat{V}_{|H_1}\,, \end{array} \end{aligned}$$

(A.2)

and

$$\begin{aligned} \begin{array}{l} \displaystyle P_{vm}=\frac{1}{mv}J_{vm}\\ \displaystyle Q_{vm}=\frac{1}{v}(I_{m}\otimes J_v)-P_{vm}\\ \displaystyle R_{vm}=I_{vm}-Q_{vm}-P_{vm}\,. \end{array} \end{aligned}$$

(A.3)

Proof

The proof is easier if we start from (A.1) and try to end with (8), by using (A.2) and (A.3) to write

$$ \begin{array}{l} \displaystyle P_{vm}\otimes \widehat{\Psi }_{1|H_1}+Q_{vm}\otimes \widehat{\Psi }_{2|H_1}+R_{vm}\otimes \widehat{\Psi }_{3|H_1}\\ \displaystyle = \frac{1}{mv}J_{vm} \otimes \left( \widehat{U}_{|H_1}+(v-1)\widehat{V}_{|H_1}+v(m-1)\widehat{M}_{|H_1}\right) ~~~~~~~~~~~~~~~~~~~~~~~~\\ \end{array}$$

$$ \begin{array}{l} \displaystyle +\left( \frac{1}{v}(I_{m}\otimes J_v)-\frac{1}{mv}J_{vm}\right) \otimes \left( \widehat{U}_{|H_1}+(v-1)\widehat{V}_{|H_1}-v\widehat{M}_{|H_1}\right) ~~~~~~\\ +\left( I_{vm}-\frac{1}{v}(I_{m}\otimes J_v)+P_{vm}-P_{vm}\right) \otimes (\widehat{U}_{|H_1}-\widehat{V}_{|H_1})\\ \displaystyle =\left( \frac{1}{mv}J_{vm} +\frac{1}{v}(I_{m}\otimes J_v)-\frac{1}{mv}J_{vm}+I_{vm}-\frac{1}{v}(I_{m}\otimes J_v)\right) \otimes \widehat{U}_{|H_1}\\ \displaystyle +\left( \frac{v\!-\!1}{mv}J_{vm}+\frac{v\!-\!1}{v}(I_{m}\!\otimes \! J_v)-\frac{v\!-\!1}{mv}J_{vm}-I_{vm}+\frac{1}{v}(I_{m}\!\otimes \! J_v)\!\right) \!\otimes \! \widehat{V}_{|H_1}\\ \displaystyle +\left( \frac{m-1}{m}J_{vm}-(I_{m}\otimes J_v)+\frac{1}{m}J_{vm}\right) \otimes \widehat{M}_{|H_1}\\ \displaystyle =I_{vm}\otimes \widehat{U}_{|H_1}+(I_m\otimes (J_v-I_v))\otimes \widehat{V}_{|H_1}+(J_{vm}-(I_m\otimes J_m))\otimes \widehat{M}_{|H_1} \end{array}$$

which is (8).    $\square$

We now prove that $P_{vm}$, $Q_{vm}$ and $R_{vm}$ in (A.3) are indeed mutually orthogonal projectors.

Result 2 : $P_{vm}$, $Q_{vm}$ and $R_{vm}$ in (A.3) are mutually orthogonal projectors with ${\mathrm{rank}}(P_{vm})=1$, ${\mathrm{rank}}(Q_{vm})=m-1$, ${\mathrm{rank}}(R_{vm})=m(v-1)$.

Proof

$P_{vm}$, $Q_{vm}$ and $R_{vm}$ in (A.3) are idempotent, since

$$\begin{aligned} P_{vm}P_{vm}= & {} \frac{1}{mv}J_{vm}\frac{1}{mv}J_{vm}=\frac{1}{(mv)^2}mvJ_{vm}=\frac{1}{mv}J_{vm}=P_{vm},\\[5pt] Q_{vm}Q_{vm}= & {} \left( \frac{1}{v}(I_{m}\otimes J_v)-P_{vm}\right) \left( \frac{1}{v}(I_{m}\otimes J_v)-P_{vm}\right) \\= & {} \frac{1}{v^2}(I_{m}\!\otimes \! J_v)(I_{m}\!\otimes \! J_v)-\frac{1}{v}(I_{m}\!\otimes \! J_v)P_{vm}-\frac{1}{v}P_{vm}(I_{m}\!\otimes \! J_v)+P_{vm} \\= & {} \frac{1}{v}(I_{m}\otimes J_v)-\frac{2}{vm}J_{vm}+\frac{1}{vm}J_{vm}=\frac{1}{v}(I_{m}\otimes J_v)-P_{vm}=Q_{vm}, \end{aligned}$$

$$\begin{aligned} R_{vm}R_{vm}= & {} (I_{vm}-Q_{vm}-P_{vm})(I_{vm}-Q_{vm}-P_{vm})=I_{vm}-Q_{vm}-P_{vm}\\&-Q_{vm}+Q_{vm}Q_{vm}+Q_{vm}P_{vm}-P_{vm}+P_{vm}Q_{vm}+P_{vm}P_{vm}\\ {}= & {} I_{vm}-Q_{vm}-P_{vm}-Q_{vm}+Q_{vm}-P_{vm}+P_{vm}\\= & {} I_{vm}-Q_{vm}-P_{vm}=R_{vm}, \end{aligned}$$

and they are mutually orthogonal since

$$ \begin{array}{rcl} \displaystyle P_{vm}Q_{vm}&{}=&{}P_{vm}\left( \frac{1}{v}(I_{m}\otimes J_v)-P_{vm}\right) =\frac{1}{v}P_{vm}(I_{m}\otimes J_v)-P_{vm}\\ &{}=&{}\displaystyle \frac{1}{vm}J_{vm}\left( \frac{1}{v}(I_{m}\otimes J_v)-I_{vm}\right) =0_{vm},\\ R_{vm}P_{vm}&{}=&{}(I_{vm}-Q_{vm}-P_{vm})P_{vm}=P_{vm}-P_{vm}=0_{vm},\\ R_{vm}Q_{vm}&{}=&{}(I_{vm}-Q_{vm}-P_{vm})Q_{vm}=Q_{vm}-Q_{vm}=0_{vm}. \end{array}$$

Since $P_{vm}$, $Q_{vm}$ and $R_{vm}$ are idempotent, we have

$$ \begin{array}{l} \displaystyle {\mathrm{rank}}(P_{vm})=\mathrm{tr}(P_{vm})=\frac{1}{vm}\mathrm{tr}(I_{vm})=1\\ \displaystyle {\mathrm{rank}}(Q_{vm})=\mathrm{tr}(Q_{vm})=\frac{1}{v}\mathrm{tr}(I_{m}\otimes J_v)\!-\!\mathrm{tr}(P_{vm})=\frac{mv}{v}\!-\!1=m\!-\!1\\ \displaystyle {\mathrm{rank}}(R_{vm})=\mathrm{tr}(R_{vm})=\mathrm{tr}(I_{vm})\!-\!\mathrm{tr}(Q_{vm})\!-\!\mathrm{tr}(P_{vm})=vm\!-\!m=(v\!-\!1)m. \end{array}$$

   $\square$

Setting together the previous two results with the next one, we will finally be able to prove (16).

Result 3 : For the matrix $A^{**}$ in (14), we may write

$$\begin{aligned} \widehat{\Psi }_{1|H_1}= & {} A_1^{**},\\ \widehat{\Psi }_{2|H_1}= & {} \frac{1}{m-1}\sum _{j=1}^{m-1}A_{jv+1}^{**},\\ \widehat{\Psi }_{3|H_1}= & {} \frac{1}{m(v-1)}\sum _{j=1}^m\sum _{k=1}^{v-1}A^{**}_{(j-1)v+k+1}\,. \end{aligned}$$

Proof

Let ${\mathrm{BTr}}_r(A)$ denote the sum of all $r{\scriptstyle \times }r$ diagonal blocks of A. Then, for $U_{|H_1}$, $V_{|H_1}$ and $W_{|H_1}$ in (9)–(11) we may write

$$\begin{aligned} \widehat{U}_{|H_1}= & {} \frac{1}{mv}{\mathrm{BTr}}_r(A),\\ \widehat{V}_{|H_1}= & {} \frac{1}{mv(v-1)}\left( {\mathrm{BTr}}_r((I_m\otimes J_v\otimes I_r)A)-{\mathrm{BTr}}_r(A)\right) ,\\ \widehat{M}_{|H_1}= & {} \frac{1}{mv^2(m-1)}\left( {\mathrm{BTr}}_r((J_{vm}\otimes I_r)A)-{\mathrm{BTr}}_r((I_m\otimes J_v\otimes I_r)A)\right) . \end{aligned}$$

Then, since from (14) we may write

$$\begin{aligned} A=\Gamma ' A^{**}\Gamma \,, \end{aligned}$$

(A.4)

from (A.2), and from the definition of the matrix $\Gamma$ in (5), we have

$$\begin{aligned} \widehat{\Psi }_{1|H_1}= & {} \widehat{U}_{|H_1}+(v-1)\widehat{V}_{|H_1}+v(m-1)\widehat{M}_{|H_1}\\[5pt]= & {} \frac{1}{mv}{\mathrm{BTr}}_r(A)+\frac{1}{mv}\left( {\mathrm{BTr}}_r((I_m\otimes J_v\otimes I_r)A)-{\mathrm{BTr}}_r(A)\right) \\&+\frac{1}{mv}\left( {\mathrm{BTr}}_r((J_{vm}\otimes I_r)A)-{\mathrm{BTr}}_r((I_m\otimes J_v\otimes I_r)A)\right) \\[5pt]= & {} \frac{1}{mv}{\mathrm{BTr}}_r((J_{vm}\otimes I_r)A)={\mathrm{BTr}}_r((P_{vm}\otimes I_r)A)\\[5pt]= & {} {\mathrm{BTr}}_r((P_{vm}\otimes I_r)\Gamma ' A^{**}\Gamma ) ={\mathrm{BTr}}_r(\Gamma (P_{vm}\otimes I_r)\Gamma ' A^{**})\\= & {} {\mathrm{BTr}}_r((\,\underbrace{((\Gamma _m\otimes \Gamma _v)P_{vm}(\Gamma _m\otimes \Gamma _v)')}_{=diag(\underbrace{1,0,\ldots ,0}_{vm})}\otimes I_r) A^{**})=A_1^{**}\,, \end{aligned}$$

where $A_1^{**}$ represents the first diagonal block of $A^{**}$ of dimensions $r{\scriptstyle \times }r$, since, given the definition of a Helmert matrix, $\Gamma _m\otimes \Gamma _v$ is a $vm{\scriptstyle \times }vm$ matrix whose first row is equal to $\frac{1}{\sqrt{vm}}\underline{1}'_{vm}$, where $\underline{1}_{vm}$ denotes a vector of 1’s of dimension vm, and as such $(\Gamma _m\otimes \Gamma _v)P_{vm}$ is a $vm{\scriptstyle \times }vm$ matrix whose first row is equal to $\frac{1}{\sqrt{vm}}\underline{1}'_{vm}$ and all other rows are null, so that

$$\begin{aligned} (\Gamma _m\otimes \Gamma _v)P_{vm}(\Gamma _m\otimes \Gamma _v)'=diag(\,\underbrace{1,0,\ldots ,0}_{vm}\,)\,, \end{aligned}$$

(A.5)

what implies

$$ ((\Gamma _m\otimes \Gamma _v)P_{vm}(\Gamma _m\otimes \Gamma _v)')\otimes I_r=bdiag(I_r,0_{v(m-1){\scriptstyle \times }v(m-1)})\,. $$

Using (A.4), and once again the definition of the matrix $\Gamma$ in (5), we also have

$$\begin{aligned} \widehat{\Psi }_{2|H_1}= & {} \widehat{U}_{|H_1}+(v-1)\widehat{V}_{|H_1}-v\widehat{M}_{|H_1}\\= & {} \frac{1}{mv}{\mathrm{BTr}}_r(A)+\frac{1}{mv}\left( {\mathrm{BTr}}_r((I_m\otimes J_v\otimes I_r)A)-{\mathrm{BTr}}_r(A)\right) \\&-\frac{1}{mv(m-1)}\left( {\mathrm{BTr}}_r((J_{vm}\otimes I_r)A)-{\mathrm{BTr}}_r((I_m\otimes J_v\otimes I_r)A)\right) \\= & {} \frac{1}{mv}\left( 1+\frac{1}{m-1}\right) {\mathrm{BTr}}_r((I_m\otimes J_v\otimes I_r)A)\\&-\frac{1}{mv(m-1)}{\mathrm{BTr}}_r((J_{vm}\otimes I_r)A)\\= & {} \frac{1}{m-1}{\mathrm{BTr}}_r\left( \left( \left( \frac{1}{v}(I_m\otimes J_v)-\frac{1}{mv}J_{vm}\right) \otimes I_r\right) A\right) \\ {}= & {} \frac{1}{m-1}{\mathrm{BTr}}_r\left( \left( Q_{vm}\otimes I_r\right) A\right) =\frac{1}{m-1}{\mathrm{BTr}}_r((Q_{vm}\otimes I_r)\Gamma ' A^{**}\Gamma )\\= & {} \frac{1}{m-1}{\mathrm{BTr}}_r(\Gamma (Q_{vm}\otimes I_r)\Gamma ' A^{**})\\= & {} \frac{1}{m-1}{\mathrm{BTr}}_r((((\Gamma _m\otimes \Gamma _v)Q_{vm}(\Gamma _m\otimes \Gamma _v))\otimes I_r) A^{**})\\= & {} \frac{1}{m-1}\sum _{j=1}^{m-1}A_{jv+1}^{**}\,, \end{aligned}$$

where $A_j^{**}$ represents the j-th diagonal block of $A^{**}$ of dimensions $r{\scriptstyle \times }r$, since

$$\begin{aligned} (\Gamma _m\otimes \Gamma _v)Q_{vm}(\Gamma _m\otimes \Gamma _v)'=diag(\,\underbrace{0,\ldots ,0}_{v},\underbrace{\underbrace{1,0,\ldots ,0}_{v},\ldots ,\underbrace{1,0,\ldots ,0}_{v}}_{m-1 \hbox {~times}}\,) \end{aligned}$$

(A.6)

and as such,

$$ \begin{array}{l} \displaystyle ((\Gamma _m\otimes \Gamma _v)Q_{vm}(\Gamma _m\otimes \Gamma _v)')\otimes I_r\\ \displaystyle =bdiag(I_{vr{\scriptstyle \times }vr},\underbrace{\underbrace{I_r, 0_{(v-1)r{\scriptstyle \times }(v-1)r}},\dots ,\underbrace{I_r,0_{(v-1)r{\scriptstyle \times }(v-1)r}}}_{m-1})\,. \end{array}$$

This is so because, given the definition of a Helmert matrix, for a $v{\scriptstyle \times }v$ Helmert matrix $\Gamma _v$, we have that $\Gamma _vJ_v$ is a matrix whose first row is equal to $\frac{1}{\sqrt{v}}\underline{1}'_v$, and as such we have

$$ \Gamma _vJ_v\Gamma _v'=diag(\,\underbrace{v,0,\dots ,0}_{v}\,)\,, $$

so that we have

$$ \begin{array}{l} \displaystyle \frac{1}{v}\left( I_m\otimes \Gamma _v J_v\Gamma '_v\right) \\ \displaystyle = bdiag(\underbrace{diag(\,\underbrace{1,0,\dots ,0}_{v}\,),diag(\,\underbrace{1,0,\dots ,0}_{v}\,),\dots ,diag(\,\underbrace{1,0,\dots ,0}_{v}\,)}_{m})\\ \displaystyle = diag(\,\underbrace{\underbrace{1,0,\dots ,0}_{v},\underbrace{1,0,\dots ,0}_{v},\dots ,\underbrace{1,0,\dots ,0}_{v}}_{m}\,) \end{array}$$

and as such we may write, using the relation in (A.5),

$$\begin{aligned}&\displaystyle (\Gamma _m\otimes \Gamma _v)Q_{vm}(\Gamma _m\otimes \Gamma _v)'\\&\displaystyle = (\Gamma _m\otimes \Gamma _v)\left( \frac{1}{v}(I_m\otimes J_v)-p_{vm}\right) (\Gamma _m\otimes \Gamma _v)'\\&\displaystyle = \left( \frac{1}{v}(\Gamma _m\otimes \Gamma _v J_v)-(\Gamma _m\otimes \Gamma _v)P_{vm}\right) (\Gamma _m\otimes \Gamma _v)'\\&\displaystyle = \frac{1}{v}(\Gamma _m\otimes \Gamma _v J_v)(\Gamma _m\otimes \Gamma _v)'-(\Gamma _m\otimes \Gamma _v)P_{vm}(\Gamma _m\otimes \Gamma _v)'\\&\displaystyle = \frac{1}{v}(\Gamma _m\Gamma _m'\otimes \Gamma _v J_v\Gamma _v')-(\Gamma _m\otimes \Gamma _v)P_{vm}(\Gamma _m\otimes \Gamma _v)'~~~~~~~~~~~~~~~~~\\&\displaystyle = \frac{1}{v}(I_m\otimes \Gamma _v J_v\Gamma _v'))-diag(\,\underbrace{1,0,\dots ,0}_{vm}\,)\\&\displaystyle = diag(\,\underbrace{\underbrace{1,0,\dots ,0}_{v},\underbrace{1,0,\dots ,0}_{v},\dots ,\underbrace{1,0,\dots ,0}_{v}}_{m}\,)-diag(\,\underbrace{1,0,\dots ,0}_{vm}\,)\\&\displaystyle = diag(\,\underbrace{0,\ldots ,0}_{v},\underbrace{\underbrace{1,0,\ldots ,0}_{v},\ldots ,\underbrace{1,0,\ldots ,0}_{v}}_{m-1\hbox {~times}}\,)\,. \end{aligned}$$

And finally, using once again (A.4), and the definition of $R_{vm}$ in (A.3), we have

$$\begin{aligned} \widehat{\Psi }_{3|H_1}= & {} \widehat{U}_{|H_1}-\widehat{V}_{|H_1}\\= & {} \frac{1}{mv}{\mathrm{BTr}}_r(A)-\frac{1}{mv(v-1)}\left( {\mathrm{BTr}}_r((I_m\otimes J_v\otimes I_r)A)-{\mathrm{BTr}}_r(A)\right) \\= & {} \frac{1}{mv}\left( 1+\frac{1}{v-1}\right) {\mathrm{BTr}}_r(A)-\frac{1}{mv(v-1)}{\mathrm{BTr}}_r((I_m\otimes J_v\otimes I_r)A)\\= & {} \frac{1}{m(v-1)}\left( {\mathrm{BTr}}_r(A)-\frac{1}{v}{\mathrm{BTr}}_r((I_m\otimes J_v\otimes I_r)A)\right) \\= & {} \frac{1}{m(v-1)}{\mathrm{BTr}}_r\left( (I_{mvr}-\frac{1}{v}(I_m\otimes J_v\otimes I_r))A\right) \\= & {} \frac{1}{m(v-1)}{\mathrm{BTr}}_r\left( ((I_{mv}-\frac{1}{v}(I_m\otimes J_v))\otimes I_r)A\right) \\= & {} \frac{1}{m(v-1)}{\mathrm{BTr}}_r\left( (R_{vm}\otimes I_r)A\right) \\= & {} \frac{1}{m(v-1)}{\mathrm{BTr}}_r\left( (R_{vm}\otimes I_r)\Gamma ' A^{**} \Gamma \right) \\= & {} \frac{1}{m(v-1)}{\mathrm{BTr}}_r\left( \Gamma (R_{vm}\otimes I_r)\Gamma ' A^{**} \right) \\= & {} \frac{1}{m(v-1)}{\mathrm{BTr}}_r((((\Gamma _m\otimes \Gamma _v) R_{vm}(\Gamma _m\otimes \Gamma _v)')\otimes I_r) A^{**} )\\= & {} \frac{1}{m(v-1)}\sum _{j=1}^m\sum _{k=1}^{v-1}A^{**}_{(j-1)v+k+1}\,,\\[-15pt] \end{aligned}$$

where, once again, $A^{**}_j$ represents the j-th diagonal block of $A^{**}$ of dimensions $r{\scriptstyle \times }r$.

This is so given that from the definition of $R_{vm}$ in (A.3) and from (A.5) and (A.6) it is clear that we have

$$ \begin{array}{l} \displaystyle (\Gamma _m\otimes \Gamma _v)R_{vm}(\Gamma _m\otimes \Gamma _v)'\\ \displaystyle =(\Gamma _m\otimes \Gamma _v)(\Gamma _m\otimes \Gamma _v)'-(\Gamma _m\otimes \Gamma _v)Q_{vm}(\Gamma _m\otimes \Gamma _v)'\\ \displaystyle -(\Gamma _m\otimes \Gamma _v)P_{vm}(\Gamma _m\otimes \Gamma _v)'~~~\\ \displaystyle =I_{vm}-diag(\,\underbrace{0,\ldots ,0}_{v},\underbrace{\underbrace{1,0,\ldots ,0}_{v},\ldots ,\underbrace{1,0,\ldots ,0}_{v}}_{m-1\hbox {~times}}\,)-diag(\,\underbrace{1,0,\ldots ,0}_{vm}\,)\\ \displaystyle =diag(0,\underbrace{\underbrace{1,\ldots ,1}_{v-1},0,\underbrace{1,\ldots ,1}_{v-1},\ldots ,0, \underbrace{1,\ldots ,1}_{v-1}}_{m\hbox {~times}}\,)\,, \end{array}$$

so that

$$ \begin{array}{l} \displaystyle ((\Gamma _m\otimes \Gamma _v)R_{vm}(\Gamma _m\otimes \Gamma _v)')\otimes I_r\\ \displaystyle =bdiag(\,\underbrace{0_{r{\scriptstyle \times }r},I_{(v-1)r},0_{r{\scriptstyle \times }r},I_{(v-1)r},\dots ,0_{r{\scriptstyle \times }r},I_{(v-1)r}}_{m\hbox {~times}}\,). \end{array}$$

   $\square$

Finally, expression (16) holds since from Result 1 and the mutual orthogonality of $P_{vm}$, $Q_{vm}$ and $R_{vm}$ proven in Result 2 and yet the rank results therein, we may write

$$ |A^*|=|\widehat{\Psi }_{1|H_1}|\,|\widehat{\Psi }_{2|H_1}|^{m-1}\,|\widehat{\Psi }_{3|H_1}|^{m(v-1)}, $$

which given the expressions obtained in Result 3 for $\widehat{\Psi }_{1|H_1}$, $\widehat{\Psi }_{2|H_1}$ and $\widehat{\Psi }_{3|H_1}$ yields (16).

Expression (17) would be proven in a completely similar way.

Appendix B: The GIG, EGIG and GNIG distributions

In this appendix we define the GIG (Generalized Integer Gamma), EGIG (Exponentiated Generalized Integer Gamma) and GNIG (Generalized Near-Integer Gamma) distributions and establish the notation used concerning their p.d.f.’s and c.d.f.’s.

We say that the r.v. (random variable) X has a Gamma distribution with shape parameter ${r\,(>0)}$ and rate parameter ${\lambda \,(>0)}$, and we will denote this fact by $X\sim \Gamma (r,\lambda )$, if the p.d.f. of X is

$$ f^{}_X(x)=\frac{\lambda ^r}{\Gamma (r)}\,e^{-\lambda x}\,x^{r-1}~~~~(x>0)\,. $$

Let $X_j\sim \Gamma (r_j,\lambda _j)$ ${(j=1,\dots ,p)}$ be a set of p independent r.v.’s and consider the r.v.

$$ W=\sum ^p_{j=1}X_j\,. $$

In case all the $r_j\in \mathbb {N}$, the distribution of W is what we call a GIG distribution (Coelho 1998).   If all the $\lambda _j$ are different, W has a GIG distribution of depth p, with shape parameters $r_j$ and rate parameters $\lambda _j$, with p.d.f.

$$ f^{}_W(w)=f^{{ \,GIG}}\Bigl (w\,\bigl |\,\{r_j\}_{j=1:p};\{\lambda _j\}_{j=1:p};p\Bigr )=K\sum ^p_{j=1} P_j(w)\,e^{-\lambda _j w}\,, $$

and c.d.f.

$$ F^{}_W(w)=F^{{ \,GIG}}\Bigl (w\,\bigl |\,\{r_j\}_{j=1:p};\{\lambda _j\}_{j=1:p};p\Bigr )=1-K\sum ^p_{j=1}P^*_j(w)\,e^{-\lambda _jw}\,, $$

for ${w>0}$, where

$$\begin{aligned} K=\prod ^p_{j=1}\lambda _j^{r_j}\,,~~~~~~P_j(w)=\sum ^{r_j}_{k=1} c_{j,k}\,w^{k-1} \end{aligned}$$

(B.1)

and

$$\begin{aligned} P^*_j(w)=\sum ^{r_j}_{k=1}c_{j,k}(k-1)!\sum ^{k-1}_{i=0}\frac{w^i}{i!\,\lambda _j^{k-i}}\,, \end{aligned}$$

(B.2)

with

$$\begin{aligned} c_{j,r_j}=\frac{1}{(r_j-1)!}\mathop {\prod ^p_{i=1}}_{i\ne j}(\lambda _i-\lambda _j)^{-r_i}\,,~~~~j=1,\dots ,p\,, \end{aligned}$$

(B.3)

and, for $k=1,\dots ,r_j-1$ and $j=1,\dots ,p$,

$$\begin{aligned} c_{j,r_j-k}=\frac{1}{k}\sum _{i=1}^k\frac{(r_j-k+i-1)!}{(r_j-k-1)!}\,R(i,j,p)c_{j,r_j-(k-i)}\,, \end{aligned}$$

(B.4)

where

$$\begin{aligned} R(i,j,p)=\mathop {\sum ^p_{k=1}}_{k\ne j}r_k(\lambda _j-\lambda _k)^{-i}~~~~(i=1,\dots ,r_j-1)\,. \end{aligned}$$

(B.5)

In case some of the $\lambda _j$ assume the same value as other $\lambda _j$’s, the distribution of W still is a GIG distribution, but in this case with a reduced depth. In this more general case, let $\widetilde{\{}\lambda _\ell \widetilde{\}}_{\ell =1:g(\le p)}$ be the set of different $\lambda _j$’s and let ${\mathop {\{}\limits ^{\approx }}\!\!r_\ell \!\!{\mathop {\}}\limits ^{\approx }}_{\ell =1:g(\le p)}$ be the set of the corresponding shape parameters, with $r_\ell$ $(\ell =1,\dots ,g)$ being the sum of all $r_j$ ${(j\in \{1,\dots ,p\})}$ which correspond to the $\lambda _j$ assuming the value $\lambda _\ell$. In this case W will have a GIG distribution of depth g, with shape parameters $r_\ell$ and rate parameters $\lambda _\ell$ ${(\ell =1,\dots ,g)}$.

The r.v. ${Z=e^{-W}}$ has then what Arnold et al. (2013) call an Exponentiated Generalized Integer Gamma (EGIG) distribution of depth g, with p.d.f.

$$\begin{aligned} \begin{array}{rcl} \displaystyle f^{}_Z(z) &{} = &{} \displaystyle f^{{ EGIG}}\biggl (z~\bigl |\,{\mathop {\{}\limits ^{\approx }}\!\!r_\ell \!\!{\mathop {\}}\limits ^{\approx }}_{\ell =1:g};\widetilde{\{}\lambda _\ell \widetilde{\}}_{\ell =1:g};g\biggr )\\ &{} = &{} \displaystyle f^{{ \,GIG}}\biggl (-\log \,z~\bigl |\,{\mathop {\{}\limits ^{\approx }}\!\!r_\ell \!\!{\mathop {\}}\limits ^{\approx }}_{j=1:g};\widetilde{\{}\lambda _\ell \widetilde{\}}_{\ell =1:g};g\biggr )\,\frac{1}{z}\\ &{} = &{} \displaystyle K^*\sum _{\ell =1}^g P_\ell (-\log \,z)\,z^{\lambda _\ell -1}~~~~~~~~~~(0<z<1) \end{array} \end{aligned}$$

(B.6)

and c.d.f.

$$\begin{aligned} \begin{array}{rcl} \displaystyle F^{}_Z(z) &{} = &{} \displaystyle F^{{ EGIG}}\biggl (z~\bigl |\,{\mathop {\{}\limits ^{\approx }}\!\!r_\ell \!\!{\mathop {\}}\limits ^{\approx }}_{\ell =1:g};\widetilde{\{}\lambda _\ell \widetilde{\}}_{\ell =1:g};g\biggr )\\ &{} = &{} \displaystyle 1-F^{{ \,GIG}}\biggl (-\log \,z~\bigl |\,{\mathop {\{}\limits ^{\approx }}\!\!r_\ell \!\!{\mathop {\}}\limits ^{\approx }}_{\ell =1:g};\widetilde{\{}\lambda _\ell \widetilde{\}}_{\ell =1:g};g\biggr )\\ &{} = &{} \displaystyle K^*\sum _{\ell =1}^g P^*_\ell (-\log \,z)\,z^{\lambda _\ell }~~~~~~~~~~(0<z<1)\,, \end{array} \end{aligned}$$

(B.7)

for

$$\begin{aligned} K^*=\prod _{\ell =1}^g \lambda _\ell ^{r_\ell } \end{aligned}$$

(B.8)

and $P_\ell (\,\cdot \,)$ and $P^*_\ell (\,\cdot \,)$ given by (B.1) and (B.2), with j replaced by $\ell$.

Then let W be a r.v. with a GIG distribution of depth g, with rate parameters $\widetilde{\{}\lambda _\ell \widetilde{\}}_{\ell =1:g}$ and shape parameters ${\mathop {\{}\limits ^{\approx }}\!\!r_\ell \!\!{\mathop {\}}\limits ^{\approx }}_{\ell =1:g}$ $(\in \mathbb N)$ and let $W^*\sim \Gamma (r,\lambda )$, with $r\in {\mathbb R}^+\backslash {\mathbb N}$. Let further W and $W^*$ be two independent r.v.’s. Then the r.v.

$$ Y=W+W^* $$

has a Generalized Near-Integer Gamma (GNIG) distribution (Coelho 2004) of depth ${g+1}$, with rate parameters $\widetilde{\{}\lambda _\ell \widetilde{\}}_{\ell =1:g}$ and $\lambda$ and corresponding shape parameters ${\mathop {\{}\limits ^{\approx }}\!\!r_\ell \!\!{\mathop {\}}\limits ^{\approx }}_{\ell =1:g}$ $(\in \mathbb N)$ and r, with p.d.f.

$$ \begin{array}{rcl} f_{Y}^{}(y) &{} = &{} \displaystyle f^{{ \,GNIG}}\Bigl (y\,|\,{\mathop {\{}\limits ^{\approx }}\!\!r_\ell \!\!{\mathop {\}}\limits ^{\approx }}_{\ell =1:g},r;\widetilde{\{}\lambda _\ell \widetilde{\}}_{\ell =1:g},\lambda ;g+1\Bigr )\\ &{} = &{} \displaystyle K^*\lambda ^r\sum _{\ell =1}^g e^{-\lambda _\ell y}\sum _{k=1}^{r_\ell }\left\{ c_{\ell ,k}\frac{\Gamma (k)}{\Gamma (k+r)}y^{\,k+r-1}\,\!_1F_1\bigl (r,k\!+\!r,-(\lambda \!-\!\lambda _\ell )y\bigr )\right\} \end{array}$$

and c.d.f.

$$ \begin{array}{rcl} F_{Y}^{}(y) &{} = &{} \displaystyle F^{{ \,GNIG}}\Bigl (y\,|\,{\mathop {\{}\limits ^{\approx }}\!\!r_\ell \!\!{\mathop {\}}\limits ^{\approx }}_{\ell =1:g},r;\widetilde{\{}\lambda _\ell \widetilde{\}}_{\ell =1:g},\lambda ;g+1\Bigr )\\ &{} = &{} \displaystyle \frac{\lambda ^r\,y^r}{\Gamma (r+1)}\,_1F_1\bigl (r,r+1,-\lambda y\bigr )\\ &{} &{} \displaystyle -K^{**}\! \sum _{\ell =1}^g e^{-\lambda _\ell y}\sum _{k=1}^{r_\ell }c^*_{\ell ,k}\sum _{i=0}^{k-1}\left\{ \!\frac{y^{\,r+i}\,\lambda _\ell ^i}{\Gamma (r^*\!+\!i)}\,_1F_1\bigl (r,r^*\!+\!i,-(\lambda \!-\!\lambda _\ell )y\bigr )\!\right\} \end{array}$$

where ${z>0}$, $K^{**}=K^*\lambda ^r$, for $K^*$ in (B.8), ${r^*=r+1}$, and

$$ c^*_{\ell ,k}=\frac{c_{\ell ,k}}{\lambda _\ell ^k}\,\Gamma (k) $$

for $c_{\ell ,k}$ given by (B.3)–(B.5) with j replaced by $\ell$, and

$$ _1F_1(a,b,y)=\sum _{i=0}^\infty \frac{\Gamma (a+i)}{\Gamma (b+i)}\frac{\Gamma (b)}{\Gamma (a)}\,\frac{z^i}{i!} $$

is the Kummer confluent hypergeometric function.