Obtaining the exact and near-exact distributions of the likelihood ratio statistic to test circular symmetry through the use of characteristic functions

Abstract

In this paper the authors show how through the use of the characteristic function of the negative logarithm of the likelihood ratio test (l.r.t.) statistic to test circular symmetry it is possible to obtain highly manageable expressions for the exact distribution of such statistic, when the number of variables, $p$, is odd, and highly manageable and accurate approximations for an even $p$. For the case of an even $p$, two kinds of near-exact distributions are developed for the l.r.t. statistic which correspond, for the logarithm of the l.r.t. statistic, to a Generalized Near-Integer Gamma distribution or finite mixtures of these distributions. Numerical studies conducted in order to assess the quality of these new approximations show their impressive performance, namely when compared with the only available asymptotic distribution in the literature.

Appendices

Appendix 1: Notation used in the expression of the l.r.t. statistic

In this Appendix we summarize the main results in Olkin and Press (1969) and establish the notation related with the l.r.t. statistic used in this paper.

Under $H_0$ in (1), we have that $\varvec{\Sigma }$ is circular symmetric, thus its eigenvalues are real and there exists an orthogonal matrix $\mathbf P$, such that,

$$\begin{aligned} \varvec{\Sigma }=\mathbf{ PD} _{\lambda }\mathbf P ^{\prime } \end{aligned}$$

with $\mathbf D _{\lambda }=diag\left(\lambda _{1},\ldots ,\lambda _{p}\right).$ The columns of the matrix $\mathbf P =[\mathbf u _{jk}]$ are the eigenvectors of $\varvec{\Sigma }$, corresponding to $\lambda _{1},\ldots ,\lambda _{p},$ and may be given by

$$\begin{aligned} \mathbf u _{jk}=\frac{1}{\sqrt{p}}\left\{ \,cos\left[\frac{2\pi }{p}(j{-}1)(k{-}1)\right]\!+\!sin\left[\frac{2\pi }{p}(j-1)(k\!-\!1)\right]\right\} ,\ \ j,k=1,\ldots ,p. \end{aligned}$$

We may note that the $\mathbf u _{jk}$ do not depend on the elements of $\varvec{\Sigma }$, only the eigenvalues $\lambda _1,\ldots ,\lambda _p$ do (see Olkin and Press 1969 for details).

Consider a random sample of size $N=n+1$ from the distribution $N_p\left(\varvec{\mu }, \varvec{\Sigma } \right)$ and let $\mathbf X _{N\times p}$ be the sample matrix. Let $\mathbf E _{N1}$ be an ${N\times 1}$ unitary vector. Let

$$\begin{aligned} \overline{\mathbf{X }}=\left[\overline{\mathbf{X }}_{1}\ldots \overline{\mathbf{X }}_{p}\right]=\frac{1}{N}\mathbf X ^{\prime }\mathbf E _{N1} \end{aligned}$$

be the vector of sample means, and

$$\begin{aligned} \mathbf S =\left(\mathbf X -\mathbf E _{N1}\overline{\mathbf{X }}^{\prime }\right)^{\prime }\left(\mathbf X -\mathbf E _{N1}\overline{\mathbf{X }}^{\prime }\right)\,. \end{aligned}$$

Let us take $\mathbf{y =\frac{1}{\sqrt{N}}\overline{\mathbf{X }}\mathbf P }$ and $\mathbf{V =\mathbf P ^{\prime }\mathbf SP }$. Then, $\mathbf y$ and $\mathbf V =[v_{ij}]$ are independently distributed with

$$\begin{aligned} \mathbf y \sim N_p\left(\frac{1}{\sqrt{N}}\,\varvec{\mu }\,\mathbf P ,\overline{\varvec{\Sigma }}\right)\ \ \ \ \ \ \ \ \ \ \ \mathbf V \sim W_{p}\left(N-1,\overline{\varvec{\Sigma }}\right) \end{aligned}$$

(32)

where $\overline{\varvec{\Sigma }}=\mathbf P ^{\prime }\varvec{\Sigma }\mathbf P .$ If $\varvec{\Sigma }$ is circular then

$$\begin{aligned} \overline{\varvec{\Sigma }}=\mathbf D _{\lambda }=diag\left(\lambda _{1},\ldots ,\lambda _{p}\right),\ \ \lambda _{j}=\lambda _{p-j+2}\,\quad j=2,\ldots ,p\,. \end{aligned}$$

We then define, for even $p$,

$$\begin{aligned} v_{j}=\left\{ \begin{array}{lll} v_{jj},&~~&j=1 \text{ or} j=m+1 \\ v_{jj}+v_{p-j+2,p-j+2},&\,&j=2,\ldots ,m, \end{array} \right. \end{aligned}$$

(33)

while for odd $p$,

$$\begin{aligned} v_{j}=\left\{ \begin{array}{lll} v_{jj},&~~&j=1 \\ v_{jj}+v_{p-j+2,p-j+2},&\,&j=2,\ldots ,m+1, \end{array} \right. \end{aligned}$$

(34)

with ${v_{p-j+2}=v_j}$ for ${(j=2,\ldots ,p)}$.

Appendix 2: Proof of theorems 1, 2 and 3

1.1 Proof of Theorem 1

Proof

From the expression of the c.f. of $W$ in (5) and considering that for an odd $p$ we have $m=\left\lfloor \frac{p}{2}\right\rfloor =\frac{p-1}{2}$, we may write the c.f. of $W$ as

$$\begin{aligned} \nonumber \varPhi _{W}(t)&= \left(\frac{\Gamma \left(\frac{N-1}{2}\right)\Gamma \left(\frac{N}{2}- \mathrm{i}t\right)}{\Gamma \left(\frac{N-1}{2}-\mathrm{i}t\right)\Gamma \left(\frac{N}{2}\right)}\right)^{\frac{p-1}{2}}\times \prod ^{p-1}_{j=1}\frac{\Gamma \left(\frac{N}{2}\right)\Gamma \left(\frac{N}{2}-\frac{j+1}{2}-\mathrm{i}t\right)}{\Gamma \left(\frac{N}{2}-\frac{j+1}{2}\right)\Gamma \left(\frac{N}{2}-\mathrm{i}t\right)}\\ \nonumber&= \left(\frac{\Gamma \left(\frac{N-1}{2}\right)\Gamma \left(\frac{N}{2}- \mathrm{i}t\right)}{\Gamma \left(\frac{N-1}{2}-\mathrm{i}t\right)\Gamma \left(\frac{N}{2}\right)}\right)^{\frac{p-1}{2}}\times \mathop {\prod ^{p-2}_{j=1}}_{\text{ step} 2}\frac{\Gamma \left(\frac{N}{2}-\frac{j+1}{2}+\frac{j+1}{2}\right)\Gamma \left(\frac{N}{2}-\frac{j+1}{2}-\mathrm{i}t\right)}{\Gamma \left(\frac{N}{2}-\frac{j+1}{2}\right)\Gamma \left(\frac{N}{2}-\frac{j+1}{2}+\frac{j+1}{2}-\mathrm{i}t\right)}\\ \nonumber&\times \mathop {\prod ^{p-1}_{j=2}}_{\text{ step} 2}\frac{\Gamma \left(\frac{N}{2}-\frac{j+1}{2}+\frac{j}{2}\right)\Gamma \left(\frac{N}{2}-\frac{j+1}{2}-\mathrm{i}t\right)}{\Gamma \left(\frac{N}{2}-\frac{j+1}{2}\right)\Gamma \left(\frac{N}{2}-\frac{j+1}{2}+\frac{j}{2}-\mathrm{i}t\right)}\mathop {\prod ^{p-1}_{j=2}}_{\text{ step} 2}\frac{\Gamma \left(\frac{N}{2}\right)\Gamma \left(\frac{N}{2}-\frac{j+1}{2}+\frac{j}{2}-\mathrm{i}t\right)}{\Gamma \left(\frac{N}{2}-\frac{j+1}{2}+\frac{j}{2}\right)\Gamma \left(\frac{N}{2}-\mathrm{i}t\right)}\\ \nonumber&= \left(\frac{\Gamma \left(\frac{N-1}{2}\right)\Gamma \left(\frac{N}{2}- \mathrm{i}t\right)}{\Gamma \left(\frac{N-1}{2}-\mathrm{i}t\right)\Gamma \left(\frac{N}{2}\right)}\right)^{\frac{p-1}{2}}\times \mathop {\prod ^{p-2}_{j=1}}_{\text{ step} 2}\prod _{k=0}^{\frac{j+1}{2}-1}\left(\frac{N-j-1}{2}+k\right)\left(\frac{N-j-1}{2}+k-\mathrm{i}t\right)^{-1}\\ \nonumber&\times \mathop {\prod ^{p-1}_{j=2}}_{\text{ step} 2}\prod _{k=0}^{\frac{j}{2}-1}\left(\frac{N-j-1}{2}+k\right)\left(\frac{N-j-1}{2}+k-\mathrm{i}t\right)^{-1}\left(\frac{\Gamma \left(\frac{N}{2}\right)\Gamma \left(\frac{N}{2}-\frac{1}{2}-\mathrm{i}t\right)}{\Gamma \left(\frac{N}{2}-\frac{1}{2}\right)\Gamma \left(\frac{N}{2}-\mathrm{i}t\right)}\right)^{\frac{p-1}{2}} \end{aligned}$$

noticing that

$$\begin{aligned} \left\lfloor \frac{j+1}{2} \right\rfloor =\displaystyle \left\{ \begin{array}{ll} \frac{j+1}{2}&\quad \text{ if} j \text{ is} \text{ odd}\\ \frac{j}{2}&\quad \text{ if} j \text{ is} \text{ even} \end{array} \right. \end{aligned}$$

(35)

we may write

$$\begin{aligned} \varPhi _{W}(t)&= \prod ^{p-1}_{j=1}\prod _{k=0}^{\left\lfloor \frac{j+1}{2}-1\right\rfloor }\left(\frac{N-j-1}{2}+k\right)\left(\frac{N-j-1}{2}+k-\mathrm{i}t\right)^{-1}\end{aligned}$$

(36)

$$\begin{aligned}&= \prod ^{p}_{j=2}\left(\frac{N}{2}-\frac{j}{2}\right)^{1+\left\lfloor \frac{p-j}{2}\right\rfloor }\left(\frac{N}{2}-\frac{j}{2}-\mathrm{i}t\right)^{-1-\left\lfloor \frac{p-j}{2}\right\rfloor }\,. \end{aligned}$$

(37)

Since the c.f. in (36) corresponds to the c.f. of the sum of independent Exponential r.v.’s, counting the number of Exponential distributions with the same rate parameter we obtain the representation in (37) for the c.f. of $W$. $\square$

1.2 Proof of Theorem 2

Proof

When the number of variables, $p$, is even we have that $m=\left\lfloor \frac{p}{2}\right\rfloor =\frac{p}{2}$, and then we may rewrite the c.f. of $W$ in (5) in the form

$$\begin{aligned} \nonumber \varPhi _{W}(t)=\left(\frac{\Gamma \left(\frac{N-1}{2}\right)\Gamma \left(\frac{N}{2}- \mathrm{i}t\right)}{\Gamma \left(\frac{N-1}{2}-\mathrm{i}t\right)\Gamma \left(\frac{N}{2}\right)}\right)^{\frac{p}{2}}\times \prod ^{p-1}_{j=1}\frac{\Gamma \left(\frac{N}{2}\right)\Gamma \left(\frac{N}{2}-\frac{j+1}{2}-\mathrm{i}t\right)}{\Gamma \left(\frac{N}{2}-\frac{j+1}{2}\right)\Gamma \left(\frac{N}{2}-\mathrm{i}t\right)}, \end{aligned}$$

or,

$$\begin{aligned} \nonumber \varPhi _{W}(t)&= \left(\frac{\Gamma \left(\frac{N-1}{2}\right)\Gamma \left(\frac{N}{2}- \mathrm{i}t\right)}{\Gamma \left(\frac{N-1}{2}-\mathrm{i}t\right)\Gamma \left(\frac{N}{2}\right)}\right)^{\frac{p}{2}}\\ \nonumber&\times \left\{ \mathop {\prod ^{p-1}_{j=3}}_{\text{ step} 2}\frac{\Gamma \left(\frac{N}{2}-\frac{j+1}{2}+\frac{j+1}{2}\right)\Gamma \left(\frac{N}{2}-\frac{j+1}{2}-\mathrm{i}t\right)}{\Gamma \left(\frac{N}{2}-\frac{j+1}{2}\right)\Gamma \left(\frac{N}{2}-\frac{j+1}{2}+\frac{j+1}{2}-\mathrm{i}t\right)}\right\} \times \frac{\Gamma \left(\frac{N}{2}\right)\Gamma \left(\frac{N}{2}-1-\mathrm{i}t\right)}{\Gamma \left(\frac{N}{2}-1\right)\Gamma \left(\frac{N}{2}-\mathrm{i}t\right)}\\ \nonumber&\times \mathop {\prod ^{p-2}_{j=2}}_{\text{ step} 2}\frac{\Gamma \left(\frac{N}{2}-\frac{j+1}{2}+\frac{j}{2}\right)\Gamma \left(\frac{N}{2}-\frac{j+1}{2}-\mathrm{i}t\right)}{\Gamma \left(\frac{N}{2}-\frac{j+1}{2}\right)\Gamma \left(\frac{N}{2}-\frac{j+1}{2}+\frac{j}{2}-\mathrm{i}t\right)}\mathop {\prod ^{p-2}_{j=2}}_{\text{ step} 2}\frac{\Gamma \left(\frac{N}{2}\right)\Gamma \left(\frac{N}{2}-\frac{j+1}{2}+\frac{j}{2}-\mathrm{i}t\right)}{\Gamma \left(\frac{N}{2}-\frac{j+1}{2}+\frac{j}{2}\right)\Gamma \left(\frac{N}{2}-\mathrm{i}t\right)}\\ \nonumber&= \left(\frac{\Gamma \left(\frac{N-1}{2}\right)\Gamma \left(\frac{N}{2}- \mathrm{i}t\right)}{\Gamma \left(\frac{N-1}{2}-\mathrm{i}t\right)\Gamma \left(\frac{N}{2}\right)}\right)^{\frac{p}{2}}\\ \nonumber&\times \left\{ \mathop {\prod ^{p-1}_{j=3}}_{\text{ step} 2}\prod _{k=0}^{\frac{j+1}{2}-1}\left(\frac{N-j-1}{2}+k\right)\left(\frac{N-j-1}{2}+k-\mathrm{i}t\right)^{-1}\right\} \times \frac{\Gamma \left(\frac{N}{2}\right)\Gamma \left(\frac{N}{2}-1-\mathrm{i}t\right)}{\Gamma \left(\frac{N}{2}-1\right)\Gamma \left(\frac{N}{2}-\mathrm{i}t\right)}\\ \nonumber&\times \left\{ \mathop {\prod ^{p-2}_{j=2}}_{\text{ step} 2}\prod _{k=0}^{\frac{j}{2}-1}\left(\frac{N-j-1}{2}\!+\!k\right)\left(\frac{N-j-1}{2}\!+\!k-\mathrm{i}t\right)^{-1}\right\} \left(\frac{\Gamma \left(\frac{N}{2}\right)\Gamma \left(\frac{N}{2}\!-\!\frac{1}{2}\!-\!\mathrm{i}t\right)}{\Gamma \left(\frac{N}{2}\!-\!\frac{1}{2}\right)\Gamma \left(\frac{N}{2}\!-\!\mathrm{i}t\right)}\!\right)^{\!\frac{p-2}{2}}, \end{aligned}$$

which, after some simplifications and using the equality in (35), may be written as

$$\begin{aligned} \nonumber \varPhi _{W}(t)&= \prod ^{p-1}_{j=2}\prod _{k=0}^{\left\lfloor \frac{j+1}{2}-1\right\rfloor }\left(\frac{N-j-1}{2}+k\right)\left(\frac{N-j-1}{2}+k-\mathrm{i}t\right)^{-1}\\ \nonumber&\times \frac{\Gamma \left(\frac{N-1}{2}\right)\Gamma \left(\frac{N}{2}-1-\mathrm{i}t\right)}{\Gamma \left(\frac{N}{2}-1\right)\Gamma \left(\frac{N-1}{2}-\mathrm{i}t\right)}\nonumber \\&= \underbrace{\prod ^{p}_{j{=}2}\theta _{j}^{r_{j}^*}\left(\theta _{j}-\mathrm{i}t\right)^{-r_{j}^*}}_{\varPhi _{W_{1}}(t)}{\times }\underbrace{\frac{\Gamma \left(\frac{N{-}1}{2}\right)\Gamma \left(\frac{N}{2}-1-\mathrm{i}t\right)}{\Gamma \left(\frac{N}{2}{-}1\right)\Gamma \left(\frac{N-1}{2}{-}\mathrm{i}t\right)}}_{\varPhi _{W_{2}}(t)}\, \end{aligned}$$

with $r_{j}^*$ and $\theta _{j}$ given in (11) and (7). $\square$

1.3 Proof of Theorem 3

Proof

In this proof we will consider only the case of $h=6$, since the cases $h=2$ and $h=4$ are derived in a similar way.

If in the c.f. of $W$ given in (10) we replace $\varPhi _{W_{2}}(t)$ by

$$\begin{aligned} \varPhi _{M_G}(t)=\sum ^{3}_{k=1} \omega _k\,\lambda ^{s_k}\,(\lambda -\mathrm{i}t)^{-s_k}\,, \end{aligned}$$

we obtain

$$\begin{aligned} \nonumber \varPhi _{W}(t)&\approx \varPhi _{W_{1}}(t)\times \underbrace{\sum ^{3}_{k=1} \omega _k\,\lambda ^{s_k}\,(\lambda -\mathrm{i}t)^{-s_k}}_{\varPhi ^*_{3}(t)}\\ \nonumber&\approx \sum ^{3}_{k=1} \omega _k\, \underbrace{\underbrace{\varPhi _{W_{1}}(t)}_{\text{ GIG} \text{ distribution}}\times \,\underbrace{\lambda ^{s_k}\,(\lambda -\mathrm{i}t)^{-s_k}}_{\text{ Gamma} \text{ distribution}}}_{\text{ GNIG} \text{ distribution}} \end{aligned}$$

that is the c.f. of the mixture of three GNIG distributions of depth $p$ with cdf’s and pdf’s given in (24) and (23). The parameters $\,p_{\nu }$, $s_{\nu }$ and $\lambda$ are defined in such a way that

$$\begin{aligned} \left.\frac{d^j}{dt^j}\,\varPhi _{M_G}(t)\right|_{t=0}=\left.\frac{d^j}{dt^j}\,\varPhi _{W_{2}}(t)\right|_{t=0}\,,~~~~j=1,\ldots ,6\,, \end{aligned}$$

what gives rise to a near-exact distribution that matches the first six exact moments of $W$. By simple transformation it’s easy to derive the near-exact c.d.f’s and pdf’s for $\Lambda ^{2/N}$. $\square$

Appendix 3: The Gamma, Generalized Integer Gamma (GIG) and Generalized Near-Integer Gamma (GNIG) distributions

We will use this Appendix to establish some notation concerning distributions used in the paper, as well as to give the expressions for the pdf’s and cdf’s of the GIG and GNIG distributions.

We will say that the r.v. $X$ has a Gamma distribution with rate parameter $\lambda >0$ and shape parameter $r>0$, if its pdf may be written as

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

and we will denote this fact by

$$\begin{aligned} X\sim \Gamma (r,\lambda )\,. \end{aligned}$$

Let

$$\begin{aligned} X_j\,\sim \,\Gamma (r_j,\lambda _j)~~~~~j=1,\ldots ,p \end{aligned}$$

be $p$ independent r.v.’s with Gamma distributions with shape parameters $r_j\in I\!\!N$ and rate parameters $\lambda _j>0$, with $\lambda _j\ne \lambda _{j^{\prime }}$, for all $j,j^{\prime }\in \{1,\ldots ,p\}$. We will say that then the r.v.

$$\begin{aligned} Y\,=\,\sum ^p_{j=1}X_j \end{aligned}$$

has a GIG distribution of depth $p$, with shape parameters $r_j$ and rate parameters $\lambda _j$, $(j=1,\ldots ,p)$, and we will denote this fact by

$$\begin{aligned} Y\,\sim \,GIG(r_j,\lambda _j;\,p)\,. \end{aligned}$$

The pdf and cdf of $Y$ are respectively given by (Coelho 1998)

$$\begin{aligned} f^{\scriptscriptstyle GIG}(y|r_1,\ldots ,r_p;\lambda _1,\ldots ,\lambda _p;p)\,=\,K\sum ^p_{j=1}P_j(y)\,e^{-\lambda _j\,y}\,,~~~~(y>0) \end{aligned}$$

(38)

and

$$\begin{aligned} F^{\scriptscriptstyle GIG}(y|r_1,\ldots ,r_j;\lambda _1,\ldots ,\lambda _p;p)\,=\,1-K\sum ^p_{j=1}P^*_j(y)\,e^{-\lambda _j\,y}\,,~~~~(y>0)\qquad \end{aligned}$$

(39)

where $K$ is given by (5) in Coelho (1998), and $P_j(y)$ and $P_j^*(y)$ are given by (7) and (16) in the same reference.

The GNIG distribution of depth $p+1$ (see Coelho 2004) is the distribution of the r.v.

$$\begin{aligned} Z=Y_1+Y_2 \end{aligned}$$

where $Y_1$ and $Y_2$ are independent, $Y_1$ having a GIG distribution of depth $p$ and $Y_2$ with a Gamma distribution with a non-integer shape parameter $r$ and rate parameter $\lambda \ne \lambda _j$ $(j=1,\ldots ,p)$. The pdf of $Z$ is given by

$$\begin{aligned} \begin{array}{l} \displaystyle \!\!\!\!\!\!\!\!\! f^{\scriptscriptstyle GNIG} (z|r_1,\ldots ,r_p,r;\,\lambda _1,\ldots ,\lambda _p,\lambda ;p+1) =\\ \!\!\!\displaystyle K\lambda ^r \sum \limits _{j = 1}^p {e^{ - \lambda _j z} } \sum \limits _{k = 1}^{r_j } {\left\{ {c_{j,k} \frac{{\Gamma (k)}}{{\Gamma (k\!+\!r)}}z^{k + r - 1} {}_1F_1 (r,k\!+\!r, - (\lambda \!-\!\lambda _j )z)} \right\} }\,, (z > 0) \end{array} \end{aligned}$$

(40)

and the cdf given by

$$\begin{aligned}&\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle F^{\scriptscriptstyle GNIG} (z|r_1,\ldots ,r_p,r;\,\lambda _1,\ldots ,\lambda _p,\lambda ;p\!+\!1) = \frac{\lambda ^r \,{z^r }}{{\Gamma (r\!+\!1)}}{}_1F_1 (r,r\!+\!1, - \lambda z)\nonumber \\&\!\!\!\!\!\! \displaystyle - K\lambda ^r \sum \limits _{j = 1}^p {e^{ - \lambda _j z} } \sum \limits _{k = 1}^{r_j } {c_{j,k}^* } \sum \limits _{i = 0}^{k - 1} {\frac{{z^{r + i} \lambda _j^i }}{{\Gamma (r\!+\!1\!+\!i)}}} {}_1F_1 (r,r\!+\!1\!+\!i, - (\lambda - \lambda _j )z), (z>0)\nonumber \\ \end{aligned}$$

(41)

where

$$\begin{aligned} c_{j,k}^* = \frac{{c_{j,k} }}{{\lambda _j^k }}\Gamma (k) \end{aligned}$$

with $c_{j,k}$ given by (11) through (13) in Coelho (1998). In the above expressions $_1F_1(a,b;z)$ is the Kummer confluent hypergeometric function. This function typically has very good convergence properties and is nowadays easily handled by a number of software packages.