Distribution under elliptical symmetry of a distance-based multivariate coefficient of variation

Abstract

In the univariate setting, the coefficient of variation is widely used to measure the relative dispersion of a random variable with respect to its mean. Several extensions of the univariate coefficient of variation to the multivariate setting have been introduced in the literature. In this paper, we focus on a distance-based multivariate coefficient of variation. First, some real examples are discussed to motivate the use of the considered multivariate dispersion measure. Then, the asymptotic distribution of several estimators is analyzed under elliptical symmetry and used to construct approximate parametric confidence intervals that are compared with non-parametric intervals in a simulation study. Under normality, the exact distribution of the classical estimator is derived. As this natural estimator is biased, some bias corrections are proposed and compared by means of simulations.

Appendix

Proof

(Proposition 1) Let $\mathbf {X}=( \mathbf {X}_{1},\ldots , \mathbf {X}_{n})$ be a sequence of n independent p-variate random vectors and $g_{\mathsf {A}}$ be a transformation on $\mathbf {X}$ defined by $g_{{\mathsf {A}}}: \mathbf {X} \mapsto ({\mathsf {A}}\mathbf {X}_{1}, \ldots , {\mathsf {A}}\mathbf {X}_{n})$ where ${\mathsf {A}}$ is a $p\times p$ non-singular matrix.

First, since the location and covariance estimators $\mathbf {T}_n(\mathbf {X})$ and ${\mathsf {C}}_n(\mathbf {X})$ are affine equivariant, the estimator $V_n(\mathbf {X})$ is invariant under $g_{\mathsf {A}}$ as detailed below:

$$\begin{aligned} V_n(g_{\mathsf {A}}(\mathbf {X}))&= \frac{1}{\sqrt{\mathbf {T}_n(\mathbf {X})^t {\mathsf {A}}^t ({\mathsf {A}}^t)^{-1} {\mathsf {C}}_n(\mathbf {X}) {\mathsf {A}}^{-1}{\mathsf {A}}\mathbf {T}_n(\mathbf {X})}} = V_n(\mathbf {X}) \end{aligned}$$

Now, let $F_{\varvec{\mu }, {\mathsf {\Sigma }}}$ and $F_{\varvec{\mu }',{\mathsf {\Sigma }}'}$ be two distributions belonging to $\mathscr {F}_h$ and having the same theoretical coefficient of variation $\gamma = \gamma '$. As $\sqrt{\varvec{\mu }^t {\mathsf {\Sigma }}^{-1} \varvec{\mu }}=\Vert {\mathsf {\Sigma }}^{-1/2} \varvec{\mu }\Vert$ where $\Vert .\Vert$ is the Euclidean norm, the equality $\gamma = \gamma '$ implies the equality of the two norms $\Vert {\mathsf {\Sigma }}^{-1/2} \varvec{\mu }\Vert =\Vert {\mathsf {\Sigma }}'^{-1/2} \varvec{\mu }'\Vert$. Therefore, there exists an orthogonal matrix ${\mathsf {B}}$ such that ${\mathsf {\Sigma }} ^{-1/2}\varvec{\mu }= {\mathsf {B}} {\mathsf {\Sigma }}'^{-1/2}\varvec{\mu }'.$

Take as matrix ${\mathsf {A}}$ the non-singular matrix ${\mathsf {\Sigma }}^{1/2} {\mathsf {B}} {\mathsf {\Sigma }}'^{-1/2}$. It follows directly that $F_{\varvec{\mu },{\mathsf {\Sigma }}} = F_{{\mathsf {A}}\varvec{\mu }', {\mathsf {A}} {\mathsf {\Sigma }}'{\mathsf {A}}^{t}}$. From there, it comes

$$\begin{aligned} \text{ G }_{F_{\varvec{\mu }, {\mathsf {\Sigma }}}} \left[ V_n(\mathbf {X}) \right] = \text{ G }_{F_{{\mathsf {A}} \varvec{\mu }', {\mathsf {A}} {\mathsf {\Sigma }}'{\mathsf {A}}^t}} \left[ V_n(\mathbf {X}) \right] = \text{ G }_{F_{\varvec{\mu }', {\mathsf {\Sigma }}'}} \left[ V_n(g_{\mathsf {A}} (\mathbf {X}))\right] = \text{ G }_{F_{\varvec{\mu }',{\mathsf {\Sigma }}'}} \left[ V_n(\mathbf {X}) \right] \end{aligned}$$

where $\text{ G }_{F}\left[ . \right]$ corresponds to the distribution of [.] computed under the assumption that $\mathbf {X} _{i} \sim F$ for $i= 1,\ldots ,n$. This concludes the proof. $\square$

Proof

(Proposition 2) For affine-equivariant estimators $\mathbf {T}_n:=\mathbf {T}_n(\mathbf {X})$ and ${\mathsf {C}}_n:={\mathsf {C}}_n(\mathbf {X})$ satisfying (A1), (A2) and (A3), their joint asymptotic distribution is given by

$$\begin{aligned} N_{p^2+p}( (\varvec{\mu }, \text{ vec }{\mathsf {\Sigma }})^t, V)&\text { with }&V = \begin{pmatrix} \tau {\mathsf {\Sigma }} &{} 0_{p\times p^2}\\ 0_{p\times p^2}^t &{} {\mathsf {\Xi }}\\ \end{pmatrix} \end{aligned}$$

The delta method for the function f defined by $f: \mathbb {R}^{p+p^2} \rightarrow \mathbb {R}: W=(\mathbf {T}_n, \text{ vec }{\mathsf {C}}_n ) \mapsto (\mathbf {T}_n^t {\mathsf {C}}_n^{-1}\mathbf {T}_n)^{-1/2}= V_n$ allows to say that

$$\begin{aligned} \sqrt{n}( V_n - \gamma ) \mathop {\longrightarrow }\limits ^{\mathscr {L}} N \left( 0, \nabla f(\varvec{\mu }, \text{ vec }{\mathsf {\Sigma }})^t ~V ~\nabla f (\varvec{\mu }, \text{ vec }{\mathsf {\Sigma }}) \right) \end{aligned}$$

where $\nabla f$ denotes the vector of partial derivatives of f.

The following identities can be derived from properties of the $\text{ vec }$ operator and the Kronecker product (see for instance Magnus and Neudecker 2007):

$$\begin{aligned} \left. \frac{\partial f}{\partial \mathbf {T}_n}\right| _{\varvec{\mu },\text{ vec }{\mathsf {\Sigma }}}&= -\gamma ^3 {\mathsf {\Sigma }}^{-1}\varvec{\mu }\\ \left. \frac{\partial f}{\partial \text{ vec }{\mathsf {C}}_n}\right| _{\varvec{\mu },\text{ vec }{\mathsf {\Sigma }}}&= \frac{\gamma ^3}{2}(\varvec{\mu }^t \otimes \varvec{\mu }^t)({\mathsf {\Sigma }}^{-1} \otimes {\mathsf {\Sigma }}^{-1}), \end{aligned}$$

which allows to obtain the following expression for the asymptotic variance of $V_n$:

$$\begin{aligned} \gamma ^6 \tau \varvec{\mu }^t {\mathsf {\Sigma }}^{-1} \varvec{\mu }+ \frac{\gamma ^6}{4}(\varvec{\mu }^t \otimes \varvec{\mu }^t)({\mathsf {\Sigma }}^{-1} \otimes {\mathsf {\Sigma }}^{-1})^t ~{\mathsf {\Xi }}~ (\varvec{\mu }^t \otimes \varvec{\mu }^t)({\mathsf {\Sigma }}^{-1} \otimes {\mathsf {\Sigma }}^{-1}). \end{aligned}$$

(25)

Since the asymptotic distribution of $V_n$ depends on $\varvec{\mu }$ and ${\mathsf {\Sigma }}$ only through $\gamma$, it suffices to compute expression (25) for any parameters satisfying $(\varvec{\mu }^t {\mathsf {\Sigma }}^{-1} \varvec{\mu })^{-1/2} = \gamma$. Taking for instance $\varvec{\mu }_0 = (1/\gamma ) \mathbf {e}_1$ and ${\mathsf {\Sigma }}_0 = {\mathsf {I}}_{p}$ allows to conclude. $\square$

Proof

(Lemma 1) The non-central F distribution function with degrees of freedom $d_1$ and $d_2$ and non-centrality parameter $\delta$ evaluated in a fixed x can be expressed as a function of $t=\delta /2$

$$\begin{aligned} G(t)= e^{-t} \sum _{j=0}^{+\infty } \frac{t^j}{j!} C_j \end{aligned}$$

(26)

where $C_j := I\left( \left. d_1x / (d_2+d_1x) \right| d_1/2 +j , d_2/2\right)$ with I the regularized incomplete Beta function. Since the series converges uniformly on any compact of $]0;+\infty [$, the function G is continuous in t. The idea is to examine the sign of the derivative of G with respect to t. As a power series in t with convergence domain $[0,+\infty [$, G can be differentiated easily to obtain

$$\begin{aligned} G'(t) = e^{-t}\sum _{j=0}^{+\infty } \frac{t^j}{j!} \left( C_{j+1} -C_j\right) \end{aligned}$$

Using properties of the regularized incomplete Beta function, this derivative can be shown to be strictly negative, which concludes the proof. $\square$

Proof

(Proposition 3) Let $\mathscr {A}$ be the event

$$\begin{aligned} \left\{ F^{-1}_{p,n-p,\delta }(\beta ) \le T \le F^{-1}_{p,n-p,\delta }(1- \alpha +\beta ) \right\} , \end{aligned}$$

where $T = \frac{n-p}{p} \frac{1}{V_n^2}$ is a random variable following a non-central F distribution with degrees of freedom p and $n-p$ and non-centrality parameter $\delta =n/\gamma ^2$.

As a consequence of Lemma 1 and by definition of $I(V_n)$, the following events are equivalent

$$\begin{aligned} \mathscr {A} \cap \left\{ V_n \le C_{p,n-p} \right\} = \left\{ \gamma \in I(V_n) \right\} \cap \left\{ V_n \le C_{p,n-p} \right\} . \end{aligned}$$

The proof can then be concluded thanks to the inequalities

$$\begin{aligned} 1-\alpha&= \mathbb {P}_\gamma \left[ \mathscr {A} \right] \ge \mathbb {P}_{\gamma } \left[ \mathscr {A} \cap \left\{ V_n \le C_{p,n-p} \right\} \right] \\&\ge \mathbb {P}_\gamma \left[ \mathscr {A} \right] - \epsilon = 1 - \alpha - \epsilon . \end{aligned}$$

$\square$

Proof

(Proposition 4) First, let us show that the function g is strictly increasing in $\gamma$. This function simplifies as follows, provided that $0<p<n$,

$$\begin{aligned} g(\gamma )&= \frac{\Gamma \left( \frac{n-p}{2}+\frac{1}{2} \right) }{\Gamma \left( \frac{n-p}{2}\right) }e^{-{\frac{n}{2\gamma ^2}}} \sum _{j=0}^{+\infty }\left( \frac{n}{2\gamma ^2}\right) ^j \frac{1}{j!}\frac{\Gamma \left( \frac{p}{2}+j - \frac{1}{2} \right) }{\Gamma \left( \frac{p}{2}+j\right) }. \end{aligned}$$

(27)

Since the series is uniformly convergent on any compact of $]0;+\infty [$, this function is continuous. The idea is to examine the sign of the derivative of g with respect to $\gamma$. As a power series in $\gamma$ whose convergence domain is $]0; +\infty [$, the series in (27) can be easily differentiated to obtain

$$\begin{aligned} g'(\gamma )&=\frac{\Gamma \left( \frac{n-p+1}{2}\right) }{\Gamma \left( \frac{n-p}{2}\right) } \frac{n}{\gamma ^3}e^{-{\frac{n}{2\gamma ^2}}} \sum ^{+\infty }_{j=0}\left( \frac{n}{2\gamma ^2}\right) ^j \frac{1}{j!} \frac{\Gamma \left( \frac{p-1}{2} +j\right) }{\Gamma \left( \frac{p}{2} +j \right) } \left( 1- \frac{\frac{p+1}{2}+j}{\frac{p}{2}+j}\right) , \end{aligned}$$

which is strictly positive for every $\gamma \in ]0; +\infty [$. Moreover, noting that

$$\begin{aligned} \lim _{\gamma \rightarrow +\infty }g(\gamma )&= \frac{\Gamma \left( \frac{n-p+1}{2}\right) \Gamma \left( \frac{p-1}{2}\right) }{\Gamma \left( \frac{n-p}{2}\right) \Gamma \left( \frac{p}{2}\right) }\\ \lim _{\gamma \rightarrow 0^{+}} g(\gamma )&=0 \end{aligned}$$

concludes the proof. $\square$