A formulation for continuous mixtures of multivariate normal distributions

1 Continuous mixtures of normal distributions

In the last few decades, a number of formulations have been put forward, in the context of distribution theory, where a multivariate normal variable represents the basic constituent but with the superposition of another random component, either in the sense that the normal mean value or the variance matrix or both these components are subject to the effect of another random variable of continuous type. We shall refer to these constructions as ‘mixtures of normal variables’; the matching phrase ‘mixtures di normal distributions’ will also be used.

To better focus ideas, recall a few classical instances of the delineated scheme. Presumably, the best-known such formulation is represented by scale mixtures of normal variables, which can be expressed as

Y = ξ + V^{1 / 2} X

(1)

where $X \sim N_{d} (0, Σ)$ , $V$ is an independent random variable on $R^{+}$ , and $ξ \in R^{d}$

is a vector of constants. Scale mixtures

1 provide a stochastic representation of a wide subset of the class of elliptically contoured distributions, often called briefly elliptical distributions. For a standard account of elliptical distributions, see for instance Fang et al. (1990); specifically, their Section 2.6 examines the connection with scale mixtures of normal variables. A very importance instance occurs when

1 / V \sim χ_{ν}^{2} / ν

, which leads to the multivariate Student’s

t

distribution.

Another very important construction is the normal variance-mean mixture proposed by Barndorff-Nielsen (1977, 1978) and extensively developed by subsequent literature, namely

Y = ξ + V γ + V^{1 / 2} X

(2)

where $γ \in R^{d}$ is a vector of constants and $V$ is assumed to have a generalized inverse Gaussian (GIG) distribution. In this case $Y$ turns out to have a generalized hyperbolic (GH) distribution, which will recur later in the paper.

Besides 1 and 2, there exists a multitude of other constructions which belong to the idea of normal mixtures delineated in the opening paragraph. Many of these formulations will be recalled in the subsequent pages, to illustrate the main target of the present contribution, which is to present a general formulation for normal mixtures. Our proposal involves an additional random component, denoted $U$ , and the effect of $U$ and $V$ is regulated by two functions, non-linear in general. As we shall see, this construction encompasses a large number of existing constructions in a unifying scheme, for which we develop various general properties.

The role of this activity is to highlight the relative connections of the individual constructions, with an improved understanding of their nature. As a side-effect, the presentation of the individual formulations plays also the role of a review of this stream of literature. Finally, the proposed formulation can facilitate the conception of additional proposals with specific aims. The emphasis is primarily on the multivariate context.

Since it moves a step towards generality, we mention beforehand the formulation of Tjetjep & Seneta (2006) where $V$ and $V^{1 / 2}$ in 2 are replaced by two linear functions of them, which allows to incorporate a number of existing families. Their construction is, however, entirely within the univariate domain. A number of multivariate constructions aiming at some level of generality do exist, and will examined in the course of the discussion.

In the next section, our proposed general scheme is introduced, followed by the derivation of a number of general properties. The subsequent sections show how to frame a large number of existing constructions within the proposed scheme. In the final section, we indicate some directions for even more general constructions.

2 Generalized mixtures of normal distributions

2.1 Notation and other formal preliminaries

As already effectively employed, the notation $W \sim N_{d} (μ, Σ)$ indicates that $W$ is a $d$ -dimensional normal random variable with mean vector $μ$ and variance matrix $Σ$ . The density function and the distribution function of $W$ at $x \in R^{d}$ are denoted by $φ_{d} (x; μ, Σ)$ and $Φ_{d} (x; μ, Σ)$ . Hence, specifically, we have

φ_{d} (x; μ, Σ) = \frac{1}{det (2 π Σ)^{1 / 2}} exp {- \frac{1}{2} (x - μ)^{⊤} Σ^{- 1} (x - μ)}

if $Σ > 0$ . When $d = 1$ , we drop the subscript $d$ . When $d = 1$ and, in addition, $μ = 0$ and $Σ = 1$ , we use the simplified notation $φ (\cdot)$ and $Φ (\cdot)$ for the density function and the distribution function.

A quantity arising in connection with the multivariate normal distribution, but not only there, is the Mahalanobis distance, defined (in the non-singular case) as

∥ x ∥_{Σ} = {(x^{⊤} Σ^{- 1} x)}^{1 / 2}

(3)

which is written in the simplified form $∥ x ∥$ when $Σ$

is the identity matrix.

A function which will appear in various expressions is the inverse Mills ratio

ζ (t) = \frac{φ (t)}{Φ (t)}, t \in R .

(4)

A positive continuous random variable

V

has a GIG distribution if its density function can be written as

g (v; λ, χ, ψ) = \frac{{(\sqrt{ψ / χ})}^{λ}}{2 K_{λ} (\sqrt{χ ψ})} v^{λ - 1} exp (- \frac{1}{2} (χ v^{- 1} + ψ v)), v > 0,

(5)

where $λ \in R$ , $ψ > 0$ , $χ > 0$ and $K_{λ}$ denotes the modified Bessel function of the third kind. In this case, we write $V \sim N^{-} (λ, χ, ψ)$ . The numerous properties of the GIG distribution and interconnections with other parametric families are reviewed by Jørgensen (1982). We recall two basic properties: both the distribution of $1 / V$ and of $c V$ for $c > 0$

are still of GIG type. A fact to be used later is that the Gamma distribution is obtained when

λ > 0

and

χ \to 0

A result in matrix theory which will be used repeatedly is the Sherman-Morrison formula for matrix inversion, which states

(A + b d^{⊤})^{- 1} = A^{- 1} - \frac{1}{1 + d^{⊤} A^{- 1} b} A^{- 1} b d^{⊤} A^{- 1}

(6)

provided that the square matrix $A$ and the vectors $b, d$ have conformable dimensions, and the inverse matrices exist.

2.2 Definition and basic facts

Consider a $d$ -dimensional random variable $X \sim N_{d} (0, Σ)$ and univariate random variables $U$ and $V$

with joint distribution function

G (u, v)

, such that

(X, U, V)

are mutually independent; hence

G

can be factorized as

G (u, v) = G_{U} (u) G_{V} (v)

. We assume

Σ > 0

to avoid technical complications and concentrate on the constructive process. These definitions and assumptions will be retained for the rest of the paper.

Given any real-valued function $r (u, v)$ , a positive-valued function $s (u, v)$ , and vectors $ξ$ and $γ$ in $R^{d}$ , we shall refer to

	$Y$	$=$	$ξ + r (U, V) γ + s (U, V) X$		(7)
		$=$	$ξ + R γ + S X$		(8)

as a generalized mixture of normal (GMN) variables; we have written $R = r (U, V)$ and $S = s (U, V)$ with independence of $(R, S)$ from $X$ . Denote by $H$ the joint distribution function of $(R, S)$ implied by $r, s, G$ . The distribution of $Y$ is identified by the notation $Y \sim {G M N}_{d} (ξ, Σ, γ, H)$ .

For certain purposes, it is useful to think of $Y$ as generated by the hierarchical construction

\begin{matrix} (Y | U = u, V = v) & \sim & N_{d} (ξ + r (u, v) γ, s (u, v)^{2} Σ), (U, V) & \sim & G_{U} \times G_{V} . \end{matrix}

(9)

For instance, this representation is convenient for computing the mean vector and the variance matrix as

$E {Y}$	$=$	$E {E {Y \| U, V}}$	(10)
	$=$	$E {ξ + r (U, V) γ}$
	$=$	$ξ + E {R} γ$

provided $E {R}$ exists, and

$var {Y}$	$=$	$var {E {Y \| U, V}} + E {% v a r {Y \| U, V}}$	(11)
	$=$	$var {ξ + r (U, V) γ} + E {s (U, V)^{2} Σ}$
	$=$	$var {R} γ γ^{⊤} + E {S^{2}} Σ$

provided $var {R}$ and $E {S^{2}} > 0$ exist. Another use of representation 9

is to facilitate the development of some EM-type algorithm for parameter estimation.

Similarly, by a conditioning argument, it is simple to see that the characteristic function of

Y

c (t) = exp (i t^{⊤} ξ) E {c_{N} (t; r (U, V) γ, s (U, V)^{2} Σ)}, t \in R,

where $c_{N} (t; μ, Σ)$ denotes the characteristic function of a $N (μ, Σ)$ variable. Also, the distribution function of $Y$ is

F (y) = E {Φ_{d} (y; ξ + r (U, V) γ, s (U, V)^{2} Σ)} .

(12)

Consider the density function of $Y$ , $f (y)$ , in the case that $S = s (U, V)$ is a non-null constant. From 9 it follows that

f (y) = E_{G} {φ_{d} (y; ξ + r (U, V) γ, s (U, V)^{2} Σ)} = E_{H} {φ_{d} (y; ξ + R γ, S^{2} Σ)}

where the first expected value is taken with respect to the distribution $G$ , the second one with respect to $H$ . Assume further that the distribution $H$ of $(R, S)$ is absolutely continuous with density function $h (r s,)$ , and note that the transformation from $(R, S, X)$ to $(R, S, Y)$ is invertible, so that a standard computation for densities of transformed variables yields, in an obvious notation,

$f_{R, S, Y} (r, s, y)$	$=$	$s^{- d} f_{R, S, X} (r, s, s^{- 1} (y - ξ - r γ))$
	$=$	$h (r, s) s^{- d} φ_{d} (s^{- 1} (y - ξ - r γ); 0, Σ)$
	$=$	$h (r, s) φ_{d} (y; ξ + r γ, s^{2} Σ)$

taking into account the independence of $(R, S)$ and $X$ . Hence we arrive at

f (y) = \int_{R \times R_{+}} φ_{d} (y; ξ + r γ, s^{2} Σ) d H (r, s) .

(13)

An alternative route to obtain this expression would be via differentiation of the distribution function 12 with exchange of the integration and differentiation signs.

For statistical work, it is often useful to consider constructions of type 7 where the distributions of $U$ and $V$ belong to some parametric family. In these cases, care must be taken to avoid overparameterization. Given the enormous variety of specific instances embraced by 7, it seems difficult to establish general suitable condition, and we shall then discuss this issue within specific families or classes of distributions.

In the above passage, as well as in the rest of the paper, the term ‘family’ refers to the set of distributions obtained by a given specification of the variables $(X, U, V)$ when their parameters vary in some admissible space, while keeping the other ingredients fixed. Broader sets, generated for instance when the distributions of $U$ and $V$ vary across various parametric families, constitute ‘classes’.

A clarification is due about the use of the notation in 7–8 and some derived expressions to be presented later on. When we shall examine a certain family belonging to the general construction, that notation will translate into a certain parameterization, which often is not the most appropriate for inferential or for interpretative purposes, and its use here must not be intended as a recommendation for general usage. This scheme is adopted merely for uniformity and simplicity of treatment in the present investigation.

2.3 Affine transformations and other distributional properties

For the random variable $Y$ introduced by 7-8, consider an affine transformation $W = b + B^{⊤} Y$ , for a $q$ -dimensional vector $b$ and a full-rank matrix $B$ of dimension $d \times q$ , with $q \leq d$ ; denote these assumptions as ‘the b-B conditions’. It is immediate that

W = b + B^{⊤} Y = b + B^{⊤} ξ + r (U, V) B^{⊤} γ + s (U, V) B^{⊤} X

is still of type 7–8 with the same mixing variables $(R, S)$ and modified numerical parameters. We have then reached the following conclusion.

Proposition 1

If $Y \sim {G M N}_{d} (ξ, Σ, γ, H)$ and $b, B$ satisfy the b-B conditions introduced above, it follows that

b + B^{⊤} Y \sim {G M N}_{q} (b + B^{⊤} ξ, B^{⊤} Σ B, B^{⊤} γ, H)

(14)

is still a member of the GMN class, with the same mixing distribution of $Y$ .

Partition now $Y$ in two sub-vectors of sizes $d_{1}, d_{2}$ , such that $d_{1} + d_{2} = d$ , with corresponding partitions of the parameters in blocks of matching sizes, as follows

Y = (\begin{matrix} Y_{1} Y_{2} \end{matrix}), ξ = (\begin{matrix} ξ_{1} ξ_{2} \end{matrix}), γ = (\begin{matrix} γ_{1} γ_{2} \end{matrix}), Σ = (\begin{matrix} Σ_{11} & Σ_{12} Σ_{21} & Σ_{22} \end{matrix}) .

(15)

To establish the marginal distributions of $Y_{1}$ , we use Proposition 1 with $b = 0$ and $B$ equal to a matrix formed by $I_{d_{1}}$ in the top $d_{1}$ rows and a block of $0$ s in the bottom $d_{2}$ rows. For $Y_{2}$ , we proceed similarly, but setting the bottom $d_{2}$ rows of $B$ equal to $I_{d_{2}}$ . We then arrive at the following conclusion.

Proposition 2

If $Y \sim {G M N}_{d} (ξ, Σ, γ, H)$ is partitioned as indicated in 15, then

Y_{1} \sim {G M N}_{d_{1}} (ξ_{1}, Σ_{11}, γ_{1}, H), Y_{2} \sim {G M N}_{d_{2}} (ξ_{2}, Σ_{22}, γ_{2}, H) .

(16)

We now want examine conditions which ensure independence of $Y_{1}$ and $Y_{2}$ . From 9 it is clear that, if $Σ_{12} = Σ_{21}^{⊤} = 0$ , $Y_{1}$ and $Y_{2}$ are conditionally independent given $(U, V)$ , with conditional distribution

(Y_{j} | U = u, V = v) \sim N_{d} (ξ_{j} + r (u, v) γ_{j}, s (u, v)^{2} Σ_{j j}), j = 1, 2,

(17)

where $(U, V) \sim G_{U} \times G_{V} .$ Moreover, if $s (U, V) \equiv 1$ (constant) and one of the marginal distributions is symmetric, i.e., $γ_{1} = 0$ or $γ_{2} = 0$ , then $Y_{1}$ and $Y_{2}$ are independent. The notation $s (U, V) \equiv 1$

and similar ones later on must be intended ‘with probability 1’; we shall not replicate this specification subsequently.

A more detailed argument is as follows, where we take $ξ = 0$ for mere simplicity of notation. without affecting the generality of the argument. From 9, we have that the conditional joint characteristic function of $(Y_{1}, Y_{2})$ , given $U = u$ and $V = v$ (or, equivalently, given $R = r$ and $S = s$ ), is

so that the joint characteristic function of $(Y_{1}, Y_{2})$ is

$c (t_{1}, t_{2})$	$=$	$E {e^{i t_{1}^{⊤} Y_{1} + i t_{2}^{⊤} Y_{2}}}$	(18)
	$=$	$E {E {e^{i t_{1}^{⊤} Y_{1} + i t_{2}^{⊤} Y_{2}} \| U, V}}$
	$=$	$E {e^{i r (U, V) (t_{1}^{⊤} γ_{1} + t_{2}^{⊤} γ_{2}) - \frac{1}{2} s (U, V)^{2} (t_{1}^{⊤} Σ_{11} t_{1} + 2 t_{1}^{⊤} Σ_{12} t_{2} + t_{2}^{⊤} Σ_{22} t_{2})}}$
	$=$	$E {e^{i r (U, V) t_{1}^{⊤} γ_{1} - \frac{1}{2} s (U, V)^{2} t_{1}^{⊤} Σ_{11} t_{1}} e^{i r (U, V) t_{2}^{⊤} γ_{2} - \frac{1}{2} s (U, V)^{2} t_{2}^{⊤} Σ_{22} t_{2}} e^{- s (U, V)^{2} t_{1}^{⊤} Σ_{12} t_{2}}} .$

In analogous way, by 17 the marginal characteristic functions are

$c_{j} (t_{j})$	$=$	$E {e^{i t_{j}^{⊤} Y_{j}}}$	(19)
	$=$	$E {E {e^{i t_{j}^{⊤} Y_{j}} \| U, V}}$
	$=$

Note that, if $γ_{j} = 0$ and $s (U, V) \equiv 1$ , then by 19 $c_{j} (t_{j})$ reduces to the centred normal characteristic function $c_{N, j} (t_{j}) = e^{- \frac{1}{2} t_{j}^{⊤} Σ_{j j} t_{j}}$ for $j = 1, 2$ . We have then reached the following conclusion.

Proposition 3

Given partition 15, the components $Y_{1}, Y_{2}$ are independent provided $s (U, V) \equiv 1$ , $Σ_{12} = 0$ and at least one of $γ_{1}$ and $γ_{0}$ is $0$ , with the following implications:

if $γ_{1} = 0$ , the joint characteristic function 18 reduces to $c_{N, 1} (t_{1}) c_{2} (t_{2})$ ,
if $γ_{2} = 0$ , the joint characteristic function 18 reduces to $c_{1} (t_{1}) c_{N, 2} (t_{2})$ .

If both $γ_{1}$ and $γ_{2}$ are $0$ , the distribution reduces to the case of independent normal variables.

In essence, under the conditions of Proposition 3, one of $Y_{1}$ and $Y_{2}$ has a plain normal distribution and the other one falls under the construction discussed later in Section 3.

Outside the conditions of Proposition 3, the structure of 18 does not appear to be suitable for factorization as the product of two legitimate characteristic functions, and we conjecture that, in general, independence between $Y_{1}$ and $Y_{2}$ cannot be achieved.

Examine now the conditional distributions associated to partition 15. Factorize the joint density of $Y$ as $f (y_{1}, y_{2}) = f_{1 | 2} (y_{1} | y_{2}) f_{2} (y_{2})$ where $f_{1 | 2} (y_{1} | y_{2})$ is the conditional density of $(Y_{1} | Y_{2} = y_{2})$ and $f_{2} (y_{2})$ is the marginal density of $Y_{2}$ . For simplicity of treatment, suppose that $(R, S)$ is absolutely continuous, with density $h (r, s)$ . Then, by 13 and the properties of the multivariate normal density, write

f_{1 | 2} (y_{1} | y_{2}) f_{2} (y_{2}) = \int_{R \times R_{+}} φ_{d_{1}} (y_{1}; ξ_{1 | 2} + γ_{1 | 2} r, s^{2} Σ_{1 | 2}) φ_{d_{2}} (y_{2}; ξ_{2} + γ_{2} r, s^{2} Σ_{22}) h (r, s) d r d s

where

ξ_{1 | 2} = ξ_{1} + Σ_{12} Σ_{22}^{- 1} (y_{2} - ξ_{2}), γ_{1 | 2} = γ_{1} - Σ_{12} Σ_{22}^{- 1} γ_{2}, Σ_{11 | 2} = Σ_{11} - Σ_{12} Σ_{22}^{- 1} Σ_{21}

having assumed that the conditioning operation and integration can be exchanged. Hence, for the conditional density of $Y_{1}$ given $y_{2}$ we have

f_{1 | 2} (y_{1} | y_{2}) = \frac{1}{f_{2} (y_{2})} \int_{R \times R_{+}} φ_{d_{1}} (y_{1}; ξ_{1 | 2} + γ_{1 | 2} r, s^{2} Σ_{1 | 2}) φ_{d_{2}} (y_{2}; ξ_{2} + γ_{2} r, s^{2} Σ_{22}) h (r, s) d r d s .

Now, from the Bayes’s rule, we obtain that the conditional density of $(R, S)$ given $Y_{2} = y_{2}$ is

h_{c} (r, s | y_{2}) = \frac{φ_{d_{2}} (y_{2}; ξ_{2} + γ_{2} r, s^{2} Σ_{22}) h (r, s)}{f_{2} (y_{2})} .

(20)

Using this fact in the last integral, we can re-write

f_{1 | 2} (y_{1} | y_{2}) = \int_{R \times R_{+}} φ_{d_{1}} (y_{1}; ξ_{1 | 2} + γ_{1 | 2} r, s^{2} Σ_{11 | 2}) h_{c} (r, s | y_{2}) d r d s

(21)

which exhibits the same structure of 13. Therefore we can conclude that

(Y_{1} | Y_{2} = y_{2}) \sim {G M N}_{d_{1}} (ξ_{1 | 2}, Σ_{11 | 2}, γ_{1 | 2}, H_{c (y_{2})})

where $H_{c (y_{2})}$ denotes the distribution function associated to the conditional density 20.

For many GMN constructions 7–8, the density function of $Y$ is likely to be known in explicit form; in these cases, the same holds true for $Y_{2}$ , recalling 16. Then, a convenient aspect of expression 20 is that it indicates how to compute the conditional density once the joint unconditional distribution $H (r, s)$ is available explicitly. Clearly, this is especially amenable in those constructions where $(R, S)$ is really a univariate variable, as in Sections 3 and 4 below.

In one of the appendices, we illustrate the use of 20–21 in the case of a multivariate $t$ distribution.

2.4 On quadratic forms

For use in the next result, but also in the rest of the paper, define the quantities

Ω = Σ + γ γ^{⊤}, η = {(1 + γ^{⊤} Σ^{- 1} γ)}^{- 1 / 2} Σ^{- 1} γ, α^{2} = ∥ γ ∥_{Σ}^{2} = γ^{⊤} Σ^{- 1} γ, δ^{2} = \frac{α^{2}}{1 + α^{2}}

(22)

such that $α^{2} \in [0, \infty)$ and $δ^{2} \in [0, 1)$ . For notational convenience, we introduce the notation

μ_{h k} = E {R^{h} S^{k}}, k = 0, 1, \dots

(23)

when the named expectation exists.

Proposition 4

For a random variable $Y_{0}$ having distribution of type 7–8 with $ξ = 0$ , the following facts hold:

$S^{- 2} (Y_{0} - R γ)^{⊤} Σ^{- 1} (Y_{0} - R γ)$	$\sim$	$χ_{d}^{2},$	(24)
$E {Y_{0}^{⊤} Σ^{- 1} Y_{0}}$	$=$	$d E {S^{2}} + α^{2} E {R^{2}} = d μ_{02} + α^{2} μ_{20},$	(25)
$E {Y_{0}^{⊤} Ω^{- 1} Y_{0}}$	$=$	$d E {S^{2}} + δ^{2} (E {R^{2}} - E {S^{2}}) = d μ_{02} + δ^{2} (μ_{20} - μ_{02}),$	(26)

provided $E {R^{2}}$ and $E {S^{2}}$ exist, using the quantities defined in 22 and 23.

Proof: From 8, write $(Y_{0} - R γ)^{⊤} Σ^{- 1} (Y_{0} - R γ) = S^{2} X^{⊤} Σ^{- 1} X$ , where $X^{⊤} Σ^{- 1} X \sim χ_{d}^{2}$ is independent of $S$ ; this yields result 24. For equality 25, expand the initial identity of this proof as

Y_{0}^{⊤} Σ^{- 1} Y_{0} - 2 R γ^{⊤} Σ^{- 1} Y_{0} + R^{2} γ^{⊤} Σ^{- 1} γ = S^{2} X^{⊤} Σ^{- 1} X

and take expectation on both sides of this equality. We obtain

E {R Y_{0}} = E {R E {Y_{0} | U, V}} = E {R E {R γ + S X) | U, V}} = E {R (R γ + S E {X | U, V})} = E {R^{2}} γ,

bearing in mind that $E {X | U, V} = E {X} = 0$ , by the independence assumption between $X$ and $(U, V)$ . This leads to 25.

For 26, write $Q = Y_{0}^{⊤} Ω^{- 1} Y_{0}$ , and $E{Q}=\rm tr(Ω−1E{Y0Y⊤0})$ . Using 10 and 11, we obtain $E {Y_{0} Y_{0}^{⊤}} = var {Y_{0}} + E {Y_{0}} E {Y_{0}^{⊤}} = E {R^{2}} γ γ^{⊤} + E {S^{2}} Σ$ , so that

E{Q}=E{R2}γ⊤Ω−1γ+E{S2}\rm tr(Ω−1Σ).

By using the Sherman-Morrison equality 6, we conclude the proof. qed

In the subsequent pages, the matrix $Ω$ defined in 22 and the associated quadratic form $Q = Y_{0}^{⊤} Ω^{- 1} Y_{0}$ will appear repeatedly. A connected relevant question is: under which conditions is 26 free of $γ$ ? Equivalently, under which conditions

E {Q} = E {Y_{0}^{⊤} Ω^{- 1} Y_{0}} = d E {S^{2}} ?

(27)

This equality represents a form of invariance which is known to hold in some cases to be recalled later on, but we want to examine it more generally. One setting where equality 27 holds is given by $R = U V^{1 / 2}$ , $S = V^{1 / 2}$ , where $V > 0$ , and $E {U^{2}} = 1$ . It is then immediate to see that $E {R^{2}} = E {S^{2}}$ , so that the final term of 26 is zero.

The conditions $R = U V^{1 / 2}$ and $S = V^{1 / 2}$ are in turn achieved when $Z = U γ + X$ and $Y_{0} = V^{1 / 2} Z$ . In this case $Q = V Q_{0}$ , where $Q_{0} = Z^{⊤} Ω^{- 1} Z$ which is independent of $V$ . Hence, $E {Q} = E {V} E {Q_{0}}$ , where

E{Q0}=\rm tr(Ω−1E{ZZ⊤})=E{U2}γ⊤Ω−1γ+\rm tr% (Ω−1Σ)

since $E {Z} = E {U} γ$ and $var {Z} = var {U} γ γ^{⊤} + Σ$ and so $E {Z Z^{⊤}} = E {U^{2}} γ γ^{⊤} + Σ$ . Thus, if $E {U^{2}} = 1$ , then by using 6 it clearly follows that $E {Q_{0}} = d$ . We shall return to this issue later on.

2.5 Mardia’s measures of multivariate asymmetry and kurtosis

For a multivariate random variable $Z$ such that $μ_{Z} = E {Z}$ and $Σ_{Z} = var {Z}$ , Mardia (1970, 1974)

has introduced measures of multivariate skewness and kurtosis, defined as

β_{1, d} = E {{[(Z - μ_{Z})^{⊤} Σ_{Z}^{- 1} (Z^{'} - μ_{Z})]}_{Z}^{3}}, β_{2, d} = E {{[(Z - μ_{Z})^{⊤} Σ_{Z}^{- 1} (Z - μ_{Z})]}_{Z}^{2}},

(28)

where $Z^{'}$ is an independent copy of $Z$ , provided these expected values exist. These measures represent extensions of corresponding familiar quantities for the univariate case:

β_{1} = \frac{E {(Z - μ_{Z})^{3}}^{2}}{} var {Z}^{3} = γ_{1}^{2}, β_{2} = \frac{E {(Z - μ_{Z})^{2}}}{% var {Z}^{2}} = γ_{2} + 3,

(29)

in the sense that $β_{1, 1} = β_{1}$ and $β_{2, 1} = β_{2}$ .

We want to find expressions for 28 in the case of a random variable $Y$ of type 7–8. Recall the expressions for $μ_{Y}$ and $Σ_{Y}$ given in 10 and 11, and the notation defined in 23

for the moments of

(R, S)

, and write

R_{0} = R - μ_{10}, Y - μ_{Y} = R_{0} γ + S X .

assuming that the involved mean values exist. Taking into account the invariance of $β_{d, 1}$ and $β_{d, 2}$ with respect to non-singular affine transformations, it is convenient to work with the transformed quantities

X_{0} = Σ^{- 1 / 2} X \sim N_{d} (0, I_{d}), γ_{0} = Σ^{- 1 / 2} γ, Y_{0} = Σ^{- 1 / 2} (Y - μ_{Y}) = R_{0} γ_{0} + S X_{0}

where any form of the square root matrix $Σ^{1 / 2}$ can be adopted.

The subsequent development involves extensive algebra of which we report here only the summary elements; detailed computations are provided in an appendix. Recall $α^{2}$ introduced in 22 and define

{¯ μ}_{20} = μ_{20} - μ_{10}^{2} = var {R} ρ = \frac{{¯ μ}_{20}}{μ_{02}}, ¯ ρ = \frac{ρ α^{2}}{1 + ρ α^{2}} = \frac{α^{2} {¯ μ}_{20}}{μ_{02} + α^{2} {¯ μ}_{20}} .

Introduce the auxiliary random variables $T_{0} = α^{- 1} γ_{0}^{⊤} X_{0} \sim N (0, 1)$ , which is independent of $(R, S)$ , and $Z_{0} = α R_{0} + S T_{0}$ . We need to compute the following expectations:

$E {S^{2} Z_{0}}$	$=$	$α (μ_{12} - μ_{10} μ_{02}),$
$E {S^{2} Z_{0}^{2}}$	$=$	$α^{2} (μ_{22} - 2 μ_{12} μ_{10} + μ_{10}^{2} μ_{02}) + μ_{04},$
$E {Z_{0}^{3}}$	$=$	$α^{3} (μ_{30} - 3 μ_{20} μ_{10} + 2 μ_{10}^{3}) + 3 α (μ_{12} - μ_{10} μ_{02}),$
$E {Z_{0}^{4}}$	$=$	$α^{4} (μ_{40} - 4 μ_{30} μ_{10} + 6 μ_{20} μ_{10}^{2} - 3 μ_{10}^{4}) + 6 α^{2} (μ_{22} - 2 μ_{12} μ_{10} + μ_{10}^{2} μ_{02}) + 3 μ_{04},$

assuming the existence of moments of $(R, S)$ up to the fourth order. With these ingredients, the Mardia’s measures for the GMN construction can be expressed as

	$β_{1, d}$	$=$			(30)
	$β_{2, d}$	$=$	$μ_{02}^{- 2} ((d + 1) (d - 1) μ_{04} + 2 (d - 1) (1 - ¯ ρ) E {S^{2} Z_{0}^{2}} + (1 - ¯ ρ)^{2} E {Z_{0}^{4}}) .$		(31)

Considering the complexity that typically involves the explicit specification of 28 outside the normal family, the above expressions appear practically manageable. They are further simplified when one specializes them to a given family or to a certain subclass of the GMN construction. For a given choice of the distribution $H$ , we need to work out the following ingredients: (i) the marginal moments of $R$ , $μ_{h 0}$ , up to order 4, (ii) the marginal moments $μ_{02}$ and $μ_{04}$ of $S$ , (ii) the cross moments $μ_{12} = E {R S^{2}}$ and $μ_{22} = E {R^{2} S^{2}}$ . The working is illustrated next for the GH family; additional illustrations will appear later.

Mardia’s measures for the GH family

For the GH family with representation 2, there is a single mixing variable $V \sim N^{-} (λ, χ, ψ)$ with density 5 and $R = V$ , $S = V^{1 / 2}$ . General expressions for $E {V^{h}} = μ_{h 0}$ are given in Section 2.1 of Jørgensen (1982), among others. These expressions also provide $μ_{02} = E {V}$ and $μ_{04} = E {V^{2}}$ . The two other required quantities are $μ_{12} = E {V^{2}}$ and $μ_{22} = E {V^{3}}$ which are still ordinary moments of $V$ . We can now compute

$E {S^{2} Z_{0}}$	$=$	$α(E{V2}−(E{V})2)=ασ2V,% \leavevmode\nobreak\ say,$
$E {S^{2} Z_{0}^{2}}$	$=$	$α^{2} (E {V^{3}} - 2 E {V^{2}} E {V} + (E {V})^{2}) + E {V^{2}},$
$E {Z_{0}^{3}}$	$=$	$α^{3} (E {V^{3}} - 3 E {V^{2}} E {V} + 2 (E {V})^{3}) + 3 α var {V}$
	$=$	$(α σ_{V})^{3} β_{1} (V) + 3 α σ_{V}^{2},$
$E {Z_{0}^{4}}$	$=$	$α^{4} (E {V^{4}} - 4 E {V^{3}} E {V} + 6 E {V^{2}} (E {V})^{2} - 3 (E {V})^{4})$
		$+ 6 α^{2} (E {V^{3}} - 2 E {V^{2}} E {V} + (E {V})^{3}) + 3 E {V^{2}}$
	$=$

where $σ_{V}^{2} = var {V}$ and $β_{1} (V)$ , $β_{2} (V)$ are the univariate measures of skewness and kurtosis in 29 evaluated for $V$ . Plugging the above quantities in 30 and 31 completes the computation.

Remark

There exists an interesting way of re-writing 30 and 30 which will turn out useful later on. Since $Z_{0} = α R_{0} + S T_{0} \sim {G M N}_{1} (- α μ_{10}, 1, α, H))$ with zero mean and

var {Z_{0}} = α^{2} {¯ μ}_{20} + μ_{02} = (1 - ¯ ρ)^{- 1} μ_{02}

we can introduced an univariate standardized GMN-type variable

{~ Z}_{0} = \frac{α R + S T_{0} - α μ_{10}}{\sqrt{α^{2} {¯ μ}_{20} + μ_{02}}} \sim {G M N}_{1} (- \frac{μ_{10}}{\sqrt{α^{2} {¯ μ}_{20} + μ_{02}}}, \frac{1}{α^{2} {¯ μ}_{20} + μ_{02}}, \frac{α}{\sqrt{α}}

A formulation for continuous mixtures of multivariate normal distributions

Authors

On mean and/or variance mixtures of normal distributions

Identifiability of two-component skew normal mixtures with one known component

The bivariate K-finite normal mixture "blanket" copula: an application to driving patterns

Maximal skewness projections for scale mixtures of skew-normal vectors

Normal variance mixtures: Distribution, density and parameter estimation

On formulations of skew factor models: skew errors versus skew factors

BAMBI: An R package for Fitting Bivariate Angular Mixture Models

1 Continuous mixtures of normal distributions

2 Generalized mixtures of normal distributions

2.1 Notation and other formal preliminaries

2.2 Definition and basic facts

2.3 Affine transformations and other distributional properties

Proposition 1

Proposition 2

Proposition 3

2.4 On quadratic forms

Proposition 4

2.5 Mardia’s measures of multivariate asymmetry and kurtosis

Mardia’s measures for the GH family

Remark

A formulation for continuous mixtures of multivariate normal distributions

Authors

Related Research

On mean and/or variance mixtures of normal distributions

Identifiability of two-component skew normal mixtures with one known component

The bivariate K-finite normal mixture "blanket" copula: an application to driving patterns

Maximal skewness projections for scale mixtures of skew-normal vectors

Normal variance mixtures: Distribution, density and parameter estimation

On formulations of skew factor models: skew errors versus skew factors

BAMBI: An R package for Fitting Bivariate Angular Mixture Models

This week in AI

1 Continuous mixtures of normal distributions

2 Generalized mixtures of normal distributions

2.1 Notation and other formal preliminaries

2.2 Definition and basic facts

2.3 Affine transformations and other distributional properties

Proposition 1

Proposition 2

Proposition 3

2.4 On quadratic forms

Proposition 4

2.5 Mardia’s measures of multivariate asymmetry and kurtosis

Mardia’s measures for the GH family

Remark