3.1 Multivariate t-distribution

They arise, for instance, in the multivariate regression analysis in a construction of confidence regions for regression coefficients.

Let X = (X_{1}, …, X_{d})′ ∼ N_{d}(00 0, R), where R = (ρ_{ij}) is such that ρ_{ii} = 1, i = 1, …, d. Let a random variable Y be independent of X and has a χ^{2}(n) distribution.

Definition 1. A random vector t = (t_{1}, …, t_{d})′, t_{i} = X_{i}/\sqrt{Y/n}, i = 1, …, d, is said to have a multivariate t-distribution with parameters R and n, in symbols, t ∼ t_{d}(R, n). Its density function is of the form (R > 0)

f_{t}(x) = \frac{Γ((d + n)/2)}{(nπ)^{d/2}Γ(n/2)|R|^{1/2}} (1 + \frac{1}{n}x′R^{−1}x)^{−(d+n)/2} , x ∈ ℝ^{d}.

The multivariate t-distribution is non-singular iff R > 0.

For n ≥ 3: (i) Et = 00 0; (ii) V ar t_{i} = n/(n − 2), i = 1, …, d; (iii) Cov (t_{i}, t_{j}) = nρ_{ij}/(n − 2), i ≠ j; (iv) Corr (t_{i}, t_{j}) = ρ_{ij}, i ≠ j.

For fixed k < d consider the partitions of t and R:

t = ( \begin{matrix}t_{1} \\ t_{2} \end{matrix} ) , R = ( \begin{matrix}R_{11} & R_{12} \\ R′_{12} & R_{22} \end{matrix} ) ,

where t_{1} = (t_{1}, …, t_{k})′, t_{2} = (t_{k+1}, …, t_{d})′, R_{ii} is the correlation matrix of t_{i}, i = 1, 2, and R_{12} = (Corr (t_{i}, t_{j})). Then the marginal distributions are: t_{1} ∼ t_{k}(R_{11}, n), t_{2} ∼ t_{d−k}(R_{22}, n).

If t ∼ t_{d}(R, n) and y = Ct, where C is any given m × d real matrix of the rank m ≤ d, then y ∼ t_{m}(CRC′, n).

If t ∼ t_{d}(R, n) then t′R^{−1}t/d ∼ F (d, n).

3.2 Other multivariate t-distributions

Definition 2. Assume that X_{i} ∼ N(0, 1), Y_{i} ∼ χ^{2}(n_{i}), i = 1, …, d, and all the random variables are mutually independent. The distribution of a random vector (X_{1}/\sqrt{Y_{1}/n_{1}}, …, X_{d}/\sqrt{Y_{d}/n_{d}})′ is called a multivariate t-distribution.

Definition 3. Let a random vector X be as in Definition 1 while random variables Y_{i}, i = 1, …, d, be as in Definition 2. If X, Y_{1}, …, Y_{d} are independent then the distribution of a random vector (X_{1}/\sqrt{Y_{1}/n_{1}}, …, X_{d}/\sqrt{Y_{d}/n_{d}})′ is called a multivariate t-distribution.

3.3 The Dirichlet distribution

It arises, for instance, in regression and analysis of variance problems for tests of hypotheses about regression coefficients. Definition 4. Assume that X_{i} ∼ χ^{2}(2n_{i}), i = 0, 1, …, d, and they are independent. The distribution of a random vector D = (X_{1}/ ∑_{j=0}^{d} X_{j}, …, X_{d}/ ∑_{j=0}^{d} X_{j})′ is called the Dirichlet distribution with the parameters n_{1}, …, n_{d}, n_{0}. Its density function is of the form

f_{D}(x) = \frac{Γ(∑_{j=0}^{d} n_{j})}{∏_{j=0}^{d} Γ(n_{j})} ∏_{j=1}^{d} x_{j}^{n_{j}−1} (1 − ∑_{j=1}^{d} x_{j})^{n_{0}−1} , x ∈ 𝕊,

𝕊 = {(x_{1}, …, x_{d})′ ∈ ℝ^{d} : x_{i} ≥ 0, i = 1, …, d, ∑_{i=1}^{d} x_{i} ≤ 1} .

Properties: (i) E(D_{i}) = n_{i}/N; (ii) V ar (D_{i}) = n_{j}(N − n_{j})/(N^{2}(N + 1)); (iii) Cov (D_{i}, D_{j}) = −n_{i}n_{j}/(N^{2}(N + 1)); (iv) Corr (D_{i}, D_{j}) = −\sqrt{n_{i}n_{j}/((N − n_{i})(N − n_{j}))}, where N = ∑_{j=0}^{d} n_{j}.

For fixed k < d the marginal distribution of (D_{1}, …, D_{k})′ is again the Dirichlet distribution with the parameters n_{1}, …, n_{k}, N − ∑_{j=1}^{k} n_{j}.

3.4 Multivariate F -distribution

It is required, for instance, for testing hypotheses about variances and covariances in normal populations.

Definition 5. Assume that X_{i} ∼ χ^{2}(n_{i}), i = 0, 1, …, d, and they are independent. The distribution of a random vector Y = ((X_{1}/n_{1})(X_{0}/n_{0})^{−1}, …, (X_{d}/n_{d})(X_{0}/n_{0})^{−1})′ is called the multivariate F -distribution with the parameters n_{1}, …, n_{d}, n_{0}. Its density function is of the form

f_{Y}(x) = \frac{Γ(∑_{j=0}^{d} n_{j}/2) ∏_{j=0}^{d} n_{j}^{n_{j}/2} ∏_{j=1}^{d} x_{j}^{n_{j}/2−1}}{∏_{j=0}^{d} Γ(n_{j}/2)(n_{0} + ∑_{j=1}^{d} n_{j}x_{j})^{∑_{j=0}^{d} n_{j}/2}}, x_{1} > 0, …, x_{d} > 0.

Properties: (i) E(Y_{i}) = n_{0}/(n_{0} − 2), n_{0} > 2;

(ii) V ar (Y_{i}) = \frac{2n_{0}^{2}(n_{i} + n_{0} − 2)}{n_{i}(n_{0} − 2)^{2}(n_{0} − 4)}, n_{0} > 4;

(iii) Cov (Y_{i}, Y_{j}) = 2n_{0}^{2}/((n_{0} − 2)^{2}(n_{0} − 4)); (iv) Corr (Y_{i}, Y_{j}) = \sqrt{n_{i}n_{j}/((n_{i} + n_{0} − 2)(n_{j} + n_{0} − 2))}.

For fixed k < d the marginal distribution of (Y_{1}, …, Y_{k})′ is again the multivariate F -distribution with the parameters n_{1}, …, n_{k}, n_{0}.

3.5 Multivariate log-normal distribution

This distribution is of great interest in business and economic applications where many positive random variables appear to follow the lognormal distribution quite closely.

Definition 6. Let X ∼ N_{d}(μ, Σ). The distribution of a random vector Y = (e^{X_{1}}, …, e^{X_{d}})′ is called the multivariate log-normal distribution with the parameters μ and Σ. Its density function is of the form

f_{Y}(y) = \frac{1}{∏_{j=1}^{d} y_{j}} f(log y; μ, Σ)

= \frac{1}{(2π)^{d/2}|Σ|^{1/2} ∏_{j=1}^{d} y_{j}} exp \{−(log y − μ)′Σ^{−1}(log y − μ)/2\} ,

y_{1} > 0, …, y_{d} > 0, where log y = (log y_{1}, …, log y_{d})′.

Properties: (i) E(Y_{i}) = exp{μ_{i} + σ_{i}^{2}/2};

(ii) V ar (Y_{i}) = exp{2μ_{i} + 2σ_{i}^{2}} − exp{2μ_{i} + σ_{i}^{2}};

(iii) Cov (Y_{i}, Y_{j}) = exp{μ_{i} + μ_{j} + (σ_{i}^{2} + σ_{j}^{2})/2 + ρ_{ij}σ_{i}σ_{j}} − exp{μ_{i} + μ_{j} + (σ_{i}^{2} + σ_{j}^{2})/2}.

3.6 Multivariate uniform distributions

We meet those distributions in different applications. Also they are particular cases of some other distributions.

Definition 7. A random vector X taking values in a set 𝕊 ⊂ ℝ^{d} with finite volume is said to have a uniform distribution on 𝕊 if

P (X ∈ ℂ) = \frac{volume (ℂ)}{volume (𝕊)} for any ℂ ⊂ 𝕊.

The density function of X equals 1/volume (𝕊) on 𝕊 and zero otherwise.

Interesting properties helping in simulation.

1. If X_{1}, X_{2}, … is a sequence of independent uniformly distributed on 𝕊 random vectors and Y is the first vector among X_{j} that belongs to a set 𝔸 ⊂ 𝕊 (volume (A) > 0), then Y has a uniform distribution on 𝔸.

2. Let X is distributed in ℝ^{d} with the density function f while a random variable U is uniformly [0, 1] distributed, and X and U are independent. Then the random vector (X′, Uf(X))′ has a uniform distribution on 𝕊 = {(x′, u) : x ∈ ℝ^{d}, 0 ≤ u ≤ f(x)} ⊂ ℝ^{d+1}.

3. If X = (X_{1}, …, X_{d}) ∼ N_{d}(00 0, I_{d}) then the random vector (X_{1}/|X|, …, X_{d}/|X|)′ has a uniform distribution on the unit sphere 𝕊^{d−1} = {x ∈ ℝ^{d} : |x| = 1} in ℝ^{d}.

4. If X = (X_{1}, …, X_{d})′ has a uniform distribution on the unit sphere 𝕊^{d−1} and R is a random variable with the density function h(u) = du^{d−1}, u ∈ [0, 1], and X and R are independent, then the random vector Y = (RX_{1}, …, RX_{d})′ has a uniform distribution on the unit ball {x ∈ ℝ^{d} : |x| ≤ 1}.

5. If U_{1}, …, U_{d} are independent and uniformly [0, 1] distributed random variables then a random vector X = (X_{1}, …, X_{d})′ such that

X_{1} = U_{1:d}, X_{2} = U_{2:d} − U_{1:d}, …, X_{d} = U_{d:d} − U_{d−1:d},

where U_{1:d} ≤ U_{2:d} ≤ … ≤ U_{d:d} are the order statistics, is uniformly distributed on the symplex

{(x_{1}, …, x_{d})′ ∈ ℝ^{d} : x_{i} ≥ 0, i = 1, …, d, ∑_{j=1}^{d} x_{j} ≤ 1}.

6. If U_{1}, …, U_{d−1} are independent and uniformly [0, 1] distributed random variables then a random vector X = (X_{1}, …, X_{d})′ such that X_{1} = U_{1:d−1}, X_{2} = U_{2:d−1} − U_{1:d−1}, …, X_{d−1} = U_{d−1:d−1} − U_{d−2:d−1}, X_{d} = 1 − U_{d−1:d−1} (see property 5), is uniformly distributed on the symplex surface

{(x_{1}, …, x_{d})′ ∈ ℝ^{d} : x_{i} ≥ 0, i = 1, …, d, ∑_{j=1}^{d} x_{j} = 1}.

3.7 Exercises

1. Show that when d = 1, the multivariate t-distribution given in Definition 1 reduces to one-dimensional t-distribution with the density function

f(x) = \frac{Γ((n + 1)/2)}{\sqrt{nπ}Γ(n/2)} (1 + \frac{x^{2}}{n})^{−(n+1)/2} , x ∈ ℝ.

2. Write down a formula for the density function of the random vector y = at + b, where t ∼ t_{d}(R, n), a, b ∈ ℝ, a ≠ 0.

3. Write down a formula for the density function of the multivariate t-distribution given in Definition 2 and compare it with that from Definition 1.

4. Show that the multivariate t-distribution given in Definition 2 is a particular case of that given in Definition 3.

5. Show that when d = 1, the Dirichlet distribution reduces to onedimensional Beta(α, β) distribution with the density function

f(x) = \frac{Γ(α + β)}{Γ(α)Γ(β)}x^{α−1}(1 − x)^{β−1}, x ∈ [0, 1], α > 0, β > 0.

6. Let C be a d×d real orthogonal matrix. Show that if a random vector D has the Dirichlet distribution then CD does not have the Dirichlet distribution.

7. Show that when d = 1, the multivariate F -distribution reduces to one-dimensional F (m, n) distribution with the density function

f(x) = \frac{Γ((m + n)/2)m^{m/2}n^{n/2}x^{m/2−1}}{Γ(m/2)Γ(n/2)(n + mx)^{(m+n)/2}}, x > 0.

8. Find a transformation that lead from the Dirichlet distribution to the multivariate F -distribution (take for simplicity the cases d = 1 and d = 2).

9. Suppose X = (X_{1}, X_{2})′ is uniformly distributed over the triangle {(x, y) : 0 < x < y < 1}. Find: (i) the density of X; (ii) the marginal densities of X_{1} and X_{2}. Are X_{1} and X_{2} independent?

10. Two points are picked independently and uniformly at random from the region inside a circle of the unit radius. Let R_{1} and R_{2} be the distances of these points from the center of the circle. Find P (R_{2} ≤ R_{1}/2).

11. Let X = min{U, V } and Y = max{U, V } for independent uniform [0, 1] variables U and V. Find the distribution of: (i) X; (ii) 1 − Y; (iii) Y − X.