NonNorMvtDist (version 1.0.2)

MvtMardiaPareto1: Mardia's Multivariate Pareto Type I Distribution

Description

Calculation of density function, cumulative distribution function, equicoordinate quantile function and survival function, and random numbers generation for Mardia's multivariate Pareto Type I distribution with a scalar parameter parm1 and a vector of parameters parm2.

Usage

dmvmpareto1(x, parm1 = 1, parm2 = rep(1, k), log = FALSE)
pmvmpareto1(q, parm1 = 1, parm2 = rep(1, k))
qmvmpareto1(
  p,
  parm1 = 1,
  parm2 = rep(1, k),
  interval = c(max(1/parm2) + 1e-08, 1e+08)
)
rmvmpareto1(n, parm1 = 1, parm2 = rep(1, k))
smvmpareto1(q, parm1 = 1, parm2 = rep(1, k))

Arguments

x

vector or matrix of quantiles. If \(x\) is a matrix, each row vector constitutes a vector of quantiles for which the density \(f(x)\) is calculated (for \(i\)-th row \(x_i\), \(f(x_i)\) is reported).

parm1

a scalar parameter, see parameter \(a\) in Details.

parm2

a vector of parameters, see parameters \(\theta_i\) in Details.

log

logical; if TRUE, probability densities \(f\) are given as \(log(f)\).

q

a vector of quantiles.

p

a scalar value corresponding to probability.

interval

a vector containing the end-points of the interval to be searched. Default value is set as c(max(1 / parm2) + 1e-8, 1e8) according to \(x_i > 1 / \theta_i, \theta_i>0, i=1,\cdots, k\).

n

number of observations.

k

dimension of data or number of variates.

Value

dmvmpareto1 gives the numerical values of the probability density.

pmvmpareto1 gives the cumulative probability.

qmvmpareto1 gives the equicoordinate quantile.

rmvmpareto1 generates random numbers.

smvmpareto1 gives the value of survival function.

Details

Multivariate Pareto type I distribution was introduced by Mardia (1962) as a joint probability distribution of several nonnegative random variables \(X_1, \cdots, X_k\). Its probability density function is given by $$f(x_1, \cdots, x_k) = \frac{[ \prod_{i=1}^{k} \theta_i] a(a+1) \cdots (a+k-1)}{(\sum_{i=1}^{k} \theta_i x_i - k + 1)^{a+k}},$$ where \(x_i > 1 / \theta_i, a >0, \theta_i>0, i=1,\cdots, k\).

Cumulative distribution function \(F(x_1, \dots, x_k)\) is obtained by the following formula related to survival function \(\bar{F}(x_1, \dots, x_k)\) (Joe, 1997) $$F(x_1, \dots, x_k) = 1 + \sum_{S \in \mathcal{S}} (-1)^{|S|} \bar{F}_S(x_j, j \in S),$$ where the survival function is given by $$\bar{F}(x_1, \cdots, x_k) = \left( \sum_{i=1}^{k} \theta_i x_i - k + 1 \right)^{-a}.$$

Equicoordinate quantile is obtained by solving the following equation for \(q\) through the built-in one dimension root finding function uniroot: $$\int_{0}^{q} \cdots \int_{0}^{q} f(x_1, \cdots, x_k) dx_k \cdots dx_1 = p,$$ where \(p\) is the given cumulative probability.

Random numbers \(X_1, \cdots, X_k\) from Mardia's multivariate Pareto type I distribution can be generated through linear transformation of multivariate Lomax random variables \(Y_1, \cdots, Y_k\) by letting \(X_i = Y_i + 1/\theta_i, i = 1, \cdots, k\); see Nayak (1987).

References

Joe, H. (1997). Multivariate Models and Dependence Concepts. London: Chapman & Hall.

Mardia, K. V. (1962). Multivariate Pareto distributions. Ann. Math. Statist. 33, 1008-1015.

Nayak, T. K. (1987). Multivariate Lomax Distribution: Properties and Usefulness in Reliability Theory. Journal of Applied Probability, Vol. 24, No. 1, 170-177.

See Also

uniroot for one dimensional root (zero) finding.

Examples

# NOT RUN {
# Calculations for the Mardia's multivariate Pareto Type I with parameters:
# a = 5, theta1 = 1, theta2 = 2, theta3 = 3
# Vector of quantiles: c(2, 1, 1)

dmvmpareto1(x = c(2, 1, 1), parm1 = 5, parm2 = c(1, 2, 3)) # Density

pmvmpareto1(q = c(2, 1, 1), parm1 = 5, parm2 = c(1, 2, 3)) # Cumulative Probability

# Equicoordinate quantile of cumulative probability 0.5
qmvmpareto1(p = 0.5, parm1 = 5, parm2 =  c(1, 2, 3))

# Random numbers generation with sample size 100
rmvmpareto1(n = 100, parm1 = 5, parm2 = c(1, 2, 3)) 

smvmpareto1(q = c(2, 1, 1), parm1 = 5, parm2 = c(1, 2, 3)) # Survival function

# }