MvtLomax {NonNorMvtDist}

Multivariate Lomax (Pareto Type II) Distribution

Description

Calculation of density function, cumulative distribution function, equicoordinate quantile function and survival function, and random numbers generation for multivariate Lomax (Pareto Type II) distribution with a scalar parameter parm1 and vector parameter parm2.

Usage

dmvlomax(x, parm1 = 1, parm2 = rep(1, k), log = FALSE)

pmvlomax(q, parm1 = 1, parm2 = rep(1, k))

qmvlomax(p, parm1 = 1, parm2 = rep(1, k), interval = c(0, 1e+08))

rmvlomax(n, parm1 = 1, parm2 = rep(1, k))

smvlomax(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(0, 1e8).

n

number of observations.

k

dimension of data or number of variates.

Details

Multivariate Lomax (Pareto type II) distribution was introduced by Nayak (1987) as a joint probability distribution of several skewed positive random variables X_1, X_2, \cdots, X_k. Its probability density function is given by

f(x_1, x_2, \dots, x_k) = \frac{[ \prod_{i=1}^{k} \theta_i] a(a+1) \cdots (a+k-1)}{(1+\sum_{i=1}^{k} \theta_i x_i)^{a+k}},

where x_i > 0, a>0, \theta_i>0, i=1,\dots,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, \dots, x_k) = ( 1+\sum_{i=1}^{k} \theta_i x_i )^{-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 from multivariate Lomax distribution can be generated by simulating k independent exponential random variables having a common environment parameter following gamma distribution with shape parameter a and scale parameter 1; see Nayak (1987).

Value

dmvlomax gives the numerical values of the probability density.

pmvlomax gives the cumulative probability.

qmvlomax gives the equicoordinate quantile.

rmvlomax generates random numbers.

smvlomax gives the value of survival function.

References

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

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

# Calculations for the multivariate Lomax with parameters: 
# a = 5, theta1 = 1, theta2 = 2 and theta3 = 3.
# Vector of quantiles: c(3, 2, 1)

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

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

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

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

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