# MvtGlomx: Generalized Multivariate Lomax (Pareto Type II) Distribution In NonNorMvtDist: Multivariate Lomax (Pareto Type II) and Its Related Distributions

## Description

Calculation of density function, cumulative distribution function, equicoordinate quantile function and survival function, and random numbers generation for generalized multivariate Lomax distribution with a scalar parameter parm1 and vectors of parameters parm2 and parm3.

## Usage

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 dmvglomax(x, parm1 = 1, parm2 = rep(1, k), parm3 = rep(1, k), log = FALSE) pmvglomax( q, parm1 = 1, parm2 = rep(1, k), parm3 = rep(1, k), algorithm = c("numerical", "MC"), nsim = 1e+07 ) qmvglomax( p, parm1 = 1, parm2 = rep(1, k), parm3 = rep(1, k), interval = c(1e-08, 1e+08), algorithm = c("numerical", "MC"), nsim = 1e+06 ) rmvglomax(n, parm1 = 1, parm2 = rep(1, k), parm3 = rep(1, k)) smvglomax( q, parm1 = 1, parm2 = rep(1, k), parm3 = rep(1, k), algorithm = c("numerical", "MC"), nsim = 1e+07 ) 

## 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 θ_i in Details. parm3 a vector of parameters, see parameters l_i in Details. log logical; if TRUE, probability densities f are given as log(f). q a vector of quantiles. algorithm method to be used for calculating cumulative probability. Two options are provided as (i) numerical using adaptive multivariate integral and (ii) MC using Monte Carlo method. Recommend algorithm numerical for (k <= 4) dimension and MC for (k > 4) dimension based on running time consumption. Default option is set as numerical. nsim number of simulations used in algorithm MC. 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(1e-8, 1e8). n number of observations. k dimension of data or number of variates.

## Details

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

f(x_1, \cdots, x_k) = \frac{[ ∏_{i=1}^{k} θ_i^{l_i}] Γ(∑_{i=1}^{k} l_i + a ) ∏_{i=1}^{k} x_i^{l_i-1}}{Γ(a)[ ∏_{i=1}^{k} Γ(l_i)] (1+∑_{i=1}^{k} θ_i x_i )^{∑_{i=1}^{k} l_i + a}},

where x_i>0, a,θ_i, l_i>0, i=1,\cdots,k.

Cumulative distribution function F(x_1, …, x_k) is obtained by multiple integral

F(x_1, …, x_k) = \int_{0}^{x_1} \cdots \int_{0}^{x_k} f(y_1, \cdots, y_k) dy_k \cdots dy_1.

This multiple integral is calculated by either adaptive multivariate integration using hcubature in package cubature (Narasimhan et al., 2018) or via Monte Carlo method.

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.

The survival function \bar{F}(x_1, \cdots, x_k) is obtained either by the following formula related to cumulative distribution function F(x_1, …, x_k) (Joe, 1997)

\bar{F}(x_1, \cdots, x_k) = 1 + ∑_{S \in \mathcal{S}} (-1)^{|S|} F_S(x_j, j \in S),

or via Monte Carlo method.

Random numbers from generalized multivariate Lomax distribution can be generated by simulating k independent gamma random variables having a common parameter following gamma distribution with shape parameter a and scale parameter 1; see Nayak (1987).

## Value

dmvglomax gives the numerical values of the probability density.

pmvglomax gives a list of two items:

\quad value cumulative probability

\quad error the estimated relative error by algorithm = "numerical" or the estimated standard error by algorithm = "MC"

qmvglomax gives the equicoordinate quantile. NaN is returned for no solution found in the given interval. The result is seed dependent if Monte Carlo algorithm is chosen (algorithm = "MC").

rmvglomax generates random numbers.

smvglomax gives a list of two items:

\quad value the value of survial function

\quad error the estimated relative error by algorithm = "numerical" or the estimated standard error by algorithm = "MC"

## References

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

Narasimhan, B., Koller, M., Johnson, S. G., Hahn, T., Bouvier, A., Kiêu, K. and Gaure, S. (2018). cubature: Adaptive Multivariate Integration over Hypercubes. R package version 2.0.3.

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

uniroot for one dimensional root (zero) finding.
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 # Calculations for the generalized multivariate Lomax with parameters: # a = 5, theta1 = 1, theta2 = 2, l1 = 4, l2 = 5 # Vector of quantiles: c(5, 6) dmvglomax(x = c(5, 6), parm1 = 5, parm2 = c(1, 2), parm3 = c(4, 5)) # Density # Cumulative Probability using adaptive multivariate integral pmvglomax(q = c(5, 6), parm1 = 5, parm2 = c(1, 2), parm3 = c(4, 5)) # Cumulative Probability using Monte Carlo method pmvglomax(q = c(5, 6), parm1 = 5, parm2 = c(1, 2), parm3 = c(4, 5), algorithm = "MC") # Equicoordinate quantile of cumulative probability 0.5 qmvglomax(p = 0.5, parm1 = 5, parm2 = c(1, 2), parm3 = c(4, 5)) # Random numbers generation with sample size 100 rmvglomax(n = 100, parm1 = 5, parm2 = c(1, 2), parm3 = c(4, 5)) smvglomax(q = c(5, 6), parm1 = 5, parm2 = c(1, 2), parm3 = c(4, 5)) # Survival function