NonNorMvtDist (version 1.1.0)

MvtGlomx: Generalized 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 generalized multivariate Lomax distribution with a scalar parameter parm1 and vectors of parameters parm2 and parm3.

Usage

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
)

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"

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.

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 $$f(x_1, \cdots, x_k) = \frac{[ \prod_{i=1}^{k} \theta_i^{l_i}] \Gamma(\sum_{i=1}^{k} l_i + a ) \prod_{i=1}^{k} x_i^{l_i-1}}{\Gamma(a)[ \prod_{i=1}^{k} \Gamma(l_i)] (1+\sum_{i=1}^{k} \theta_i x_i )^{\sum_{i=1}^{k} l_i + a}},$$ where \(x_i>0, a,\theta_i, l_i>0, i=1,\cdots,k\).

Cumulative distribution function \(F(x_1, \dots, x_k)\) is obtained by multiple integral $$F(x_1, \dots, x_k) = \int_{0}^{x_1} \cdots \int_{0}^{x_k} f(y_1, \cdots, y_k) dy_k \cdots dy_1.$$ 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 \(F(x_1, \dots, x_k)\) (Joe, 1997) $$\bar{F}(x_1, \cdots, x_k) = 1 + \sum_{S \in \mathcal{S}} (-1)^{|S|} F_S(x_j, j \in S),$$ 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).

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.

See Also

uniroot for one dimensional root (zero) finding.

Examples

# 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))

# \donttest{
# Cumulative Probability using Monte Carlo method
pmvglomax(q = c(5, 6), parm1 = 5, parm2 = c(1, 2), parm3 = c(4, 5), algorithm = "MC")# }

# \donttest{
# 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