MvtBurr: Multivariate Burr Distribution

Description

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

Usage

dmvburr(x, parm1 = 1, parm2 = rep(1, k), parm3 = rep(1, k), log = FALSE)
pmvburr(q, parm1 = 1, parm2 = rep(1, k), parm3 = rep(1, k))
qmvburr(
  p,
  parm1 = 1,
  parm2 = rep(1, k),
  parm3 = rep(1, k),
  interval = c(0, 1e+08)
)
rmvburr(n, parm1 = 1, parm2 = rep(1, k), parm3 = rep(1, k))
smvburr(q, parm1 = 1, parm2 = rep(1, k), parm3 = rep(1, k))

Value

dmvburr gives the numerical values of the probability density.

pmvburr gives the cumulative probability.

qmvburr gives the equicoordinate quantile.

rmvburr generates random numbers.

smvburr gives the value of survival function.

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 $d_i$ in Details.
parm3: a vector of parameters, see parameters $c_i$ 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 Burr distribution (Johnson and Kotz, 1972) is a joint distribution of positive random variables $X_1, \cdots, X_k$. Its probability density is given as $$f(x_1, \cdots, x_k) = \frac{[ \prod_{i=1}^{k} c_i d_i] a(a+1) \cdots (a+k-1) [ \prod_{i=1}^{k} x_i^{c_i-1}]}{(1 + \sum_{i=1}^{k} d_i x_i^{c_i})^{a+k}},$$ where $x_i >0, a,c_i,d_i>0, i=1,\cdots, k$.

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 $$\bar{F}(x_1, \cdots, x_k) = \left( 1+\sum_{i=1}^{k} d_i x_i^{c_i} \right)^{-a}.$$

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 multivariate Burr distribution can be generated through transformation of multivariate Lomax random variables $Y_1, \cdots, Y_k$ by letting $X_i=(\theta_i Y_i/d_i)^{1/c_i}, i = 1, \cdots, k$; see Nayak (1987).

References

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

Johnson, N. L. and Kotz, S. (1972). Distribution in Statistics: Continuous Multivariate Distributions. New York: John Wiley & Sons, INC.

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

Examples

Run this code

# Calculations for the multivariate Burr with parameters:
# a = 3, d1 = 1, d2 = 3, d3 = 5, c1 = 2, c2 = 4, c3 = 6
# Vector of quantiles: c(3, 2, 1)

dmvburr(x = c(3, 2, 1), parm1 = 3, parm2 = c(1, 3, 5), parm3 = c(2, 4, 6)) # Density

pmvburr(q = c(3, 2, 1), parm1 = 3, parm2 = c(1, 3, 5), parm3 = c(2, 4, 6)) # Cumulative Probability

# Equicoordinate quantile of cumulative probability 0.5
qmvburr(p = 0.5, parm1 = 3, parm2 = c(1, 3, 5), parm3 = c(2, 4, 6))

# Random numbers generation with sample size 100
rmvburr(n = 100, parm1 = 3, parm2 = c(1, 3, 5), parm3 = c(2, 4, 6))

smvburr(q = c(3, 2, 1), parm1 = 3, parm2 = c(1, 3, 5), parm3 = c(2, 4, 6)) # Survival function

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