MvtBurr {NonNorMvtDist} | R Documentation |
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))
Arguments
x |
vector or matrix of quantiles. If |
parm1 |
a scalar parameter, see parameter |
parm2 |
a vector of parameters, see parameters |
parm3 |
a vector of parameters, see parameters |
log |
logical; if TRUE, probability densities |
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 |
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).
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.
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.
See Also
uniroot
for one dimensional root (zero) finding.
Examples
# 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