MvtUniform {NonNorMvtDist}

Cook-Johnson's Multivariate Uniform Distribution

Description

Calculation of density function, cumulative distribution function, equicoordinate quantile function and survival function, and random numbers generation for Cook-Johnson’s multivariate uniform distribution with a scalar parameter parm.

Usage

dmvunif(x, parm = 1, log = FALSE)

pmvunif(q, parm = 1)

qmvunif(p, parm = 1, dim = k, interval = c(0, 1))

rmvunif(n, parm = 1, dim = 1)

smvunif(q, parm = 1)

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).

parm

a scalar parameter, see parameter a 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.

dim

dimension of data or number of variates (k).

interval

a vector containing the end-points of the interval to be searched. Default value is set as c(0, 1).

n

number of observations.

Details

Multivariate uniform distribution of Cook and Johnson (1981) is a joint distribution of uniform variables over (0,1] and its probability density is given as

f(x_1, \cdots, x_k) = \frac{\Gamma(a+k)}{\Gamma(a)a^k}\prod_{i=1}^{k} x_i^{(-1/a)-1} \left[\sum_{i=1}^{k} x_i^{-1/a} - k +1 \right]^{-(a+k)},

where 0 < x_i <=1, a>0, i=1,\cdots, k. In fact, Cook-Johnson's uniform distribution is also called Clayton copula (Nelsen, 2006).

Cumulative distribution function F(x_1, \dots, x_k) is given as

F(x_1, \cdots, x_k) = \left[ \sum_{i=1}^{k} x_i^{-1/a} - k + 1 \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.

The survival function \bar{F}(x_1, \cdots, x_k) is obtained 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).

Random numbers X_1, \cdots, X_k from Cook-Johnson’s multivariate uniform distribution can be generated through transformation of multivariate Lomax random variables Y_1, \cdots, Y_k by letting X_i = (1+\theta_i Y_i)^{-a}, i = 1, \cdots, k; see Nayak (1987).

Value

dmvunif gives the numerical values of the probability density.

pmvunif gives the cumulative probability.

qmvunif gives the equicoordinate quantile.

rmvunif generates random numbers.

smvunif gives the value of survival function.

References

Cook, R. E. and Johnson, M. E. (1981). A family of distributions for modeling non-elliptically symmetric multivariate data. J.R. Statist. Soc. B 43, No. 2, 210-218.

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.

Nelsen, R. B. (2006). An Introduction to Copulas, Second Edition. New York: Springer.

See Also

uniroot for one dimensional root (zero) finding.

Examples

# Calculations for the Cook-Johnson's multivariate uniform distribution with parameters:
# a = 2, dim = 3
# Vector of quantiles: c(0.8, 0.5, 0.2)

dmvunif(x = c(0.8, 0.5, 0.2), parm = 2) # Density

pmvunif(q = c(0.8, 0.5, 0.2), parm = 2) # Cumulative Probability

# Equicoordinate quantile of cumulative probability 0.5
qmvunif(p = 0.5, parm = 2, dim = 3)

# Random numbers generation with sample size 100
rmvunif(n = 100, parm = 2, dim = 3)

smvunif(q = c(0.8, 0.5, 0.2), parm = 3) # Survival function