Description Usage Arguments Details Value References See Also Examples

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`

.

1 2 3 4 5 6 7 8 9 |

`x` |
vector or matrix of quantiles. If |

`parm` |
a scalar parameter, see parameter |

`log` |
logical; if TRUE, probability densities |

`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 |

`n` |
number of observations. |

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{Γ(a+k)}{Γ(a)a^k}∏_{i=1}^{k} x_i^{(-1/a)-1} ≤ft[∑_{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, …, x_k)* is given as

*F(x_1, \cdots, x_k) = ≤ft[ ∑_{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, …, 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).*

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+θ_i Y_i)^{-a}, i = 1, \cdots, k*; see Nayak (1987).

`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.

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.

`uniroot`

for one dimensional root (zero) finding.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | ```
# 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
``` |

Embedding an R snippet on your website

Add the following code to your website.

For more information on customizing the embed code, read Embedding Snippets.