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 Mardia's multivariate Pareto Type I distribution with a scalar parameter `parm1`

and a vector of parameters `parm2`

.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 | ```
dmvmpareto1(x, parm1 = 1, parm2 = rep(1, k), log = FALSE)
pmvmpareto1(q, parm1 = 1, parm2 = rep(1, k))
qmvmpareto1(
p,
parm1 = 1,
parm2 = rep(1, k),
interval = c(max(1/parm2) + 1e-08, 1e+08)
)
rmvmpareto1(n, parm1 = 1, parm2 = rep(1, k))
smvmpareto1(q, parm1 = 1, parm2 = rep(1, k))
``` |

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

`parm1` |
a scalar parameter, see parameter |

`parm2` |
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. |

Multivariate Pareto type I distribution was introduced by Mardia (1962) as a joint probability distribution of several nonnegative random variables *X_1, \cdots, X_k*. Its probability density function is given by

*f(x_1, \cdots, x_k) = \frac{[ ∏_{i=1}^{k} θ_i] a(a+1) \cdots (a+k-1)}{(∑_{i=1}^{k} θ_i x_i - k + 1)^{a+k}},*

where *x_i > 1 / θ_i, a >0, θ_i>0, i=1,\cdots, k*.

Cumulative distribution function *F(x_1, …, x_k)* is obtained by the following formula related to survival function *\bar{F}(x_1, …, x_k)* (Joe, 1997)

*F(x_1, …, x_k) = 1 + ∑_{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) = ≤ft( ∑_{i=1}^{k} θ_i x_i - 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.

Random numbers *X_1, \cdots, X_k* from Mardia's multivariate Pareto type I distribution can be generated through linear transformation of multivariate Lomax random variables *Y_1, \cdots, Y_k* by letting *X_i = Y_i + 1/θ_i, i = 1, \cdots, k*; see Nayak (1987).

`dmvmpareto1`

gives the numerical values of the probability density.

`pmvmpareto1`

gives the cumulative probability.

`qmvmpareto1`

gives the equicoordinate quantile.

`rmvmpareto1`

generates random numbers.

`smvmpareto1`

gives the value of survival function.

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

Mardia, K. V. (1962). Multivariate Pareto distributions. *Ann. Math. Statist.* 33, 1008-1015.

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

`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 Mardia's multivariate Pareto Type I with parameters:
# a = 5, theta1 = 1, theta2 = 2, theta3 = 3
# Vector of quantiles: c(2, 1, 1)
dmvmpareto1(x = c(2, 1, 1), parm1 = 5, parm2 = c(1, 2, 3)) # Density
pmvmpareto1(q = c(2, 1, 1), parm1 = 5, parm2 = c(1, 2, 3)) # Cumulative Probability
# Equicoordinate quantile of cumulative probability 0.5
qmvmpareto1(p = 0.5, parm1 = 5, parm2 = c(1, 2, 3))
# Random numbers generation with sample size 100
rmvmpareto1(n = 100, parm1 = 5, parm2 = c(1, 2, 3))
smvmpareto1(q = c(2, 1, 1), parm1 = 5, parm2 = c(1, 2, 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.