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 multivariate Lomax (Pareto Type II) distribution with a scalar parameter `parm1`

and vector parameter `parm2`

.

1 2 3 4 5 6 7 8 9 |

`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 Lomax (Pareto type II) distribution was introduced by Nayak (1987) as a joint probability distribution of several skewed positive random variables *X_1, X_2, \cdots, X_k*. Its probability density function is given by

*f(x_1, x_2, …, x_k) = \frac{[ ∏_{i=1}^{k} θ_i] a(a+1) \cdots (a+k-1)}{(1+∑_{i=1}^{k} θ_i x_i)^{a+k}},*

where *x_i > 0, a>0, θ_i>0, i=1,…,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, …, x_k) = ( 1+∑_{i=1}^{k} θ_i x_i )^{-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 from multivariate Lomax distribution can be generated by simulating *k* independent exponential random variables having a common environment parameter following gamma distribution with shape parameter *a* and scale parameter *1*; see Nayak (1987).

`dmvlomax`

gives the numerical values of the probability density.

`pmvlomax`

gives the cumulative probability.

`qmvlomax`

gives the equicoordinate quantile.

`rmvlomax`

generates random numbers.

`smvlomax`

gives the value of survival function.

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.

`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 multivariate Lomax with parameters:
# a = 5, theta1 = 1, theta2 = 2 and theta3 = 3.
# Vector of quantiles: c(3, 2, 1)
dmvlomax(x = c(3, 2, 1), parm1 = 5, parm2 = c(1, 2, 3)) # Density
pmvlomax(q = c(3, 2, 1), parm1 = 5, parm2 = c(1, 2, 3)) # Cumulative Probability
# Equicoordinate quantile of cumulative probability 0.5
qmvlomax(p = 0.5, parm1 = 5, parm2 = c(1, 2, 3))
# Random numbers generation with sample size 100
rmvlomax(n = 100, parm1 = 5, parm2 = c(1, 2, 3))
smvlomax(q = c(3, 2, 1), parm1 = 5, parm2 = c(1, 2, 3)) # Survival function
``` |

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