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 Burr distribution with a scalar parameter `parm1`

and vectors of parameters `parm2`

and `parm3`

.

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

`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. |

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{[ ∏_{i=1}^{k} c_i d_i] a(a+1) \cdots (a+k-1) [ ∏_{i=1}^{k} x_i^{c_i-1}]}{(1 + ∑_{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, …, 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( 1+∑_{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=(θ_i Y_i/d_i)^{1/c_i}, i = 1, \cdots, k*; see Nayak (1987).

`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.

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.

`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 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
``` |

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