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 logistic distribution with vector parameter `parm1`

and vector parameter `parm2`

.

1 2 3 4 5 6 7 8 9 |

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

`parm1` |
a vector of location parameters, see parameter |

`parm2` |
a vector of scale 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. |

Bivariate logistic distribution was introduced by Gumbel (1961) and its multivariate generalization was given by Malik and Abraham (1973) as

*f(x_1, \cdots, x_k) = \frac{k! \exp{(-∑_{i=1}^{k} \frac{x_i - μ_i}{σ_i})}}{[∏_{i=1}^{p} σ_i] [1 + ∑_{i=1}^{k} \exp{(-\frac{x_i - μ_i}{σ_i})}]^{1+k}},*

where *-∞<x_i, μ_i<∞, σ_i > 0, i=1,\cdots, k*.

Cumulative distribution function *F(x_1, …, x_k)* is given as

*F(x_1, \cdots, x_k) = ≤ft[1 + ∑_{i=1}^{k} \exp(-\frac{x_i-μ_i}{σ_i})\right]^{-1}.*

Equicoordinate quantile is obtained by solving the following equation for *q* through the built-in one dimension root finding function `uniroot`

:

*\int_{-∞}^{q} \cdots \int_{-∞}^{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 multivariate logistic distribution can be generated through transformation of multivariate Lomax random variables *Y_1, \cdots, Y_k* by letting *X_i=μ_i-σ_i\ln(θ_i Y_i), i = 1, \cdots, k*; see Nayak (1987).

`dmvlogis`

gives the numerical values of the probability density.

`pmvlogis`

gives the cumulative probability.

`qmvlogis`

gives the equicoordinate quantile.

`rmvlogis`

generates random numbers.

`smvlogis`

gives the value of survival function

Gumbel, E.J. (1961). Bivariate logistic distribution. *J. Am. Stat. Assoc.*, 56, 335-349.

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

Malik, H. J. and Abraham, B. (1973). Multivariate logistic distributions. *Ann. Statist.* 3, 588-590.

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 logistic distribution with parameters:
# mu1 = 0.5, mu2 = 1, mu3 = 2, sigma1 = 1, sigma2 = 2 and sigma3 = 3
# Vector of quantiles: c(3, 2, 1)
dmvlogis(x = c(3, 2, 1), parm1 = c(0.5, 1, 2), parm2 = c(1, 2, 3)) # Density
pmvlogis(q = c(3, 2, 1), parm1 = c(0.5, 1, 2), parm2 = c(1, 2, 3)) # Cumulative Probability
# Equicoordinate quantile of cumulative probability 0.5
qmvlogis(p = 0.5, parm1 = c(0.5, 1, 2), parm2 = c(1, 2, 3))
# Random numbers generation with sample size 100
rmvlogis(n = 100, parm1 = c(0.5, 1, 2), parm2 = c(1, 2, 3))
smvlogis(q = c(3, 2, 1), parm1 = c(0.5, 1, 2), parm2 = c(1, 2, 3)) # Survival function
``` |

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