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 generalized multivariate Lomax 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 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 | ```
dmvglomax(x, parm1 = 1, parm2 = rep(1, k), parm3 = rep(1, k), log = FALSE)
pmvglomax(
q,
parm1 = 1,
parm2 = rep(1, k),
parm3 = rep(1, k),
algorithm = c("numerical", "MC"),
nsim = 1e+07
)
qmvglomax(
p,
parm1 = 1,
parm2 = rep(1, k),
parm3 = rep(1, k),
interval = c(1e-08, 1e+08),
algorithm = c("numerical", "MC"),
nsim = 1e+06
)
rmvglomax(n, parm1 = 1, parm2 = rep(1, k), parm3 = rep(1, k))
smvglomax(
q,
parm1 = 1,
parm2 = rep(1, k),
parm3 = rep(1, k),
algorithm = c("numerical", "MC"),
nsim = 1e+07
)
``` |

`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. |

`algorithm` |
method to be used for calculating cumulative probability. Two options are provided as (i) |

`nsim` |
number of simulations used in algorithm |

`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. |

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

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

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

Cumulative distribution function *F(x_1, …, x_k)* is obtained by multiple integral

*F(x_1, …, x_k) = \int_{0}^{x_1} \cdots \int_{0}^{x_k} f(y_1, \cdots, y_k) dy_k \cdots dy_1.*

This multiple integral is calculated by either adaptive multivariate integration using `hcubature`

in package **cubature** (Narasimhan et al., 2018) or via Monte Carlo method.

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 either 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),*

or via Monte Carlo method.

Random numbers from generalized multivariate Lomax distribution can be generated by simulating *k* independent gamma random variables having a common parameter following gamma distribution with shape parameter *a* and scale parameter *1*; see Nayak (1987).

`dmvglomax`

gives the numerical values of the probability density.

`pmvglomax`

gives a list of two items:

*\quad* `value`

cumulative probability

*\quad* `error`

the estimated relative error by `algorithm = "numerical"`

or the estimated standard error by `algorithm = "MC"`

`qmvglomax`

gives the equicoordinate quantile. `NaN`

is returned for no solution found in the given interval. The result is seed dependent if Monte Carlo algorithm is chosen (`algorithm = "MC"`

).

`rmvglomax`

generates random numbers.

`smvglomax`

gives a list of two items:

*\quad* `value`

the value of survial function

*\quad* `error`

the estimated relative error by `algorithm = "numerical"`

or the estimated standard error by `algorithm = "MC"`

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

Narasimhan, B., Koller, M., Johnson, S. G., Hahn, T., Bouvier, A., Kiêu, K. and Gaure, S. (2018). cubature: Adaptive Multivariate Integration over Hypercubes. R package version 2.0.3.

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 16 17 18 19 20 21 22 | ```
# Calculations for the generalized multivariate Lomax with parameters:
# a = 5, theta1 = 1, theta2 = 2, l1 = 4, l2 = 5
# Vector of quantiles: c(5, 6)
dmvglomax(x = c(5, 6), parm1 = 5, parm2 = c(1, 2), parm3 = c(4, 5)) # Density
# Cumulative Probability using adaptive multivariate integral
pmvglomax(q = c(5, 6), parm1 = 5, parm2 = c(1, 2), parm3 = c(4, 5))
# Cumulative Probability using Monte Carlo method
pmvglomax(q = c(5, 6), parm1 = 5, parm2 = c(1, 2), parm3 = c(4, 5), algorithm = "MC")
# Equicoordinate quantile of cumulative probability 0.5
qmvglomax(p = 0.5, parm1 = 5, parm2 = c(1, 2), parm3 = c(4, 5))
# Random numbers generation with sample size 100
rmvglomax(n = 100, parm1 = 5, parm2 = c(1, 2), parm3 = c(4, 5))
smvglomax(q = c(5, 6), parm1 = 5, parm2 = c(1, 2), parm3 = c(4, 5)) # Survival function
``` |

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