# MvtF: Multivariate F Distribution In NonNorMvtDist: Multivariate Lomax (Pareto Type II) and Its Related Distributions

## Description

Calculation of density function, cumulative distribution function, equicoordinate quantile function and survival function, and random numbers generation for multivariate F distribution with degrees of freedom df.

## Usage

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 dmvf(x, df = rep(1, k + 1), log = FALSE) pmvf(q, df = rep(1, k + 1), algorithm = c("numerical", "MC"), nsim = 1e+07) qmvf( p, df = rep(1, k + 1), interval = c(1e-08, 1e+08), algorithm = c("numerical", "MC"), nsim = 1e+06 ) rmvf(n, df = rep(1, k + 1)) smvf(q, df = rep(1, k + 1), algorithm = c("numerical", "MC"), nsim = 1e+07) 

## Arguments

 x vector or matrix of quantiles. If x is a matrix, each row vector constitutes a vector of quantiles for which the density f(x) is calculated (for i-th row x_i, f(x_i) is reported). df a vector of k+1 degrees of freedom, see parameter (2a, 2l_1, …, 2l_k) in Details. log logical; if TRUE, probability densities f are given as log(f). q a vector of quantiles. algorithm method to be used for calculating cumulative probability. Two options are provided as (i) numerical using adaptive multivariate integral and (ii) MC using Monte Carlo method. Recommend algorithm numerical for (k <= 4) dimension and MC for (k > 4) dimension based on running time consumption. Default option is set as numerical. nsim number of simulations used in algorithm MC. 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 c(1e-8, 1e8). n number of observations. k dimension of data or number of variates.

## Details

Multivariate F distribution (Johnson and Kotz, 1972) is a joint probability distribution of positive random variables and its probability density is given as

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

where x_i>0, a>0, l_i>0, i=1,\cdots, k. The degrees of freedom are (2a, 2l_1,…,2l_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 X_1, \cdots, X_k from multivariate F distribution can be generated through parameter substitutions in simulation of generalized multivariate Lomax distribution by letting θ_i = l_i/a, i = 1, \cdots, k; see Nayak (1987).

## Value

dmvf gives the numerical values of the probability density.

pmvf 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"

qmvf 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").

rmvf generates random numbers.

smvf 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"

## References

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.

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 # Calculations for the multivariate F with degrees of freedom: # df = c(2, 4, 6) # Vector of quantiles: c(1, 2) dmvf(x = c(1, 2), df = c(2, 4, 6)) # Density # Cumulative Probability using adaptive multivariate integral pmvf(q = c(1, 2), df = c(2, 4, 6), algorithm = "numerical") # Cumulative Probability using Monte Carlo method pmvf(q = c(1, 2), df = c(2, 4, 6), algorithm = "MC") # Equicoordinate quantile of cumulative probability 0.5 qmvf(p = 0.5, df = c(2, 4, 6)) # Random numbers generation with sample size 100 rmvf(n = 100, df = c(2, 4, 6)) smvf(q = c(1, 2), df = c(2, 4, 6)) # Survival function