R/CentralAndStandardizedStatistics.R

In R-Finance/Meucci: Collection of functionality ported from the MATLAB code of Attilio Meucci.

#' @title Compute central and standardized statistics.
#'
#' @description Compute central and standardized statistics, as described in A. Meucci 
#' "Risk and Asset Allocation", Springer, 2005.
#'
#'	Computes the central moments \deqn{ CM_1^X \equiv \mu_{X}\,, \quad CM_n^X \equiv E \{(X - E\{ X \})^{n}\}\,, \quad n=2,3,\ldots ,}
#'  and from them the standarized statistics \deqn{ \mu_{X},\sigma_{X},sk_{X},ku_{X},\gamma_{X}^{(5)}, \ldots ,\gamma_{X}^{(n)} .}
#'  where \deqn{\gamma_{X}^{(n)} \equiv E \{(X - \mu_{X})^{n}\}/\sigma_{X}^{n},\quad n\geq3 .} 
#'
#'	@param    X   [vector] (J x 1) draws from the distribution
#'	@param    N   [scalar] highest degree for the central moment
#'
#'	@return   ga  [vector] (1 x N) standardized statistics up to order N
#'	@return   mu  [vector] (1 x N) central moments up to order N
#'
#' @references
#' A. Meucci - "Exercises in Advanced Risk and Portfolio Management" \url{http://symmys.com/node/170},
#' "E 97 - Projection of skewness, kurtosis, and all standardized summary statistics".
#' See Meucci's script for "CentralAndStandardizedStatistics.m"
#'
#' Kendall, M., Stuart, A. - "The Advanced Theory of Statistics", 1969. Volume, 3rd Edition. Griffin.
#'
#' A. Meucci - "Annualization and general projection of skweness, kurtosis, and all summary statistics",
#' GARP Risk Professional August 2010, 55-56. \url{http://symmys.com/node/136}.
#' @author Xavier Valls \email{flamejat@@gmail.com}
#' @export

CentralAndStandardizedStatistics = function( X, N )
{
	if( !require("PerformanceAnalytics") ) stop("PerformanceAnalytics package required for this script");
	# compute central moments
	mu = matrix( 0, 1, N);
	mu[ 1 ] = mean(X);
	for( n in 2 : N )
	{
	    mu[ n ] = PerformanceAnalytics:::centeredmoment(X, n);
	}

	# compute standardized statistics 

	ga = mu;

	ga[ 2 ] = sqrt( mu[ 2 ]);
	for( n in 3 : N )
	{
	    ga[ n ] = mu[ n ] / (ga[ 2 ] ^ n);
	}
	
	return( list( ga = ga, mu = mu ) );

}