selectiongain (version 1.2.1.0)

calculatefromalpha: Non-equi coordinate quantile for singly-truncated multivariate normal distribution

Description

This function calculates the non-equi coordinate quantile vector $Q={q_{1},q_{2},...,q_{n}}$ for a multivariate normal distribution from a given alpha vector $\vec{\alpha}={ \alpha_{1},\alpha_{2},...,\alpha_{n} }$. It can be compared with the function qmvnorm() in R-package mvtnorm, which calculates the equi coordinate quantile $q$ for multi-variate normal distribution from a given alpha scalar. E.g. Let $f$ be the density function, for $n=2$: the equi coordinate quantile $\vec{q}={q,q,...,q}$ defined by mvtnorm is: $\alpha=\int_{q}^{\infty}\int_{q}^{\infty} f(x_1,x_2) dx_1 dx_2$ the non-equi coordinate quantile $Q={q_{1},q_{2},...,q_{n}}$ defined by us is: $\alpha_1=\int_{q_1}^{\infty} f(x_1) dx_1$ $\alpha_2=\int_{q_1}^{\infty}\int_{q_2}^{\infty} f(x_1,x_2) dx_1 dx_2$

Usage

calculatefromalpha(alpha, dim, corr, alg)

Arguments

dim

Dimension of the random variable X.

alpha

Probability vector $\vec{\alpha}$ for random variable X. In plant breeding, it is also called the selected fraction.

corr

Correlation matrix

alg

It decides which algorithm will be used, the Genz and Bretz' algorithm is used by default. The Miwa's algorithm can be optional (Mi. et al. 2009; Genz. et al. 2010)

Value

• The out come is a vector for the coordinates.

Details

There is a limitation, if the absolute value of $q_i$ is greater than 100, then the absolute value is settled to be 100.

References

W.G. Cochran. Improvent by means of selection. In: Proceedings Second Berkeley Symposium on Math Stat Prof, pp449-470. G.M. Tallis. Moment generating function of truncated multi-normal distribution. Journal of the Royal Statistical Society, Series B, 23(1):223-229, 1961. H.F. Utz. Mehrstufenselecktion in der Pflanzenzuechtung. Doctor thesis, University Hohenheim, 1969. X. Mi, T. Miwa and T. Hothorn. Implement of Miwa's analytical algorithm of multi-normal distribution, R Journal, 1:37-39, 2009. A., Genz, F., Bretz (2009). Computation of Multivariate Normal and t Probabilities. Lecture Notes in Statistics, Vol. 195, Springer-Verlag, Heidelberg. A., Genz, F., Bretz, T., Miwa, X., Mi, F., Leisch, F., Scheipl, T., Hothorn, 2010. mvtnorm: Multivariate normal and t distributions. R package version 0.9-9.

See Also

selectiongain(), qnorm()

Examples

##---- Should be DIRECTLY executable !! ----
##-- ==>  Define data, use random,
##--	or do  help(data=index)  for the standard data sets.

 dim=3
 alpha=rep(0,dim)

 alpha[1] = 0.3333
 alpha[2] = 0.25
 alpha[3] = 0.2



corr=diag(3)


corr12=0.3016
corr[1,2]=corr12
corr[2,1]=corr12

corr13=0.5630
corr[1,3]=corr13
corr[3,1]=corr13

corr23=0.5630
corr[2,3]=corr23
corr[3,2]=corr23

calculatefromalpha(alpha,dim=3,corr)