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Gradient and Hessian matrix of cumulative normal ditribution

  • To: mathgroup at smc.vnet.net
  • Subject: [mg67954] Gradient and Hessian matrix of cumulative normal ditribution
  • From: "Pratim Vakish" <pratim_usc at hotmail.com>
  • Date: Thu, 13 Jul 2006 06:55:40 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Hello,

I am very new in Mathematica, and my question might be easy very easy.

My question is related to M Lejeune' question this June and to B. Hanlon and 
R. Koopman's comments.

I am considering multivariate normal distributions.

I am interested in computing the gradient (vector of first-order partial 
derivative) and Hessian matrix of the multivariate cumulative normal 
distribution.

Could you indicate me how to proceed?
First, I would like to obtain the algebraic formulation of the gradient and 
Hessian matrix (is thre a closed-form formulatyion for their components?), 
and then, evaluate/compute the value of the gradient and Hessian at a given 
point.

Taking the bivariate normal distribution and the examole discussed by M 
Lejeune and B Hanlon,

r = {{1, 0.2}, {0.2, 1}};
ndist = MultinormalDistribution[{0, 0}, r])
cdf=CDF[ndist,{x1,x2}]

What is the gradient and Hessian matrix of cdf in algebraic formulation?

What is the values of the gradient and Hessian matrix of cdf at the point 
x1=1, x2=0.8, for example?

Many thanks,

Pratim

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