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Multivariate-T Simulation Problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg38986] Multivariate-T Simulation Problem
  • From: KyriakosChourdakis <tuxedomoon at yahoo.com>
  • Date: Thu, 23 Jan 2003 08:05:54 -0500 (EST)
  • Reply-to: k.chourdakis at qmul.ac.uk
  • Sender: owner-wri-mathgroup at wolfram.com

Dear all,

I am trying to simulate a multivariate t-distribution,
and encounter the following problem:

(* First load the packages *)
<<Statistics`MultinormalDistribution`
<<LinearAlgebra`MatrixManipulation`

NS = 2; (* the number of series *)
df = 20; (* the degrees of freedom - Since the number
is large, it should make the t similar to a normal *)

(* Create a random variance-covariance matrix *)
rS0 = RandomArray[UniformDistribution[-0.30,
0.80],{NS, NS}];
rS0 = rS0.Transpose[rS0];
rS0 = (rS0/Max[rS0]) (1 - IdentityMatrix[NS]) + 
      IdentityMatrix[NS];

(* the var-covar matrix is *)
rS0
{{1,         -0.290755},
 {-0.290755, 1        }}

(* create 10000 draws of 2 dimensional vectors *)
ET =
Transpose[RandomArray[MultivariateTDistribution[rS0,
df], 10000]];

(* the theoretical cov matrix is ok, with
var=df/[df-2]... *)
ST = N[CovarianceMatrix[MultivariateTDistribution[rS0,
df]]]
{{1.11111,  -0.323061}, 
 {-0.323061,  1.11111}}

(* ... But the simulated variance is not*)
Mean /@ ET
Variance /@ ET
{-0.00426186, 0.00824351}
{0.067534, 0.0670659}

(* On the other hand, if I draw from the corresponding
multinormal *)
EN = Transpose[RandomArray[MultinormalDistribution[{0,
0}, ST], 10000]];

(* Everything is OK *)
Mean /@ EN
Variance /@ EN
{0.0196956, -0.00675908} (* Should be 0 *)
{1.1362, 1.1099}         (* Should be 1.111 *)

I cannot understand why the simulated variance is
0.067 instead of 1.111/[20-2]. Alternatively, why
should I multiply all values with sqrt(1.111/0.067)=4.

Best,

Kyriakos


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