Documentation

ecmnfish

Fisher information matrix

Syntax

Fisher = ecmnfish(Data,Covariance,InvCovariance,MatrixFormat)

Arguments

 Data NUMSAMPLES-by-NUMSERIES matrix of observed multivariate normal data Covariance NUMSERIES-by-NUMSERIES matrix with covariance estimate of Data InvCovariance (Optional) Inverse of covariance matrix: inv(Covariance) MatrixFormat (Optional) Character vector that identifies parameters included in the Fisher information matrix. If MatrixFormat = [] or '', the default method full is used. The parameter choices are full — (Default) Compute full Fisher information matrix.meanonly — Compute only components of the Fisher information matrix associated with the mean.

Description

Fisher = ecmnfish(Data,Covariance,InvCovariance,MatrixFormat) computes a NUMPARAMS-by-NUMPARAMS Fisher information matrix based on current parameter estimates, where

NUMPARAMS = NUMSERIES*(NUMSERIES + 3)/2

if MatrixFormat = 'full' and

NUMPARAMS = NUMSERIES

if MatrixFormat = 'meanonly'.

The data matrix has NaNs for missing observations. The multivariate normal model has

NUMPARAMS = NUMSERIES + NUMSERIES*(NUMSERIES + 1)/2

distinct parameters. Therefore, the full Fisher information matrix is of size NUMPARAMS-by-NUMPARAMS. The first NUMSERIES parameters are estimates for the mean of the data in Mean and the remaining NUMSERIES*(NUMSERIES + 1)/2 parameters are estimates for the lower-triangular portion of the covariance of the data in Covariance, in row-major order.

If MatrixFormat = 'meanonly', the number of parameters is reduced to NUMPARAMS = NUMSERIES, where the Fisher information matrix is computed for the mean parameters only. In this format, the routine executes fastest.

This routine expects the inverse of the covariance matrix as an input. If you do not pass in the inverse, the routine computes it. You can obtain an approximation for the lower-bound standard errors of estimation of the parameters from

Stderr = (1.0/sqrt(NumSamples)) .* sqrt(diag(inv(Fisher)));

Because of missing information, these standard errors can be smaller than the estimated standard errors derived from the expected Hessian matrix. To see the difference, compare to standard errors calculated with ecmnhess.