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# 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`.