## Archive for **February 2013**

## Weighted Likelihood for Time-Varying Gaussian Parameter Estimation

In a previous article, we presented a weighted likelihood technique for estimating parameters of a probability density function . The motivation being that for time series, we may wish to weigh more recent data more heavily. In this article, we will apply the technique to a simple Gaussian density

In this case, the log likelihood is given by

Recall that the maximum likelihood occurs when

A simple calculation demonstrates that this occurs when

and

where .

Introducing a weighted expectation operator for a random variable with samples given by

the Gaussian parameters may be expressed in a familiar form via

and

This simple result justifies the use of weighted expectations for time varying Gaussian parameter estimation. As we will see, this is also useful for coding financial time series analysis.