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Weighted Likelihood for Time-Varying Gaussian Parameter Estimation

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In a previous article, we presented a weighted likelihood technique for estimating parameters \theta of a probability density function \rho(x|\theta). 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

\rho(x|\mu,\nu) = \frac{1}{\sqrt{\pi\nu}} \exp\left[-\frac{(x-\mu)^2}{\nu}\right].

In this case, the log likelihood is given by

\begin{aligned} \log\mathcal{L}(\mu,\nu) &= \sum_{i=1}^N w_i \log\rho(x_i|\mu,\nu) \\ &= -\frac{1}{2} \log\pi\nu - \frac{1}{\nu} \sum_{i=1}^N w_i \left(x_i - \mu\right)^2 \end{aligned}.

Recall that the maximum likelihood occurs when

\begin{aligned} \frac{\partial}{\partial\mu} \log\mathcal{L}(\mu,\nu) = \frac{\partial}{\partial\nu} \log\mathcal{L}(\mu,\nu) = 0. \end{aligned}

A simple calculation demonstrates that this occurs when

\begin{aligned} \mu = \sum_{i=1}^N w_i x_i \end{aligned}

and

\begin{aligned} \sigma^2 = \sum_{i=1}^N w_i \left(x_i - \mu\right)^2, \end{aligned}

where \sigma^2 = \nu/2.

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

\begin{aligned} E_w(X) = \sum_{i=1}^N w_i x_i, \end{aligned}

the Gaussian parameters may be expressed in a familiar form via

\mu = E_w(X)

and

\sigma^2 = E_w(X^2) - \left[E_w(X)\right]^2.

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.

 

Written by Eric

February 3, 2013 at 4:33 pm