3.1.1 Parametric Estimation
Parametric models are widely used in estimating probability density
functions for their ease of use. They require an investigator to assume
that the data comes from a certain distribution defined by a finite
number of parameters and that it follows a specific shape. Therefore, they are
considered to have high bias since the investigator has to make that
assumption which may not be true. In Fig. \ref{998686}, it can be seen that
electric load may be bimodal in nature and thus the Gaussian
distribution used in literature would not be a good fit. In this report,
we consider two parametric distributions, Gaussian and Gamma, and
compare their performance with nonparametric techniques.
The PDF of the Gaussian Distribution is:
\[f_{\text{Gaussian}}=\ \frac{1}{\sqrt{2\pi\sigma^{2}}}\ e^{\frac{-{(x-\mu)}^{2}}{2\sigma^{2}}}\nonumber \\\]
where \(x\in(0,\infty)\), \(\mu\) is the mean and \(\sigma\) is the
standard deviation. The PDF of the Gamma distribution is:
\[f_{\text{Gamma}}=\frac{\beta^{\alpha}}{\Gamma(\alpha)}\ x^{\alpha-1}e^{-\beta x}\nonumber \\\]
where \(\Gamma\) is the Gamma function, \(x\in(0,\infty)\) and \(\alpha\), \(\beta\) are the shape and rate parameter, respectively.
These models are fit using Maximum Likelihood Estimation which is the
most common method used to estimate parameters for parametric models.