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.