The likelihood function $ L(\theta) $ is defined as the joint probability density or mass function of the observed data, considered as a function of the parameter(s) $ \theta $:

\begin{equation}

L(\theta) = \prod_{i=1}^{n} f(x_i; \theta)

\end{equation}

The goal of MLE is to find the parameter value(s) $ \hat{\theta} $ that maximizes the likelihood function, or equivalently, the log-likelihood function $ \ell(\theta) $:

\begin{equation}

\ell(\theta) = \log L(\theta) = \sum_{i=1}^{n} \log f(x_i; \theta)

\end{equation}

To find $ \hat{\theta} $, we differentiate the log-likelihood function with respect to $\theta $, set the derivative(s) equal to zero, and solve for $ \hat{\theta} $ using this equation:

\begin{equation}

\frac{\partial \ell(\theta)}{\partial \theta} = 0

\end{equation}

If the log-likelihood function is concave, the critical point(s) obtained by solving the above equation corresponds to the maximum likelihood estimator(s) $ \hat{\theta} $. In some cases, it may be more convenient to work with the negative log-likelihood function, denoted as $ -\ell(\theta) $, which is equivalent to minimizing the negative log-likelihood.

The MLE properties, including consistency, asymptotic normality, and efficiency, make it one of the most widely used methods for parameter estimation in statistics.\\

\section{The Fisher Minimum $\chi^2$ Estimation}

The Fisher Minimum Chi-Square method involves minimizing the chi-square statistic between observed and expected frequencies. It provides robust parameter estimates by balancing goodness-of-fit and parameter stability. The RP-based Minimum Chi-Square method is an extension of the equiprobable minimum chi-square method, focusing on minimizing residuals using representative points. This method is particularly advantageous for small sample sizes and highly skewed data distributions.

The steps for Pearson-Fisher’s minimum Chi-square estimation are:

\noindent1. Formulate the likelihood function.\\

2. Maximize the likelihood function.\\

3. Determine parameter estimates.\\

4. Assess model fit using the minimum Chi-square statistic.\\

5. pare the minimum Chi-square statistic with critical values.\\

6. Draw conclusions about the goodness of fit of the model.\\

Let's first look at the basic formula of Equiprobable Fisher Minimum $\chi^2$ Estimation:

Suppose we have a set of observed data $x_i$, $i=1,2,...,n$, and the parameter to be estimated is $\theta$. Let $f(x;\theta)$ be the probability density function (PDF) of the data, and $F(x;\theta)$ be the cumulative distribution function (CDF). For a frequency histogram divided into $k$ intervals, let $O_i$ be the observed frequency in the $i$th interval, and $E_i$ be the expected frequency in the $i$th interval. Then the expression for the chi-square statistic is:

\begin{equation}

\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}

\end{equation}

where the expected frequency $E_i$ can be obtained by integration:

\begin{equation}

E_i = n [F(b_i;\theta) - F(a_i;\theta)]

\end{equation}

where $[a_i, b_i]$ are the boundaries of the $i$th interval.

The core idea of Equiprobable Fisher Minimum $\chi^2$ Estimation is to estimate the parameter $\theta$ by minimizing this chi-square statistic. Optimization algorithms (such as gradient descent, Newton's method, etc.) are usually used to find the parameter value $\theta$ that minimizes $\chi^2$.

第 6 章

\chapter{Monte Carlo Simulation of the Estimation}\label{cap3}

\begin{adjustwidth}{2.5cm}{1cm}

\small This Chapter explains .....

\end{adjustwidth}

\vspace{0.5cm}

In this section, four types of parametric estimation methods will be discussed. Moment estimation will be addressed first, followed by the method of Fisher minimum $\chi^2$ estimation with Equiprobable Cells. Lastly, the RP-based minimum $\chi^2$ estimation will be implemented.

\section{Assessment Criteria}

As shown in Table II-XV, in this study, we use RMSE as theparative standard for assessing estimation uracy. RMSE (Root Mean Square Error) is a statistical measure used to assess the uracy of parameter estimates in predictive models [21]. It quantifies the average discrepancy between predicted and observed values, crucial for evaluating the goodness-of-fit in regression analysis and time series forecasting [21]. RMSE helps researchers and analysts gauge the precision of parameter estimates, guiding model refinement and selection [3]. It serves as a fundamental tool in various fields, including economics, environmental science, and finance, where precise parameter estimation is essential for decision-making and policy formulation. RMSE ensures robust and reliable parameter estimation, facilitating more urate predictions and informed decisions [2].