《学术论文集锦[开错]》紫微客

33、033

　　\begin{equation}
　　\hat{\theta} = \text{argmin}_\theta \sum_{i=1}^{n} \frac{(O_i - E_i(\theta))^2}{E_i(\theta)}
　　\end{equation}\\

　　\noindent1. $\hat{\theta}$: The estimated parameter.\\
　　2. $\theta$: The parameter to be estimated.\\
　　3. $O_i$: The observed frequency or count in the (i)-th category.\\
　　4. $E_i(\theta)$: The expected frequency or count in the (i)-th category, which is a function of the parameter $\theta$.\\
　　5. $n$: The total number of categories or cells.\\

　　In this formula, we aim to find the value of the parameter $\hat{\theta}$ that minimizes the sum of squared differences between observed and expected frequencies, normalized by the expected frequencies. This is achieved by adjusting the parameter $\theta$ iteratively until convergence. \\

　　\vspace{0.5cm}

　　\section{Conclusion}

　　\vspace{0.5cm}

　　Based on the findings of this study, it is evident that parameter estimation methods play a crucial role in statistical analysis. Accurate estimation of parameters enables researchers to draw meaningful conclusions from data and make informed decisions. The significance of parameter estimation methods lies in their ability to capture the underlying characteristics of the data and account for sample size and assumptions.

　　The Monte Carlo comparison study conducted in this research provides valuable insights into the accuracy, efficiency, and robustness of different parameter estimation methods. By simulating data under controlled conditions, the study evaluates the strengths and weaknesses of each method and identifies factors influencing their performance. This comparative analysis serves as a guide for selecting appropriate parameter estimation techniques in practical applications.

　　The results of the study indicate that the selection of parameter values can significantly affect the estimation outcomes. Different parameter settings can lead to varying probability distribution shapes and characteristics, thereby influencing the accuracy and bias of parameter estimation. This highlights the importance of carefully considering the parameter values when applying estimation methods.

　　Furthermore, the study suggests that the method of moments and maximum likelihood estimation are commonly used methods for parameter estimation. However, their performance may be affected by sample size and the presence of volatility in parameter estimation. It is recommended to conduct further analysis, such as analysis of variance and graphical representation, to examine the influence of sample size on the accuracy of parameter estimates and visualize the impact of sample size on estimate stability.