The text in the image is in Chinese and appears to ask for the maximum likelihood estimation (MLE) for the parameters $\alpha$ and $\lambda$ of the Gamma distribution, denoted as $Ga(\alpha, \lambda)$, where $\alpha > 0$ and $\lambda > 0$.

Translation of the problem:

"Assume the sample $X_1, X_2, \dots, X_n$ comes from a Gamma distribution $Ga(\alpha, \lambda)$, where $\alpha > 0$ and $\lambda > 0$. Provide the estimation of parameters $\alpha$ and $\lambda$."

Solution Strategy:

For a Gamma distribution $X \sim Ga(\alpha, \lambda)$, with probability density function (PDF): $f(x; \alpha, \lambda) = \frac{\lambda^\alpha x^{\alpha - 1} e^{-\lambda x}}{\Gamma(\alpha)}$ where $\Gamma(\alpha)$ is the Gamma function, and $\alpha, \lambda$ are the shape and rate parameters, respectively.

We will use Maximum Likelihood Estimation (MLE) to estimate $\alpha$ and $\lambda$.

1. Log-likelihood function:

Given a sample $X_1, X_2, \dots, X_n$, the likelihood function is: $L(\alpha, \lambda) = \prod_{i=1}^n \frac{\lambda^\alpha X_i^{\alpha - 1} e^{-\lambda X_i}}{\Gamma(\alpha)}$ Taking the logarithm: $\log L(\alpha, \lambda) = n\alpha \log \lambda - n \log \Gamma(\alpha) + (\alpha - 1) \sum_{i=1}^n \log X_i - \lambda \sum_{i=1}^n X_i$

2. Partial derivatives and MLE equations:

We compute the partial derivatives with respect to $\alpha$ and $\lambda$, and set them to zero to solve for the parameters.

• Derivative with respect to $\lambda$: $\frac{\partial \log L}{\partial \lambda} = \frac{n\alpha}{\lambda} - \sum_{i=1}^n X_i = 0$ Solving this gives the MLE for $\lambda$: $\hat{\lambda} = \frac{n\alpha}{\sum_{i=1}^n X_i}$

• Derivative with respect to $\alpha$: $\frac{\partial \log L}{\partial \alpha} = n \log \lambda - n \frac{\Gamma'(\alpha)}{\Gamma(\alpha)} + \sum_{i=1}^n \log X_i = 0$ The expression $\frac{\Gamma'(\alpha)}{\Gamma(\alpha)}$ is known as the digamma function, $\psi(\alpha)$.

This equation can be solved numerically for $\alpha$, using numerical methods like Newton-Raphson.

Would you like a more detailed walkthrough of the numerical solution for $\alpha$, or any specific step in the derivation? Let me know if you need more details!

Further Exploration Questions:

1. What is the significance of the Gamma distribution in statistical modeling?
2. How does the Newton-Raphson method work for solving MLE equations?
3. What is the role of the digamma function $\psi(\alpha)$ in this estimation process?
4. Could you use alternative methods, such as the Method of Moments, to estimate $\alpha$ and $\lambda$?
5. How would you extend this estimation technique to a multivariate Gamma distribution?

Tip: For Gamma distribution MLE, the equation for $\alpha$ often requires iterative methods since it involves special functions (like the digamma function). Numerical solvers like scipy in Python are commonly used.