统计应用

MLE

知道一组采样 ，求一组参数 ，使得 最大.

• consistency (lim -> 0)
• Asymptotic Normality 渐近正态性

Fisher Information

We have introduced three key properties for evaluating estimators:

1. Unbiasedness: The estimator’s expected value equals the true parameter.
2. Efficiency: Among unbiased estimators, the one with the smallest variance is preferred. The Cram´ er-Rao lower bound provides a benchmark. 26
3. Consistency: As the sample size increases, the estimator converges to the true parameter.

For Maximum Likelihood Estimators, we have two important asymptotic proper- ties:

1. Consistency: The MLE converges to the true parameter as the sample size grows.
2. Asymptotic Normality: The distribution of the MLE, appropriately scaled, approaches a normal distribution. This allows for inference in large samples. These properties provide a foundation for choosing and justifying estimators in statistical practice.

Gamma Distribution

• [q] Gamma Distribution !