main content:
Point estimate:
moment estimation
maximum likelihood estimation
Criteria for Point Estimation
Interval estimate:
confidence interval
Symbol Description:
1 Parameter estimation problem
2 point estimate
2.1 Moment Estimation
The basic idea of the moment estimation method is to use the sample moment to estimate the population distribution moment according to the law of large numbers.
The parameters are then estimated using the relationship between the population moments and the parameters.
mark:
Fundamentals of Moment Estimation: The Law of Large Numbers
Example 1: Parameter estimation of a two-point distribution
Example 2: Parameter estimation for a normal distribution
2.2 Maximum Likelihood Estimation
Given the distribution and unknown parameters of a random variable, compute the likelihood function using the observed samples.
The parameter that maximizes the likelihood function is chosen as the parameter estimator.
The basic principle of the maximum likelihood function: maximize the likelihood function.
Example: Parameter estimation for a normal distribution
2.3 Criteria for point estimation
2.4 Point Estimation: Consistency
Consistency is the most basic requirement, the consistency of moment estimation is guaranteed by the law of large numbers, and the consistency of maximum likelihood estimation is also implicitly guaranteed by the law of large numbers.
2.5 Point Estimation: Unbiased
Any parameter estimate that satisfies consistency will converge to the true value of the parameter when the sample tends to infinity. But in the case of finite samples, the expectation of this estimate may not always be equal to the true value of the parameter.
2.6 Point Estimation: Validity
example:
2.7 Point Estimation: Asymptotic Normality
Asymptotic normal This involves some knowledge of matrices, especially the asymptotic normal properties of maximum likelihood estimation.
2.8 Interval Estimation: Confidence Intervals
