Three criteria for measuring the goodness of estimators

The three criteria for measuring the goodness of estimators are as follows:

Unbiasedness: Unbiasedness is one of the important properties of estimator. If the expected value of an estimator is equal to the real parameter value, the estimator is called unbiased estimation. Mathematically expressed as:

E (θ) = θ where E (θ) represents the expected value of the estimator and θ represents the value of the real parameter. Unbiasedness ensures that the average value of the estimator is close to the real parameter value in the process of repeated sampling.

2. Efficiency: Efficiency means that among all unbiased estimates, the estimate with the smallest variance is considered to be the most effective. Estimates with smaller variance are usually more accurate because they fluctuate less under different samples. Mathematically, the definition of effective estimation is: var (θ) ≤ var (θ ~)

Among them, var (θ) and Var(θ~) respectively represent the variance of two unbiased estimates. Effective estimation provides relatively small estimation error to some extent.

3. Consistency: Consistency refers to the property that the estimator gradually approaches the real parameter value with the increase of sample size. Mathematically, an estimator is called a uniform estimator, if for any positive number ε >; 0, there is a threshold of sample size, so that when the sample size is greater than this threshold, the error of the estimator is less than ε. Mathematically expressed as:

limn→∞P(∣θ^？ θ∣<； ε)= 1

This means that with the increase of sample size, the estimated value is closer to the real parameters.

Supplementary explanation:

Mean Square Error (MSE): Considering fairness and validity, MSE is often used as a standard to evaluate the quality of estimators. Mean square error is defined as the mean square error between the estimated value and the true value:

MSE(θ^)=E((θ^？ θ)2) Ideally, the smaller the mean square error of the estimator, the better.

These three criteria for evaluating estimators are interrelated, and a good estimator is usually unbiased, effective and consistent. In the actual statistical inference, it is necessary to comprehensively consider these properties and determine the appropriate estimation method according to the requirements of specific problems.