The Gaussian Distribution of Complementary Mathematics-Biased and Unbiased Estimation

The last article introduced the maximum likelihood estimation in the case of Gaussian distribution in one dimension.
We will continue from the previous article. The two parameter values ​​obtained through maximum likelihood estimation are examples to see if they are biased or unbiased. .

Let’s first look at the definitions of biased and unbiased estimates

definition

Biased estimate refers to the systematic error between the estimated value obtained from the sample value and the true value of the parameter to be estimated, and the expected value is not the true value of the parameter to be estimated.
Pay attention, its expectation is not equal to the true value of the parameter to be estimated is biased, and equal is unbiased~

Continuing the two parameters of

our maximum likelihood estimation in the previous article: Let us judge whether these two parameters are biased or unbiased, and then it becomes to judge whether the expectations of the two parameters are equal to them:

first look at μ, its expectations It is very simple. It is its mean value, so it is unbiased.

Let's look at σ again. Let's simplify the expectation of σ first:

so the σ estimated by the maximum likelihood is biased and slightly smaller than the ideal value. . . .
So the true unbiased estimate should be:

Therefore, this maximum likelihood estimate is actually a point estimate, which will cause a certain deviation~

Finally, those friends who are not very familiar with variance go back and read the textbook of probability theory~

Variance is a special expectation, which is defined as:
Var(x)=E((x−E(x))2)Var(x)=E((x−E(x)) ² )
Expanded representation: repeated Using the expected linearity, another representation of the variance can be calculated:
Var(x)=E((x−E(x)) ² )=E(x ² )−(E(x)) ²