Parameter estimation (individual popular understanding)

Problem background:

We know the distribution of the population, but we do not know the parameters of the distribution, so we need to estimate the unknown parameters.

Two estimations of the types:

1. Point estimation

2. Interval estimation

1. Point estimation

Including moment estimation and maximum likelihood estimation

1) Moment estimation:

Estimating the total moment with the sample moment

Here we can use the sample first order moment (mean) to estimate the whole first-order moment (mean), the second-order Central Moment Estimator (variance) as the whole second-order center distance (variance)

2) Maximum likelihood estimate:

Understand:

Using the known sample results, it is possible to reverse the parameter values that are most likely to cause such results.

So step:

1. By the density function of the population, the likelihood function (the multiplication of the probability density) is written out, and the likelihood function is a function of the unknown parameter (the parameters we want to estimate), which is an optimal programming problem.

2. For the maximum density value, unknown parameter value

3. In order to facilitate, the equation on both sides of the logarithm, and then to find a point, because the general is a practical problem, at the location can be achieved maximum value.

Evaluation criteria for good estimate:

1. Unbiased: The expectation of an estimate is equal to the estimated amount. Both the sample mean and the sample variance are unbiased estimates of the total mean and the total variance respectively.

2. Effectiveness: Under the condition of equal expectation, it is effective to consider variance and small variance estimator.

3. Consistency: The estimate converges to the estimated amount according to probability. (both the sample mean and the sample variance are consistent estimates of the total mean and the total variance)

2. Interval estimation:

The estimated interval of unknown parameters is obtained from the sample, and the reliability of the interval containing unknown parameters is reached to the predetermined requirement (this predetermined requirement is a confidence level, which is expressed by the Alpha Non-transposition point).

Steps:

1. Construct the appropriate statistic u with the parameters to be evaluated, and the distribution of the statistics is known.

2. According to the given confidence level, according to the P (U1<U<U2) =1-α, the U1,U2 is obtained, the range of unknown parameters is solved with U, and the final form is P (x*<μ<y*) =1-α, which is where the μ can fall in the probability for the 1-α of the interval (x*,y*).

This uses the upper Alpha Non-transposition point to find this confidence interval that can make the confidence 1-α.