Probability statistics: Seventh Chapter parameter Estimation _ probability statistics

Seventh Chapter parameter estimation

Content Summary:

One, point estimate

1, set for the overall sample, the overall distribution function is known as the parameters to be evaluated for the corresponding sample observations. The problem of point estimation is to construct an appropriate statistic and estimate the value of the parameters to be evaluated with its observations. Here is called the estimate, called the estimated value, both of which are collectively estimated. This is called the point estimate of unknown parameters for fixed-point estimation of unknown parameters.

2, the estimation of the original point moment of each order of the samples as the sum of the original points of the whole order, and the estimation method of the continuous function of each order origin moment of the sample as the whole order of the original point moment is called the moment estimation method.

3, set the overall distribution rate (or probability density), for the unknown parameter vector, set to come from the sample, then () the joint distribution rate (or joint probability density):

Or

A likelihood function called a sample.

Any observations of the sample (), if

is called the maximum likelihood estimate of the parameter, which is the maximum likelihood estimator of the parameter.

If or about differentiable, the maximum likelihood estimate of the parameter can be obtained by the equation:

Get. For the monotone function, the maximum likelihood estimate of the parameter can also be obtained by the equation:

, the solution of the latter one is often much more convenient than the former one.

Ii. Evaluation criteria for estimates

1, if the estimated mathematical expectations exist, and for arbitrary, meet:

is called the unbiased estimate of the parameter.

2, set and are the parameters of unbiased estimates, if for any, meet:

is said to be more effective.

2, if the estimation of parameters, if for any, at that time with the probability of convergence, that is, set up, then called the parameters of the consistent estimate.

Iii. interval Estimation

1, set the overall distribution function form known, for unknown parameters, if for a given, there are two statistics and, for arbitrary, meet:

It is called the confidence level of the confidence level of the random interval as the parameter, and the confidence lower limit and the confidence limit of the corresponding confidence interval, which is called the level of significance.

3, the interval estimation of normal population mean and variance is shown in the following table

Basic requirements:

1, understand the unknown parameters of the estimates, estimates, point estimation concepts;

2, grasp the unknown parameters of the moment estimation method, the maximum likelihood estimation method of principle and solution;

3, understand the unbiased nature of the estimates, effectiveness, consistency of the concept and will verify the estimation of the unbiased and effective;

4, to understand the concept of interval estimation, we will solve a single normal population and the related confidence intervals of two normal population mean and variance.

This chapter is about moment estimation, the theoretical basis of maximum likelihood estimation, and the understanding of random interval and corresponding probability in interval estimation.

This chapter focuses on: Moment estimation, maximum likelihood estimation, interval estimation, unbiased estimation, and validation of validity.

Difficulty Analysis:

1. The theoretical basis of moment estimation: Sinchin Law of large numbers

The order of the whole is present, the sample order moment is, that is, by Sinchin law of large numbers:

，

That is, regardless of the overall distribution (as long as the expectation exists), the sample order will become more and more approximate to the overall corresponding order moment with the increase of sample size.

2. Theoretical basis of maximum likelihood estimation: principle of statistical inference

In a random experiment, when an event occurs, the event should not be a small probability event.

3. Understanding the confidence level of confidence intervals:

In a large number of repeated samples, the sample values can be taken into a number of defined intervals, of which about 100 (%)% of the interval is included, and by the samples are worth a specific interval, may or may not be included.

Typical example Analysis:

Example 1: Set as the overall sample, for the corresponding sample observation value, set the general distribution law for

For unknown parameters, moment estimation and maximum likelihood estimation of unknown parameters are obtained.

Analysis: The moment estimation and maximum likelihood estimation of the unknown parameters of discrete random variables are carried out gradually.

Solution: (1) by the idea of the overall expectation of the problem, so that the average value of the sample, so that the estimated moment of unknown parameters and estimates are:

(2) A set of observed values for the sample, thus the likelihood function

Thus

Make

The maximum likelihood estimated value is: = =,

The maximum likelihood estimator is: =.

Example 2: A sample of the total Poisson distribution from the parameter is obtained, and the maximum likelihood estimation and moment estimation of the parameters are tested.

Analysis: The calculation process of moment estimation and maximum likelihood estimation is carried out according to the continuous random variable.

Solution: 1 The expectation of the overall knowledge of the problem, the mean of the sample, the order substitution, the moment estimator and the estimated value of the unknown parameter were:

(2) A set of observed values for the sample, thus the likelihood function

Thus

Make

The maximum likelihood estimate is: =, the maximum likelihood estimator is: =.

Example 3: Set the uniform distribution of the overall compliance, unknown, set as the overall sample.

Test: (1) to calculate the moment estimation of parameters;

(2) To find the maximum likelihood estimation of the parameters;

(3) Proof:, and all is unbiased estimate;

(4) Proof: more effective.

Analysis: To really understand the theoretical basis of the maximum likelihood estimation method, we should grasp the process of proving the unbiased and validity of the parameter estimator.

Solution:

(1) The expectation of the overall knowledge of the problem is set to the sample mean, and the moment estimator and the estimated value of the unknown parameters are respectively:

(2) A set of observed values for the sample, thus the likelihood function

To achieve the maximum,

Shall make and

, the maximum likelihood estimate is: =. The maximum likelihood estimator is: =.

(3) by the title,;

By the probability density function:

Know

By the probability density function:

Know

Therefore, and all are unbiased estimates.

(4) by the title:;

，；，；

and <<, and thus more effective.

Example 4: Set from the mean, the variance of the total sample size of two independent samples, respectively, is the two mean, it is proved: for arbitrary constants, are unbiased estimates, and determine the constant to achieve minimum.

Solution: From the problem set:;

；

Because

Thus:.

Thus, unbiased estimates of any constant are true.

By the independence,

Make

Get, at this time;

and

So when, when, to reach the smallest

Example 5: The net weight of the bag of a certain seasoning bag is obeyed, and the weighing of 9 bags (unit two) is

6.0,5.7,5.8,6.5,7.0,6.3,5.6,6.1,5.0

(1) According to previous experience, the confidence level of 0.95 is obtained.

2 The confidence interval of 0.95 is unknown and the confidence level is obtained.

3) The confidence interval of 0.95 is obtained for the confidence level.

Solution: 1 by the title, the overall, known,

And

Confidence interval for the known confidence level of the generation variance

The confidence interval for the confidence level of 0.95 is () = (5.608,6.392).

2) by the title, the general, unknown,

And

Confidence interval for the confidence level with unknown generation variance

The confidence interval for the confidence level of 0.95 is () = (5.558,6.442).

3) by the title, Overall,

And

Confidence interval for the confidence level of the surrogate

The confidence interval for the confidence level of 0.95 is (0.1506,1.211).

Example 6: To compare the life of two types of bulbs, a random random sampling of a model 10, B model lamp 10, Measured life (unit: Hours) as follows:

Type A: 560,590,560,570,580,570,600,550,570,550;

Type B: 620,570,650,600,630,580,570,600,600,580;

Set A, b two types of bulb life are subject to normal distribution.

(1) To set the same variance, to find the confidence interval of the two models with a confidence level of 0.95 for the difference of the life expectancy of the bulbs;

2 The confidence interval for the confidence level of the variance ratio is 0.95.

Solution: The two types of bulb life are seen as a whole, and,

1) by the title,

And

，

Confidence interval for the confidence level where the variance is unknown but equal

The confidence interval for the confidence level of 0.95 is (9,51).

1) by the title to know:,,

，

Confidence interval for the confidence level of the surrogate

The confidence interval for the confidence level of 0.95 is (0.0931,1.571).

Self-Test questions:

One, fill in the blanks question:

1, set the overall compliance on the uniform distribution, unknown, for the overall sample, then the moment is estimated, the maximum likelihood estimate is.

2, set the overall mean value, variance is, for the overall sample, then, =.

3, set the overall compliance with the exponential distribution, its probability density, which is unknown, as the overall sample, then, the estimated amount of more effective statistics.

4, set the overall, from the sample capacity of 9 of random samples, measured average of 5, the parameter confidence level of 0.95 of the confidence interval of &