Seventh Chapter parameter Estimation _ statistics

Parameter estimation is based on sampling and sampling distribution, and the general parameters are inferred according to sample statistics. 1 principle of parameter estimation

1 estimate and Estimated value
Estimate: The name of the statistic used to estimate the overall parameter in the parameter estimate, such as sample mean, sample ratio
Estimate: The numerical value of an estimate calculated from a specific sample
2 point estimation and interval estimation
The method of parameter estimation is somewhat estimated and interval estimated.
Point estimate: An estimate of the overall parameter using a value from the sample statistic. No probability measure reliability
Interval estimation: On the basis of point estimation, an interval range of the general parameters is given, which is usually obtained by the sample statistic plus and minus estimation error. A probability metric is given.
In interval estimation, the estimated interval of the general parameters constructed by the sample statistics is called the confidence interval, where the minimum value of the interval is called the lower confidence limit, and the maximum value is called the confidence cap.
If the steps of constructing the confidence interval are repeated many times, the proportion of the number of times in the confidence interval that contains the total parameter truth is called the confidence level, also known as confidence degree or confidence factor.
3 Criteria for evaluating estimates
Criteria for evaluation estimates: unbiased, effective, consistent
Unbiased: The mathematical expectation of the estimated sampling distribution is equal to the estimated overall parameter
Validity: Refers to the two unbiased estimators of the same overall parameter, and the estimator with smaller standard deviation is more effective
Consistency: Refers to the interval estimation of the parameter 21 parameters of the estimated population as the size of the sample increases, and the value of the point estimator approaches

In the study of a population, the main parameters concerned are the total mean μ, the total ratio π and the total variance σ2, etc. The following sections describe how to construct a confidence interval for a general parameter using sample statistics. Interval estimation of 1 population mean value

In the interval estimation of the population mean, it is necessary to consider whether the population is normal distribution, whether the population variance is known or not, and how to construct the estimated sample is a large sample or a small sample.
(1) Normal population, variance known, or non normal population, large sample
At this time, the sample mean X¯ obeys the normal distribution, after standardization, then obeys the standard normal distribution, namely
Z=x¯−μσ/n√∼n (0,1)
According to the above formula, we can get the confidence interval of the total mean μ at 1−α confidence level is: x¯±zα/2σn√
In the x¯+zα/2σn√ called the confidence limit, x¯−zα/2σn√ called the lower confidence limit, α is a predetermined probability value, also known as the risk value, is not included in the total mean value of the probability of confidence interval, 1−α called confidence level, ZΑ/2 is a sign The Α/2 of the upper side area of the normal distribution is the z value, and the zα/2σn√ is the estimation error when estimating the total mean value.
If the population obeys the normal distribution but the σ2 is unknown, or the whole does not obey the normal distribution, as long as it is in the large sample condition, the total variance in the σ2 can be replaced by the sample variance.
(2) Normal population, variance is unknown, small sample
If the total variance is σ2 unknown and is in the case of a small sample, then the sample variance S 2 is needed instead of σ2, at which point the normalized random variable of the sample is followed by the T distribution of the degree of Freedom (n−1), i.e.
T=x¯−μs/n√∼t (n−1)
The t distribution is needed to establish the confidence interval of the total mean μ.
According to the upper-formula available population mean μ at the 1−α confidence level, the confidence interval is:
Estimation of the total proportion of X¯±TΑ/2 sn√2

The sampling distribution of sample ratio p can be approximated by normal distribution when large sample is used. Now
z=p−μπ (1−π)/n−−−−−−−−−√∼n (0,1)
The estimation of 3 sample variance in the confidence interval can be obtained

By (n−1) s 2σ2∼χ2 (n−1) The total variance σ2 The confidence interval under 1−α confidence level is:
(n−1) s 2χ2α/2≤σ2≤ (n−1) s 2χ2 1−α/2
Interval estimation of 32 general parameters

For two of the total, the main concern parameters are the difference of the mean value of two μ1−μ2, the difference between the two population π1−π2, and the variance of two population is σ2 1/σ2 2. Interval estimation of the difference between 12 total mean values

(1) Estimation of the difference of two total mean values: independent samples
A. Estimation of large samples
If two samples are extracted independently from the two populations, the elements of one sample are independent of the elements in the other sample. becomes an independent sample. If all two of the population are normal or two are not obeyed, as long as the two samples are large samples, according to the sampling distribution, the sampling distribution of the difference between the two samples obeys the normal distribution and is standardized after:
z= (x 1¯−x 2¯) − (μ1−μ2) σ2 1 n 1 +σ2 2 n 2−−−−−−−−√∼n (0,1)
The confidence interval can be inferred from this
When the population variance is unknown, the sample variance is used instead.
B. Estimation of small samples
Suppose: (1) Two of the population are subject to normal distribution (2) Two random samples independently extracted from two total
Under the above hypothesis, the mean of both samples obeys normal distribution.
Situation 1: (1) Two total variance σ2 1, σ2 2 known, using the upper-standard normal distribution
(2) Two total variance σ2 1, σ2 2 unknown, but σ2 1 =σ2 2
According to the χ2 distribution (n−1) S 2/σ2 obeys the freedom as the n−1, the sample difference obeys the normal distribution, and the t distribution of the DOF is N 1 +n 2−2, and then the confidence interval is estimated.
(3) Two total variance σ2 1, σ2 2 unknown, and σ2 1≠σ2 2
The same structure t distribution, more complex
(2) Estimation of the difference of two total mean values: matching samples
Match sample: The data in one sample corresponds to the data in another sample. The matching sample eliminates the difference in time between the two methods due to the unfairness specified by the sample.
Under large sample condition, the standard normal distribution can be used regardless of the variance is known
Interval estimation of the difference of 22 total proportions of t distribution under small sample conditions

According to the sampling distribution, the difference of two sample proportions obeys normal distribution. After standardization, obey the standard normal distribution. Interval estimation of 32 population variance ratios

The

According to (n−1) s 2 /σ 2   obeys the χ 2   distribution of degrees of freedom n−1 , two sample variance ratios obey F (n 1  −2,n 2 −1)   Distribution, so f  distributions can be constructed to construct the confidence intervals of two variance ratios σ 2 1 /σ 2 2  .