数理统计Chapter7-2

5 / 29
Interval Estimation
Definition 2.1
Assume that the distribution function for population X is F (x; θ), θ is the unknown parameter, X1 , X2 , · · · , Xn is the sample from X . For some given α(0 < α < 1), if there exist two statistics ˆ1 = θ ˆ1 (X1 , X2 , · · · , Xn ) and θ ˆ2 = θ ˆ2 (X1 , X2 , · · · , Xn ), such θ that, for any θ, ˆ1 ≤ θ ≤ θ ˆ2 } = 1 − α Pθ {θ ˆ1 , θ ˆ2 ) is called as the two-sided Then the random interval (θ confidence interval for θ with confidence level(˜&Y ²) ˆ1 , θ ˆ2 are called the confidence lower limit and 1 − α, and θ ˆ1 , θ ˆ2 ) is called interval confidence upper limit respectively, also (θ estimation for parameter θ.
7 / 29
Remark
1. Confidence level 1 − α:
ˆ1 (X1 , X2 , · · · , Xn ), θ ˆ2 (X1 , X2 , · · · , Xn )) is a a. The interval (θ random interval, for each fixed sample observation value, there ˆ1 (x1 , x2 , · · · , xn ), θ ˆ2 (x1 , x2 , · · · , xn )), is one specific interval (θ which may contain θ, or may not contain θ.
5 / 29
Interval Estimation
ˆ(X1 , X2 , · · · , Xn ) is an estimator for In the point estimation, θ the parameter θ, for the given observation value of sample ˆ(x1 , x2 , · · · , xn ) (x1 , x2 , · · · , xn ), we can obtain one estimate θ for θ. By the randomness of sample, there exists deviation between the estimate and the real value of parameter. Sometimes, we hope to find a quantity to measure the extent of deviation. The aim of interval estimation is to determine the interval which contains the parameter in some probability based on the sample statistics.
a. The confidence level 1 − α reflects the reliability of the interval, the bigger 1 − α is, the higher the reliability is. b. While the length of the interval reflects the accuracy of the ˆ1 , θ ˆ2 ) is, the higher the accuracy is. interval, the shorter the Sample Size
3 / 29
Interval Estimation Interval Estimation Interval estimation for normal population
Interval estimation for a single normal population Interval estimation for two normal populations Interval estimation for large sample
a. The confidence level 1 − α reflects the reliability of the interval, the bigger 1 − α is, the higher the reliability is.
9 / 29
2. The reliability(Œ ‚ 5 ) and accuracy(° ( Ý) of interval estimation:
Determination of the Sample Size
4 / 29
Interval Estimation
ˆ(X1 , X2 , · · · , Xn ) is an estimator for In the point estimation, θ the parameter θ, for the given observation value of sample ˆ(x1 , x2 , · · · , xn ) (x1 , x2 , · · · , xn ), we can obtain one estimate θ for θ.
2 / 29
Interval Estimation Interval Estimation Interval estimation for normal population
Interval estimation for a single normal population Interval estimation for two normal populations Interval estimation for large sample
9 / 29
2. The reliability(Œ ‚ 5 ) and accuracy(° ( Ý) of interval estimation:
a. The confidence level 1 − α reflects the reliability of the interval, the bigger 1 − α is, the higher the reliability is. b. While the length of the interval reflects the accuracy of the ˆ1 , θ ˆ2 ) is, the higher the accuracy is. interval, the shorter of (θ c. Generally, when the sample size is fixed, the smaller α is (i.e. the bigger 1 − α is), the more longer the length of the confidence ˆ1 , θ ˆ2 ) is. That is, when the reliability increases, the interval (θ accuracy decreases, and the inverse conclusion still holds.
8 / 29
2. The reliability(Œ ‚ 5 ) and accuracy(° ( Ý) of interval estimation:
9 / 29
2. The reliability(Œ ‚ 5 ) and accuracy(° ( Ý) of interval estimation:
6 / 29
Interval Estimation
Theorem 2.1
If g (θ) is a strictly increasing (or decreasing) function, and ˆ1 , θ ˆ2 ] is a confidence interval for θ with confidence level 1 − α, [θ then ˆ1 ), g (θ ˆ2 )] (or [g (θ ˆ2 ), g (θ ˆ1 )]) is a confidence interval for g (θ) [g (θ with confidence level 1 − α.
8 / 29
Remark
1. Confidence level 1 − α:
ˆ1 (X1 , X2 , · · · , Xn ), θ ˆ2 (X1 , X2 , · · · , Xn )) is a a. The interval (θ random interval, for each fixed sample observation value, there ˆ1 (x1 , x2 , · · · , xn ), θ ˆ2 (x1 , x2 , · · · , xn )), is one specific interval (θ which may contain θ, or may not contain θ. b. In repeated sampling (with the same sample size n), when the number of sampling is sufficiently large, the proportion of the numbers that the real value of parameter lies in the interval ˆ1 (x1 , x2 , · · · , xn ), θ ˆ2 (x1 , x2 , · · · , xn )) is 100(1 − α)%. (θ