计量经济学5.

1. Point estimate
A point estimator, or a statistic, is an r.v., its value will vary from sample to sample. How can we rely on just one estimate X of the true population mean.
Although Xis the single “best” guess of the true population mean, the interval, say, from 8 to 14, most likely includes the true μχ ? This is interval estimation. Sampling or probability distribution:
X
t n1 S
mX X
t n1 S

FIGURE 5-1 The t distribution
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n- 1
Note: The interval is random, and not the parameterμx.
The confidence interval: a random interval, because it is X based on X and S / n which will vary from sample to sample. The population mean: although unknown, is some fixed number and it is not random. You should not say: the probability is 0.95(1－α) that μx lies in this interval. You should say: the probability is 0.95 that the random interval, contains the trueμx.
Chapter 5
STATISTICAL INFERENCE: ESTIMATION AND HYPOTHESES TESTING
Statistical inference draws conclusions about a population [i.e., probability density function (PDF) ] from a random sample that has supposedly been drawn from that population.
5.2 ESTIMATION AND HYPOTHESIS TESTING: TWIN
BRANCHES OF STATISTICAL INFERENCE
1. Estimation
Estimation: the first step in statistical inference. X : an estimator/statistic of the population parameter E(X), estimate: the particular numerical value of the estimator sampling variation /sampling error: the variation in estimation from sample to sample.
The usual procedure of estimation: —— to assume that we have a random sample of size n from the known probability distribution and use the sample to estimate the unknown parameters, that is, use the sample mean as an estimate of the population mean (or expected value) and the sample variance as an estimate of the population variance.
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2. Interval estimate
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n n critical t values:±n-1 t S t S confidence interval: P(X n 1 μ X X n 1 1 α n n (lower limit-upper limit) confidence coefficient: 1－α level of significance/the prob. of committing type I error:
2.Hypothesis testing
In hypothesis testing we may have a prior judgment or expectation about what value a particular parameter may assume.
5.3 ESTIMATION OF PARAMETERS
5.1 THE MEANING OF STATISTICAL INFERENCE
Statistical inference: the study of the relationship between a population and a sample drawn for that population. The process of generalizing from the sample value ( X ) to the population value E(X) is the essence of statistical inference.