计量经济学5ppt课件

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
Sampling or probability distribution:
s2 X ~ N (m x , n )
(X m )
Z
X ~ N ( 0 ,1 )
s/ n
P(-n-1 ≤t≤n-1)=1－α
t
X m X S/ n
~t (n 1)
critical t values:±n-1 confidence interval:
(lower limit-upper limit)
Xtn1nSmX
Xtn1S n
P(Xtn n 1SμXXtn n 1S1α
confidence coefficient: 1－α
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level of significance/the prob. of committing type I error:
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.
estimate: the particular numerical value of the estimator
sampling variation /sampling error: the variation in estimation from samplthesis testing
In hypothesis testing we may have a prior judgment or
expectation about what value a particular parameter may
assume.
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5.3 ESTIMATION OF PARAMETERS
The usual procedure of estimation:
FIGURE 5-1
The t distribution
n-1
n-1
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Note: The interval is random, and not the parameterμx.
The confidence interval: a random interval, because it is based on X and S / n which will vary from samX ple to sample.
supposedly been drawn from that population.
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5.1 THE MEANING OF STATISTICAL INFERENCE
Statistical inference: the study of the relationship between a population and a sample drawn for that population.
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2. Interval estimate
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.
You should say: the probability is 0.95 that the random interval, contains the trueμx.
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Interval estimation, in contrast to point estimation, provides a range of values that will include the true value with a certain degree of confidence or probability (such as 0.95).
BRANCHES OF STATISTICAL INFERENCE
1. Estimation
Estimation: the first step in statistical inference. X : an estimator/statistic of the population parameter E(X),
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.
The process of generalizing from the sample value ( X ) to the population value E(X) is the essence of statistical inference.
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5.2 ESTIMATION AND HYPOTHESIS TESTING: TWIN
—— 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.