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    188宝金博页面版: statisticalinference课件6

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    内容提示： 6Lecture 6: Hypothesis Testing6.1Assume X = (X1, X2, .., Xn) are i.i.d. from f(x, θ), θ ∈ Θ. Point estimation of θ will givean estimated value of θ which will be in general different from the true θ. In fact, if Θ wasnot a finite set but an interval (as it often happens) then the estimator and the true valuewill coincide with probability zero! This observation alone is convincing enough to claimthat it is not enough just to give a single estimated value of the parameter. The problemsof constructing...

    文档格式：PDF | 页数：11 | 浏览次数：113 | 上传日期：2013-03-21 12:21:58 | 文档星级：
    6Lecture 6: Hypothesis Testing6.1Assume X = (X1, X2, .., Xn) are i.i.d. from f(x, θ), θ ∈ Θ. Point estimation of θ will givean estimated value of θ which will be in general different from the true θ. In fact, if Θ wasnot a finite set but an interval (as it often happens) then the estimator and the true valuewill coincide with probability zero! This observation alone is convincing enough to claimthat it is not enough just to give a single estimated value of the parameter. The problemsof constructing confidence intervals (if the parameter was one-dimensional), or confidencesets (if the parameter was multi-dimensional), and the problems of testing hypothesesabout θ naturally arise.MotivationWe will mainly constrain ourselves to hypothesis testing and will avoid the thoroughdiscussion of confidence sets. Besides lack of time, the following argument can be put for-ward to defend this decision. In your introductory Statistics courses you have studied theinterrelationship between Hypothesis testing and the construction of Confidence intervals.In particular, given certain α size test of a hypothesis H0: θ = θ0, and having obtainedthe sample, we can take the set of the parameter values for which the test ”answers” withan acceptance when the sample is substituted in the test statistic. This set of parametervalues is a confidence set at level 1 − α. Symbolically, we can say that the subset in Θdefined via{θ|H0: θ = θis accepted given realization X = x of the sample }represents a confidence set at level (1 − α) for the unknown parameter θ.In other words, knowing how to construct tests, we basically also know how to con-struct confidence sets. Moreover, the usefulness of the relationship between testing hy-potheses and confidence sets is further exemplified by the fact that some optimality resultscarry over. It can be shown quite generally that the above procedure of constructing con-fidence sets leads to confidence sets with optimality properties if the hypothesis test usedin the construction was optimally designed.6.2General terminology in relation to hypothesis testing.Let us start with the case of testing a simple hypothesis against a simple alternative. Thisis the easiest case to discuss. Besides, the technique that is being used in this simple case(the Neyman-Pearson lemma below) is indeed fundamental and serves as a basis todeal with the more difficult cases, too.Assume that the unknown parameter θ can be one of the two values {θ0, θ1} only. Inother words, we are testing a simple hypothesis H0 : θ = θ0 versus a single alternativeH1: θ = θ1. A test ?(x) is defined as?(x) = P( reject H0| X = x).Generally, we would prefer deterministic decisions, i.e. we would like ?(x) to be equal toeither zero or one. Based on the observations we calculate ?(x) and according to its value,40

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