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    188宝金博页面版: statisticalinference课件3

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    内容提示： 3Lecture 3: PRINCIPLES OF DATA REDUCTIONAND INFERENCE3.1Data Reduction in Statistical InferenceGiven vector X=(X1, X2, .., Xn) of n i.i.d. random variables, each with a density f(x; θ),we are meant to conduct inference on θ ∈ Θ based on the observations x1, x2, .., xn.Let X takes values in X - the sample space. The statistician uses the information inthe observations x1, x2, .., xnto conduct the inference. His/her wish is to summarize theinformation in the sample by determining a few key features of th...

    文档格式：PDF | 页数：11 | 浏览次数：25 | 上传日期：2013-03-21 12:19:53 | 文档星级：
    3Lecture 3: PRINCIPLES OF DATA REDUCTIONAND INFERENCE3.1Data Reduction in Statistical InferenceGiven vector X=(X1, X2, .., Xn) of n i.i.d. random variables, each with a density f(x; θ),we are meant to conduct inference on θ ∈ Θ based on the observations x1, x2, .., xn.Let X takes values in X - the sample space. The statistician uses the information inthe observations x1, x2, .., xnto conduct the inference. His/her wish is to summarize theinformation in the sample by determining a few key features of the sample values throughtransforming the sample values. Calculating such transformations (i.e. functions of thesample) means to calculate a statistic. Typically , dim(T)<<n, i.e. using the statistic,we achieve the goal of data reduction: rather than reporting the entire sample x, thestatistic reports only that T(x)=t. Data reduction in terms of a particular statistic canbe thought of as a partition of the sample space. We partition X into disjoint subsetsAt = {X:T(X)=t}. If τ = {t:t=T(x) for some x ∈ X} then the sample space X isrepresented as a union of the following disjoint sets (i.e. is partitioned) : X =The ultimate goal in the data reduction is, when only using the value of the statistic T(x)instead of the whole vector x, ”not to lose information” about the parameter of interestθ. The whole information about θ will be contained in the statistic and, in particular, wewill treat as equal any two samples x and y that satisfy T(x)=T(y) even though theactual sample values may be different. That way we arrive at the definition of sufficiency.The information in X about θ can be discussed in terms of partitions of the sample space.t∈τAt.Definition 1(sufficient partition)Suppose for any set Atin a particular partition A = {At, t ∈ τ} we haveP{X=x | X ∈ At}does not depend on θ. Then A is a sufficient partition for θ.Note: We have seen above that the partition is defined through a suitable statistic. Ifthe statistic T is such that it generates a sufficient partition of the sample space then thestatistic itself is sufficient.3.2Example:X=(X1, X2, .., Xn) i.i.d. Bernoulli with parameter θ, i.e.P(Xi= xi) = θxi(1 − θ)1−xi, xi= 0, 1. The partition A = (A0, A1, . . . , An) where x ∈ Arif and only if (iff)nni=1Xiis sufficient for θ.i=1xi= r, is sufficient for θ. Correspondingly, the statistic T(X) =Proof: At lecture.Note that given the observed value t of T, we know that the observed value x of X is inthe partition set At. Sufficiency means that P(X = x | T = t) is a function of x and t only17

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