Transcription of Chapter 2. Order Statistics - 國立臺灣大學
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Chapter 2. Order Statistics - 國立臺灣大學
1 Chapter 2. Order Statistics1 The Order StatisticsFor a sample of independent observationsX1, X2, .. , Xnon a distributionF, the orderedsample valuesX(1) X(2) X(n),or, in more explicit notation,X(1:n) X(2:n) X(n:n),are called the Order Statistics . IfFis continuous, then with probability 1 the Order statisticsof the sample take distinct values (and conversely).There is an alternative way to visualize Order Statistics that, although it does notnecessarily yield simple expressions for the joint density, does allow simple derivation of manyimportant properties of Order Statistics . It can be called the quantile function function(orinverse distribution function, if you wish) is defined byF 1(y) = inf{x:F(x) y}.
continuous. Moreover, the above decomposition is unique. Let λ denote the Lebesgue measure on B, the σ-field of Borel sets in R. It follows from the Lebesgue decomposition theorem that we can write F c(x) = βF s(x)+(1−β)F ac(x) where 0 ≤ β ≤ 1, F s is singular with respect to λ, and F ac is absolutely continuous with respect to λ.
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