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    188宝金博页面版: statisticalinference课件7

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    内容提示： 7Lecture 7: Order Statistics7.1MotivationLet X = (X1, X2, . . . , Xn) denote a random sample from a population with a continuousdistribution function FX. Since FX is assumed to be continuous, the probability of anytwo of these random variables assuming the same value is zero. After reordering the nvalues we get X(1) ≤ X(2) ≤ · · · ≤ X(n)) in which, as mentioned, the ≤ sign couldalso be replaced by <. These values are collectively termed the order statistic of therandom sample X = (X1, X2, . . . , Xn...

    文档格式：PDF | 页数：5 | 浏览次数：141 | 上传日期：2013-03-21 12:22:39 | 文档星级：
    7Lecture 7: Order Statistics7.1MotivationLet X = (X1, X2, . . . , Xn) denote a random sample from a population with a continuousdistribution function FX. Since FX is assumed to be continuous, the probability of anytwo of these random variables assuming the same value is zero. After reordering the nvalues we get X(1) ≤ X(2) ≤ · · · ≤ X(n)) in which, as mentioned, the ≤ sign couldalso be replaced by <. These values are collectively termed the order statistic of therandom sample X = (X1, X2, . . . , Xn). The subject of order statistics generally deals withproperties of X(r)(r = 1, 2, . . . , n) which is called the r-th order statistic.Order statistics are particularly useful in nonparametric statistics because of the fol-lowing:Theorem 7.1. (Probability-integral transformation). Ifthe random variable X hasa continuous cdf FX then the random variable Y = FX(X) has the uniform probabilitydistribution over the interval (0,1). Further, given a sample X = (X1, X2, . . . , Xn) ofn i.i.d. random variables with cdf FX, the transformation U(r) = FX(X(r)) produces arandom variable U(r) which is the r-th order statistic from the uniform population in(0,1), regardless of what FXis, i.e. U(r)is distribution-free.Proof: It is your textbook and was also discussed in the introductory Lecture 1.Note: The above theorem has also an extremely important practical application in thegeneration (computer simulation) ofobservations from any specific continuous distribu-tion function. There are several well-developed uniform random number generatorsthat implement methods to generate sequences of uniform in (0,1) pseudo-random num-bers. These numbers are pseudo since in fact they are generated by a deterministic algo-rithm (therefore are not random) but look as random (hence the word pseudo-random)in the sense that they pass usual statistical tests about randomness of the generated se-quence. Every program system (Fortran, SPLUS, C, SAS, etc.) has such uniform randomnumber generators and we will not discuss their specific implementation here. What wewould like to discuss is how we could use these uniform random number generators togenerate random numbers with arbitrary continuous cumulative distribution functionFX. The answer is:1) Generate Y as uniformly distributed in (0,1) using the uniform random numbergenerator2) Calculate ξ = F−1X(Y).Then ξ is distributed according to FX(.) since its cumulative distribution function is:P(F−1X(Y) < x) = P(F−1X(FX(X)) < x) = P(X < x) = FX(x).Some further important obvious applications of order statistics are listed below:• X(n) is of interest in studying floods, earthquakes and other extreme phenomena,sports records, financial markets etc.51

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