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Outline. Order statistics Distribution of order variables (and extremes) Joint distribution of order variables (and extremes). Order statistics. Let X 1 , …, X n be a (random) sample and set X ( k ) = the k th smallest of X 1 , …, X n Then the ordered sample

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  1. Outline • Order statistics • Distribution of order variables (and extremes) • Joint distribution of order variables (and extremes) Probability theory 2008

  2. Order statistics Let X1, …, Xnbe a (random) sample and set X(k) = the kth smallest ofX1, …, Xn Then the ordered sample (X(1), X(2), …, X(n)) is called the order statistic of (X1, …, Xn) and X(k)the kth order variable Probability theory 2008

  3. Order variables - examples Example 1: Let X1, …, Xnbe U(0,1) random numbers. Find the probability that max(X1, …, Xn) > 1 – 1/n Example 2: Let X1, …, X100 be a simple random sample from a (finite) population with median m. Find the probability that X(40) > m. Probability theory 2008

  4. Distribution of the extreme order variables Probability theory 2008

  5. The beta distribution For integer-valued r and s, the beta distribution represents the rth highest of a sample of r+s-1 independent random variables uniformly distributed on (0,1) =r =s Probability theory 2008

  6. The gamma function Probability theory 2008

  7. Distribution of arbitrary order variables Probability theory 2008

  8. A useful identity Can be proven by backward induction Probability theory 2008

  9. Distribution of arbitrary order variables Probability theory 2008

  10. Distribution of arbitrary order variablesfrom a U(0,1) distribution Probability theory 2008

  11. Joint distribution of the extreme order variables Probability theory 2008

  12. Functions of random variables Let X have an arbitrary continuous distribution, and suppose that g is a (differentiable) strictly increasing function. Set Then and

  13. Linear functions of random vectors Let (X1, X2) have a uniform distribution on D = {(x , y); 0 < x <1, 0 < y <1} Set Then .

  14. Functions of random vectors Let (X1, X2) have an arbitrary continuous distribution, and suppose that g is a (differentiable) one-to-one transformation. Set Then where h is the inverse of g. Proof: Use the variable transformation theorem

  15. Density of the range Consider the bivariate injection Then and Probability theory 2008

  16. Density of the range Probability theory 2008

  17. The range of a sample from an exponential distribution with mean one Probabilistic interpretation of the last equation? Probability theory 2008

  18. Joint distribution of the order statistic Consider the mapping (X1, …, Xn)  (X(1), …, X(n)) or . where P is a permutation matrix Probability theory 2008

  19. Joint density of the order statistic Probability theory 2008

  20. Exercises: Chapter IV 4.2, 4.7, 4.9, 4.12, 4.14, 4.17 Probability theory 2008

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