Order Statistics

 

B.Sc. III    Semester- VI    Statistics –XIII

Probability Theory and Applications

Unit 1:  Order Statistics and Convergence

1.1   Order Statistics

        Let X1, X2, X3, …..,Xn  be a random sample of size n from a continuous distribution with p. d. f. f(x) and c .d. f. F(x).

         Let,   X(1) = min { X1, X2, X3, ….., Xn }

                  X(2) = 2nd min { X1, X2, X3, ….., Xn }

                  X(3) = 3rd min { X1, X2, X3, ….., Xn }

                  …..

                 X(n) = nth min { X1, X2, X3, ….., Xn } =  max{ X1, X2, X3, ….., Xn }

then X(1) , X(2) , ……, X(n) are  known as the order statistics corresponding to the random sample X1, X2, X3, …..,Xn. For convenience, we denote   Y1 = X(1) , Y2 = X(2) , Y3 = X(3) ,….., Yn = X(n) . Thus Y1, Y2, Y3,…..,Yn  are the order statistics corresponding to the random sample X1, X2, X3, ….., Xn.

Note that        X(1) < X(2) <……< X(n)        i.e.     Y1< Y2< Y3,…..<Yn .

Note: Let X1, X2, X3, X4, X5 be i. i. d.  random variables with a distribution F with a range of              (a, b). We   can relabel these X’s such that their labels correspond to arranging them in                 increasing order so that   X(1) ≤ X(2) ≤ X(3) ≤ X(4) ≤ X(5) .

                                   X(1)       X(2)        X(3)        X(4)        X(5)

                         

                           a       X5        X1          X4          X2          X3      b

                 In the case where the distribution F is continuous we can make the stronger statement that X(1) < X(2) < X(3) < X(4) < X(5) Since P(Xi = Xj) = 0 for all i ≠ j for continuous random variables.

Distribution of 1st order statistics Y1   i.e. X(1)  or Xmin 

        Let X1, X2, X3, …..,Xn  be a random sample of size n from a continuous distribution with p. d. f. f(x) and c .d. f. F(x). Let Y1 =  min { X1, X2, X3, ….., Xn } i.e. Y1 is first order statistics corresponding to the random sample X1, X2, X3, …..,Xn.

Let g1(y) be the p. d. f. and G1(y) is a c. d. f. of Y1.

By defn of  c. d. f., we have

                                         G1(y) = P [Y1 ≤ y] = P [min {X1, X2, X3, ….., Xn } ≤ y ]

                                                                         = 1- P [min {X1, X2, X3, ….., Xn } ＞ y ]

                                                                         = 1 - P [each of Xi’s ＞ y ]

                                                                         = 1- P [X1 ＞  y, X2 ＞  y, …,  Xn ＞ y]

                                                                         = 1- P [X1＞ y] . P[ X2 ＞y]…… P[ Xn ＞y].

                                                                         = 1-  ∏  P [Xk ＞ y]

                                                                         = 1- ∏  [1- P (Xk ≤  y)]

                                                               G1(y) = 1-∏ [1- F (y)]                                                                                                                ∴   G1(y) = 1- [1- F (y)]n

         p. d. f. of first order statistics Y1 is ,

                                                                     g1(y) =d/dy  G1(y) =  {1- [1- F (y)]n}

                                         = 0 – n  [1- F (y)]n-1 [-f(y)]

                                                                  g1(y) = n [1- F (y)]n-1 [f(y)]

Ex: Let X1, X2, X3, …..,Xn  be a random sample from U(0,1). Obtain p. d. f. of first order statistics X(1)   and  hence find E(X(1) ) .

Soln : Here X has   uniform distribution  i.e. X  ̴  U (0,1) , the p. d. f. of X is given by

                     f(x) = 1,        0 ≤ X ≤ 1

                            = 0,        otherwise

      Here X1, X2, X3, …..,Xn  is a random sample from U(0,1).

      Let, Y1 = X(1)   = min { X1, X2, X3, ….., Xn }.

      We have c. d. f. of X is, F(x) = P[X ≤ x] = ∫ f(t) dt = ∫ 1 dt =[t] (0 to x)  = x ;  0 ≤ x ≤ 1.

   p. d. f. of first order statistics Y1 is,

                                                                 g1(y) = n [1- F (y)]n-1 [f(y)]

                                                                           = n [1- y1]n-1 . 1

                                                         g1(y) = n [1- y1]n-1          0 ≤ y1≤ 1

                                                                     = 0,        otherwise

This is p. d. f. of beta distribution of first kind with parameters (1,n).

i. e.  Y1 ̴ β1 ( 1,n).

Now,

            E [X(1)] =  E [Y1] =  ∫ y1 g1(y) dy =  ∫  y1 n [1- y1]n-1 dy1  =  n ∫ y1 [1- y1]n-1 dy1 

                         = n  ∫  (y1)2-1 [1- y1]n-1 dy1 

                         = n β1 (2, n) =  1/n+1

 ***

Distribution of nth order statistics Yn   i.e. X(n)  or X max 

               Let X1, X2, X3, …..,Xn  be a random sample of size n from a continuous distribution with p. d. f. f(x) and c .d. f. F(x). Y1, Y2, Y3,…..,Yn  are the order statistics corresponding to the random sample X1, X2, X3, ….., Xn. Here Yn =  max { X1, X2, X3, ….., Xn } = X max .

Let gn (y) be the p. d. f. and Gn(y) is a  c. d. f. of Yn.

By defn of c. d. f., we have

                                         Gn(y) = P [Yn ≤  y] = P [max {X1, X2, X3, ….., Xn } ≤  y ]

                                                                         =  P [each of Xi’ s  ≤  y ]

                                                                         = P [X1 ≤  y,  X2 ≤  y,…… ,Xn ≤  y].

                                                                         =  P [X1 ≤  y] . P[ X2 ≤  y]…… P[ Xn ≤  y].

                                                                        = ∏  P [Xk ≤  y]

                                                                         =  ∏ F (y)

                                                              Gn(y) =  [ F (y)]n

       ∴     p. d. f  of  n th  order statistics Yn is ,

                                                           gn(y) =d/dy  Gn(y) = d/dy [ F (y)]n

                                      ∴   gn(y) =  n  [ F (y)]n-1 [f(y)]

                                                                               ***

Ex: Let X1, X2, X3, …..,Xn  be a random sample from U(0,1). Obtain p. d. f. of maximum( nth order statistics X(n) ) .

Soln : Here X has   uniform distribution  i.e. X  ̴  U (0,1) , the p. d. f. of X is given by

                     f(x) = 1,        0 ≤ X ≤ 1

                            = 0,        otherwise

      Here X1, X2, X3, …..,Xn  is a random sample from U(0,1).

      Let, Yn = X(n)   = max { X1, X2, X3, ….., Xn }.

      We have c. d. f. of X is, F(x) = P[X ≤ x] = ∫ f(t) dt = ∫ 1 dt = [t] (0 to x) = x ;  0 ≤ x ≤ 1.

   p.d.f. of nth  order statistics Yn is,

                                                                 Gn(y) = n [ F (y)]n-1 [f(y)]

                                                                           = n [ yn]n-1 . 1

                                                         gn(y) = n  ynn-1          0 ≤ yn ≤ 1

                                                                     = 0,        otherwise

This is p.d.f. of beta distribution of first kind with parameters (n,1).

i. e.  Yn ̴  β1 (n,1).

 

Note: If X1, X2, X3, …..,Xn  be a random sample from U(0,1), then

        i) Y1 =  min { X1, X2, X3, ….., Xn } ̴ β1 ( 1,n).

       ii) Yn =  max { X1, X2, X3, ….., Xn } ̴ β1 ( n,1).


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J  Joint p. d. f. of ith and jth order statistics:


CC Consider, 


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Distribution of range :

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