STAT410- Homework #09 Solved

Let and let x 1,X2 , X n be a random sample from a probability distribution with probability density function

                 •(2—x Y-1                        zero otherwise.

Recall:                Fx(x)  

Let Y 1 < Y2 < < Y denote the corresponding order statistics.

i)                         [Proving that Y 1 = min Xi —Y 0 is super easy, barely an inconvenience. 

  Let U n Yl = n min X i. Find the limiting distribution of U

"Hint": O                   Find the c.d.f. of Y l , FY               min X i(x).

                                     Use FY (x) tofindthe                 of U n, FU 

0                                                                 lim FU (u). If the limit exists, and if F 00 (u) is a c.d.f.

n 00  of a probability distribution, then that is the limiting distribution of U  

j)                         Proving that Y = max Xi 2 is super easy, barely an inconvenience.

             Find so that V n                      nß(2—Yn) = nß(2— max X i) converges in

distribution. Find the limiting distribution of V  

"Hint": 0                            Use Fx(x) to find the c.d.f. of Y n, FY max X i(x).

Use FY (x) tofindthe c.d.f. of v n, Fv 

0                                                                lim Fv (v). IF the limit exists and IF F T (v) is a c.d.f.

of a probability distribution, then that is the limiting distribution of V  

n a lim 1+—   e . Only "interesting" case is interesting.

1

                                                  n +00            n

8. Let — > 0 and let X 1 , X 2 , X n be a random sample from a probability distribution with probability density function

  4 11 

zero elsewhere.

Recall:                   The maximum likelihood estimator of is 

                                                                                                          n      3

E x 1 

i=l 1

                           W = X 3 has a Gamma( u = 4, 0            ) distribution.

g)            Show that is asymptotically normally distributed ( as n 00).

Find the parameters.

"Hint": O       By  CLT, 

That is, for large n,

                                                   ) is approximately N                   gw), [g kiw)]2 29. 

n

9. Let and let x I , X n be a random sample from a probability distribution with probability density function

                  x2 '                                          zero otherwise.

a)                 (i) Obtain the maximum likelihood estimator of X, X. (ii) Suppose n =4, x 1 = 5, x2 = 10, x3 =3, x 4 - — 20.

Find the maximum likelihood estimate of X.

b)                Is X a consistent estimator of X? Justify your answer.

( NOT enough to say "because it is the maximum likelihood estimator" )

c)                 Is X an unbiased estimator of X? If is not an unbiased estimator of X, construct an unbiased estimator of based on X. (Assume n 2. )