STAT410- Homework #10 Solved

Let IlJ>0 and let x l,X2 , X n be a random sample from a probability distribution with probability density function  zero otherwise.

                                 x                                                         1

Recall:                        -In 1                   has an Exponential distribution with mean 0 

2

  is an unbiased estimator of V.

k)       Suggest a confidence interval for             with ( 1 u) 100 % confidence level.

n

O Use Y = E — In 1 

2

                          If T has a Gamma(u, 0 1//X) distribution, then

2 T
//0 = 2 XT has a 1 2 (20) distribution.

l)                  Suppose n = 3, and          x I = 0.62, x 2 = 1.54, x 3 = 1.86.

Use part (k) to construct a 90% confidence interval for V.

m)             Find a sufficient statistic u (X 1 , X2,  , X n) for V.

n)                Find the Fisher information I (y ).

o)                 Isan efficient estimator of V ?

Ifis not efficient, find its efficiency.

                     O Find var( ).                     ("Hint": Recall Homework #08 problem 7 part (g). )

Find the Rao-Cramér lower bound.

Is RIJ an efficient estimator of V? Does Var( RIJ ) attain the R.C.L.B.?

If y is not efficient, find its efficiency.

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

1 4 11 

 zero elsewhere.

2

1

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

 is an unbiased estimator of .

n

i=l

h)              Suggest a confidence interval for with ( I — u) 100 % confidence level.

                         Use Y =        X

 

                          If T has a Gamma(a, 0= 1/4) distribution, then

i)        Suppose n = 5, x 1 = 0.3, x 2 = 0.6,          1.2, x 4 = 1.3, x 5 = 1.8 
Use part (h) to construct a 90% confidence interval for g.

j)                    Find a sufficient statistic u (X , X2,    , X n) for g.

k)                 Find the Fisher information I ( ).

( After you are done with part (k), glance back at Homework #09 problem 8 part (g). )

l)                     Is an efficient estimator of < ?

Ifis not efficient, find its efficiency.

                    O Find var( ).                      ("Hint": Recall Homework #08 problem 8 part (d). )

Find the Rao-Cramér lower bound.

 Is an efficient estimator of F, ? Does Var( ) attain the R.C.L.B.?

Ifis not efficient, find its efficiency.

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

 2 '    zero otherwise. x

d)          Find a sufficient statistic , x n) for X.


 
[1] T

//0 = 2 XT has a 1 2 (20) distribution.