[SOLVED] STAT410- Homework #10

Description

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.

1. 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.

1. 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.

1. Find a sufficient statistic u (X 1 , X2, , X n) for V.
2. Find the Fisher information I (y ).
3. 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

1. 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.

1. Find a sufficient statistic u (X , X2, , X n) for g.
2. Find the Fisher information I ( ).

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

1. 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

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

[1] ^T

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