Statistics theory of estimation

Egerton University
Mathematics Department
STAT 332: Theory of Estimation
Problem Set 3
1. Show that if T is an unbiased estimation of
 , then aT + b is unbiased estimator of a + b . Is T 2
an unbiased estimator
of  ?
2. Consider the random variables 𝑋1 , 𝑋2 , ⋯ , 𝑋𝑛 , where 𝑛 ≥ 10, 𝐸(𝑋𝑖 ) = 𝜇, 𝑉𝑎𝑟(𝑋𝑖 ) = 𝜎 2 > 0, and the estimator 𝜇̂ 𝑛 =
1
∑𝑛𝑖=1 𝑋𝑖 . Calculate
2
𝑛−10
a) The bias of 𝜇̂ 𝑛
b) The variance of 𝜇̂ 𝑛
c) The MSE of 𝜇̂ 𝑛
3. Let 𝑋1 , 𝑋2 , ⋯ , 𝑋𝑛 be a random sample from Bernoulli
( p) with common density f (x; p ) = p x (1 − p )1− x , x = 0,1 , where
0  p  1 is unknown.
1 n
a) Show that X =  X i is an unbiased estimator of p
n i =1
p (1 − p )
b) Show that Var ( X ) =
n
 X (1 − X ) 
 p (1 − p ) 
c) Show that E 
 = (1 − p ) 
.
n
n




d) Find the value of
4. Let X 1 , X 2 ,
c
(
so that cX 1 − X
) is an unbiased estimator of Var ( X ) =
, X n be a random sample from f ( x; ) =
1

1
p (1 − p )
n
.
−1
x  , 0  x  1, where   0 .is unknown.
1 n
a) Show that the MLE of  is ˆ = −  ln X i .
n i =1
b) Show that ˆ is mean-squared error consistent estimator of  .
5. Let X 1 , X 2 , , X n be a random sample from 𝑓(𝑥; 𝜆) = 𝑒 −(𝑥−𝜆) , 𝑥 > 𝜆, where   0 is unknown.
a) Show that the MME of 𝜆 is 𝑋⃖ − 1.
b) Is the MME consistent?
6. Let X 1 , X 2 ,
, X n be IID gamma ( ,  ) with density f (x; ,  ) =
   −1 − x
x e , x  0 , where  is known but 
( )
is
unknown.
a) Find the MLE ˆ of
b)
 .
Show that 1 ˆ is unbiased estimator of
7. Let X 1 , X 2 ,
1  ..
, X n be a random sample from f (x;  ) = e −x , x  0 , where   0 is unknown.
E (1 X ) and then construct an unbiased estimator of  .
b) Find the MSE of 1 X .
a) Compute
8. Let
Tn be a statistic such that lim E (Tn ) =  and lim  T2n = 0 . Prove that Tn is a consistent estimator of  .
n→
n→
, X n be a random sample from a uniform distribution on the interval ( , 2 ) ,where   0 is unknown.
n +1
2 X (n ) + X (1) is an unbiased estimator of  .
Shown that T =
5n + 4
9. Let
X1 , X 2 ,
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