Page 326 - Fundamentals of Probability and Statistics for Engineers

Parameter Estimation 309

(a) Is P ^ unbiased?
(b) Is P ^ consistent?
^
(c) Show that P is an MLE of p.
9.10 Let X be a random variable with mean m and variance 2 , and let X 1 , X 2 , .. ., X n be
2

independent samples of X. Suppose an estimator for , c 2 is found from the formula
1
2
2
2
c 2 ‰ X 2 X 1 † ‡ X 3 X 2 † ‡ ‡ X n X n 1 † Š:
ˆ
2 n 1†
c 2
Is an unbiased estimator? Verify your answer.
9.11 The geometrical mean X 1 X 2 X n ) 1/n is proposed as an estimator for the unknown
median of a lognormally distributed random variable X. Is it unbiased? Is it
unbiased as n !1?
9.12 Let X 1 , X 2 , X 3 be a sample of size three from a uniform distribution for which the pdf is
8
1
< ; for 0 x ;
f X x; †ˆ
0; elsewhere.
:
Suppose that aX (1) and bX (3) are proposed as two possible estimators for .
(a) Determine a and b such that these estimators are unbiased.
(b) Which one is the better of the two? In the above, X (j) is the jth-order statistic.
k
9.13 Let X 1 , ..., X n be a sample from a population whose kth moment k ˆ EfX g
exists. Show that the kth sample moment
n
1 X
M k ˆ X j k
n
jˆ1
is a consistent estimator for k .

9.14 Let be the parameter to be estimated in each of the distributions given below. For
each case, determine the CRLB for the variance of any unbiased estimator for .
1 x/
(a) f x; ) ˆ e , x 0.

(b) f x; ) ˆ x 1 ,0 x 1, > 0.
x
(c) p x; ) ˆ 1 ) 1 x , x ˆ 0, 1.
x
e
(d) p x; ) ˆ , x ˆ 0, 1, 2, ... .
x!
^
c 2
9.15 Determine the CRLB for the variances of M and , which are, respectively,
unbiased estimators for m and 2 in the normal distribution N(m, 2 ).
9.16 The method of moments is based on equating the kth sample moment M k to the
kth population moment k ; that is
M k ˆ k :
(a) Verify Equations (9.15).
(b) Show that M k is a consistent estimator for k .

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