Fundamentals of Probability and Statistics for Engineers

(a) Is unbiased?

(b) Is consistent?

(c) Show that anMLEofp.

9.10 Let X be a random variable with mean m and variance^2 ,andletX 1 ,X 2 ,...,Xn be
independent samples of X. Suppose an estimator for is found from the formula

Is an unbiased estimator? Verify your answer.

9.11 The geometrical mean is proposed as an estimator for the unknown
median of a lognormally distributed random variable X. Is it unbiased? Is it
unbiased as

9.12 Let X 1 ,X 2 ,X 3 be a sample of size three from a uniform distribution for which the pdf is

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.

9.13 Let X 1 ,...,Xn be a sample from a population whose kth moment
exists. Show that the kth sample moment

is a consistent estimator for

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.
(a)
(b)
(c)
(d)

9.15 Determine the CRLB for the variances of 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 Mk to the
kth population moment k;thatis

(a) Verify Equations (9.15).

(b) Show that Mk is a consistent estimator for k.

Parameter Estimation 309

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