Page 178 - Elements of Distribution Theory

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164 Parametric Families of Distributions

Let
n
1
T 2 (x) = x j .
n
j=1
Note that T 2 (x) takes values in and, for a > 0, T 2 (ax) = aT 2 (x); hence, T 2 (X)isan
equivariant statistic with range .Itnow follows from Theorem 5.8 that X is equivalent to
(T 1 (X), T 2 (X)). This can be verified directly by noting that
nT 2 (X)
X 1 = n
(1 + X j /X 1 )
j=2
and X j = X 1 (X j /X 1 ), j = 2,..., n,so that X is a function of (T 1 (X), T 2 (X)).
As noted above, the statistics T 1 (X) and T 2 (X) used here are not unique; for instance,
˜
consider T 2 (X) = X 1 . Clearly X 1 is an equivariant statistic with range so that X is
˜
equivalent to (T 1 (X), T 2 (X)). Also,
n n
T 2 (X) 1 j=1 X j 1 X j
˜ −1
T 2 (X) T 2 (X) = = = 1 +
˜
T 2 (X) n X 1 n X 1
j=2
is a function of T 1 (X).
5.7 Exercises

5.1 Suppose that n computer CPU cards are tested by applying power to the cards until failure. Let
Y 1 ,..., Y n denote the failure times of the cards. Suppose that, based on prior experience, it is
believed that it is reasonable to assume that each Y j has an exponential distribution with mean
λ and that the failure times of different cards are independent. Give the model for Y 1 ,..., Y n
by specifying the model function, the parameter, and the parameter space. Is the parameter
identifiable?
5.2 In the scenario considered in Exercise 5.1, suppose that testing is halted at time c so that only
those failure times less than or equal to c are observed; here c is a known positive value. For card
j, j = 1,..., n,we record X j , the time at which testing is stopped, and a variable D j such that
D j = 1ifa failure is observed and D j = 0if testing is stopped because time c is reached. Hence,
if D j = 0 then X j = c.Give the model for (X 1 , D 1 ),..., (X n , D n ), including the parameter and
the parameter space. Is the parameter identifiable?
5.3 Let X and Y denote independent random variables. Suppose that
Pr(X = 1) = 1 − Pr(X = 0) = λ, 0 <λ< 1;
if X = 1, then Y has a normal distribution with mean µ 1 and standard deviation σ, while if X = 0,
Y has a normal distribution with mean µ 0 and standard deviation σ. Here µ 0 and µ 1 each take
any real value, while σ> 0. Let Y 1 ,..., Y n denote independent, identically distributed random
variables such that Y 1 has the distribution of Y.Give the model for Y 1 ,..., Y n by specifying the
model function, the parameter, and the parameter space. Is the parameter identifiable?
5.4 As in Exercise 5.1, suppose that n CPU cards are tested. Suppose that for each card, there is a
probability π that the card is defective so that it fails immediately. Assume that, if a card does
not fail immediately, then its failure time follows an exponential distribution and that the failure
times of different cards are independent. Let R denote the number of cards that are defective; let
Y 1 ,..., Y n−R denote the failure times of those cards that are not defective. Give the model for
R, Y 1 ,..., Y n−R , along with the parameter and the parameter space. Is the parameter identifiable?