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246 CHAPTER 7 POINT ESTIMATION OF PARAMETERS

MIND-EXPANDING EXERCISES

7-68. Continuation of Exercise 7-65. Let X 1 , X 2 , p , not differ much from S when there are no unusual
X n be a random sample of an exponential random vari- observations.
ˆ
able of parameter . Derive the cumulative distribution (a) Calculate and S for the data 10, 12, 9, 14, 18, 15,
functions and probability density functions for X (1) and and 16.
X (n) . Use the result of Exercise 7-65. (b) Replace the first observation in the sample (10) with
7-69. Let X 1 , X 2 , p , X n be a random sample of a 50 and recalculate both S and . ˆ
continuous random variable with cumulative distribu- 7-72. Censored Data. A common problem in indus-
tion function F(x). Find try is life testing of components and systems. In this
problem, we will assume that lifetime has an exponen-
ˆ
E3F 1X 1n2 24 tial distribution with parameter , so ˆ 1 X is
an unbiased estimate of . When n components are tested
and until failure and the data X 1 , X 2 , p , X n represent actual
lifetimes, we have a complete sample, and X is indeed an
unbiased estimator of . However, in many situations, the
E 3F 1X 112 24
components are only left under test until r n failures
have occurred. Let Y 1 be the time of the first failure, Y 2 be
7-70. Let X be a random variable with mean and the time of the second failure, p , and Y r be the time of the
2
variance , and let X 1 , X 2 , p , X n be a random sample last failure. This type of test results in censored data.
n 1
There are n r units still running when the test is termi-
of size n from X. Show that the statistic V k g i 1
2
1X i 1 X i 2 2 is an unbiased estimator for for an nated. The total accumulated test time at termination is
appropriate choice for the constant k. Find this value
r
for k. T r a Y i 1n r2Y r
7-71. When the population has a normal distribution, i 1
the estimator
r
(a) Show that ˆ T r is an unbiased estimator for .
[Hint:You will need to use the memoryless property
ˆ median 10 X 1 X 0 , 0 X 2 X 0 ,
of the exponential distribution and the results of
p , 0 X n X 0 2 0.6745 Exercise 7-68 for the distribution of the minimum of
a sample from an exponential distribution with
is sometimes used to estimate the population standard parameter .] 2
deviation. This estimator is more robust to outliers than (b) It can be shown that V1T r r2 1 1 r2. How does
the usual sample standard deviation and usually does this compare to V1X 2 in the uncensored experiment?

IMPORTANT TERMS AND CONCEPTS
In the E-book, click on any Mean square error of an ence in two sample Statistic
term or concept below to estimator means Statistical inference
go to that subject. Minimum variance Parameter estimation Unbiased estimator
Bias in parameter unbiased estimator Point estimator
estimation Moment estimator Population or distribu- CD MATERIAL
Central limit theorem Normal distribution as tion moments Bayes estimator
Estimator versus the sampling distribu- Sample moments Bootstrap
estimate tion of a sample mean Sampling distribution Posterior distribution
Likelihood function Normal distribution as Standard error and Prior distribution
Maximum likelihood the sampling distri- estimated standard
estimator bution of the differ- error of an estimator