Applied Statistics and Probability for Engineers

246 CHAPTER 7 POINT ESTIMATION OF PARAMETERS

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Bias in parameter

estimation

Central limit theorem

Estimator versus

estimate

Likelihood function

Maximum likelihood

estimator

Mean square error of an

estimator

Minimum variance

unbiased estimator

Moment estimator

Normal distribution as

the sampling distribu-

tion of a sample mean

Normal distribution as

the sampling distri-

bution of the differ-

ence in two sample

means

Parameter estimation

Point estimator

Population or distribu-

tion moments

Sample moments

Sampling distribution

Standard error and

estimated standard

error of an estimator

Statistic

Statistical inference

Unbiased estimator

CD MATERIAL

Bayes estimator

Bootstrap

Posterior distribution

Prior distribution

IMPORTANT TERMS AND CONCEPTS

MIND-EXPANDING EXERCISES

7-68. Continuation of Exercise 7-65. Let X 1 , X 2 ,,

Xnbe a random sample of an exponential random vari-

able of parameter . Derive the cumulative distribution

functions and probability density functions for X(1)and

X(n). Use the result of Exercise 7-65.

7-69. Let X 1 , X 2 ,, Xnbe a random sample of a

continuous random variable with cumulative distribu-

tion function F(x). Find

and

7-70. Let Xbe a random variable with mean and

variance ^2 , and let X 1 , X 2 ,, Xnbe a random sample

of size nfrom X.Show that the statistic

is an unbiased estimator for ^2 for an

appropriate choice for the constant k. Find this value

for k.

7-71. When the population has a normal distribution,

the estimator

is sometimes used to estimate the population standard

deviation. This estimator is more robust to outliers than

the usual sample standard deviation and usually does

not differ much from Swhen there are no unusual

observations.

(a) Calculate and Sfor the data 10, 12, 9, 14, 18, 15,

and 16.

(b) Replace the first observation in the sample (10) with

50 and recalculate both Sand.

7-72. Censored Data.A common problem in indus-

try is life testing of components and systems. In this

problem, we will assume that lifetime has an exponen-

tial distribution with parameter , so is

an unbiased estimate of . When ncomponents are tested

until failure and the data X 1 , X 2 ,, Xnrepresent actual

lifetimes, we have a complete sample, and is indeed an

unbiased estimator of . However, in many situations, the

components are only left under test until rnfailures

have occurred. Let Y 1 be the time of the first failure, Y 2 be

the time of the second failure, , and Yrbe the time of the

last failure. This type of test results in censored data.

There are nrunits still running when the test is termi-

nated. The total accumulated test time at termination is

(a) Show that is an unbiased estimator for .

[Hint:You will need to use the memoryless property

of the exponential distribution and the results of

Exercise 7-68 for the distribution of the minimum of

a sample from an exponential distribution with

parameter .]

(b) It can be shown that How does

this compare to V 1 X 2 in the uncensored experiment?

V 1 Tr r 2  11 ^2 r 2.

ˆTr r

Tra

r

i 1

Yi 1 nr 2 Yr

p

X

p

ˆ 1 ˆX

ˆ

ˆ

p, 0 XnX 02 0.6745

ˆmedian 10 X 1 X 0 , 0 X 2 X 0 ,

1 Xi 1 Xi 22

Vkgni 11

p

E 3 F 1 X 11224

E 3 F 1 X 1 n 224

p

p

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