404 Maximum Likelihood Methods6.5.12.Finish the derivation of the LRT found in Example 6.5.3. Simplify as much
as possible.6.5.13.Show that expression (6.5.25) of Example 6.5.3 is true.6.5.14.As discussed in Example 6.5.3,Z, (6.5.27), can be used as a test statistic
provided that we have consistent estimators ofp 1 (1−p 1 )andp 2 (1−p 2 )whenH 0
is true. In the example, we discussed an estimator that is consistent under bothH 0
andH 1. UnderH 0 , though,p 1 (1−p 1 )=p 2 (1−p 2 )=p(1−p), wherep=p 1 =p 2.
Show that the statistic (6.5.24) is a consistent estimator ofp, underH 0 .Thus
determine another test ofH 0.
6.5.15.A machine shop that manufactures toggle levers has both a day and a night
shift. A toggle lever is defective if a standard nut cannot be screwed onto the threads.
Letp 1 andp 2 be the proportion of defective levers among those manufactured by the
day and night shifts, respectively. We shall test the null hypothesis,H 0 :p 1 =p 2 ,
against a two-sided alternative hypothesis based on two random samples, each of
1000 levers taken from the production of the respective shifts. Use the test statistic
Z∗given in Example 6.5.3.
(a)Sketch a standard normal pdf illustrating the critical region havingα=0.05.(b)Ify 1 =37andy 2 = 53 defectives were observed for the day and night shifts,
respectively, calculate the value of the test statistic and the approximatep-
value (note that this is a two-sided test). Locate the calculated test statistic
on your figure in part (a) and state your conclusion. Obtain the approximate
p-value of the test.6.5.16.For the situation given in part (b) of Exercise 6.5.15, calculate the tests
defined in Exercises 6.5.12 and 6.5.14. Obtain the approximatep-values of all three
tests. Discuss the results.6.6 TheEMAlgorithm............................
In practice, we are often in the situation where part of the data is missing. For
example, we may be observing lifetimes of mechanical parts that have been put
on test and some of these parts are still functioning when the statistical analysis is
carried out. In this section, we introduce the EM algorithm, which frequently can be
used in these situations to obtain maximum likelihood estimates. Our presentation
is brief. For further information, the interested reader can consult the literature in
this area, including the monograph by McLachlan and Krishnan (1997). Although,
for convenience, we write in terms of continuous random variables, the theory in
this section holds for the discrete case as well.
Suppose we consider a sample ofnitems, wheren 1 of the items are observed,
whilen 2 =n−n 1 items are not observable. Denote the observed items byX′=
(X 1 ,X 2 ,...,Xn 1 ) and unobserved items byZ′=(Z 1 ,Z 2 ,...,Zn 2 ). Assume that
theXis are iid with pdff(x|θ), whereθ∈Ω. Assume that theZjsandtheXisare