that changes in the autocorrelation struc- censoring. A transformation can be applied
ture have matching changes in the Fourier to keep CI endpoints from falling outside
or wavelet-based spectrum and can be the parameter space and improve perfor-
detected by comparing estimated spectra of mance, but the degree of improvement (if
adjacent blocks of the series. To be effective any) depends on the chosen function. To
for slowly changing spectral structure, the obtain CIs for distribution probabilities,
methods are extended to allow information some seemingly useful transformation func-
from more than one past block to be used tions will cause the estimated variance to
in determining if a change has occurred, in blow up in the tails of the distribution and
a way that minimizes computational bur- can lead to nonsensical confidence intervals.
den. The methods are found to provide This article compares different NACIs for
reliable and timely detection of changes in distribution probabilities. The results suggest
covariance structure in an online monitor- that basing these methods on a studentized
ing framework. They are illustrated using statistic, which is named the z
∧
procedure,
electroencephalogram traces and run-time has desirable properties when compared
computer performance metrics. with alternatives.
The final four papers in the issue address The gamma distribution is a popular
topics in reliability. Daniel R. Jeske, Richard choice for inference in reliability appli-
A. Lockhart, Michael A. Stephens, and Qi cations. In “Normal-Based Methods for
Zhang consider the problem of making a Gamma Distribution: Prediction and
inference about impending software failures. Tolerance Intervals and Stress-Strength
Higher-precision inferences can be achieved Reliability,” Kalimuthu Krishnamoorthy,
when the same software reliability model Thomas Mathew, and Shubhabrata
fits failure data from both the test interval Mukherjee propose inferential procedures
and field interval. In their article, “Cramer- for the gamma distribution using the
von Mises Tests for the Compatibility of Wilson-Hilferty (WH) normal approxima-
Two Software Operating Environments,” tion. Using the result that the cubic root of
they develop a Cramer-von Mises (CvM) a gamma random variable is approximately
test procedure within the context of a well- normally distributed, they propose normal-
known nonhomogeneous Poisson process based approaches for a gamma distribution
software reliability model. The CvM test for (a) constructing prediction limits, one-
has a compelling advantage over an existing sided tolerance limits, and tolerance inter-
likelihood ratio test (LRT0) because it does vals; (b) obtaining upper prediction limits
not require specification of the class of alter- for at least l of m observations at each of
natives, which are frequently unknown for r locations; and (c) assessing the reliabil-
real-life problems. Moreover, there are exis- ity of a stress-strength model involving
tence issues with LRT0, whereas the CvM two independent gamma random vari-
test always exists. An asymptotic approxi- ables. For each problem, a normal-based
mation for the p-value of the CvM test is approximate procedure is outlined, and its
derived and an algorithm for a small-sample applicability and validity for the gamma
bootstrap approximation is presented. A distribution are studied using Monte
simulation study shows the CvM test works Carlo simulation. The approximate pro-
well for the class of alternatives for which cedures have satisfactory properties for all
LRT0 also would work well and continues these problems. The results are illustrated
to work well for other alternatives for which using several practical examples.
LRT0 has no statistical motivation or oth- Francesc Carreras, Josep Freixas, and
erwise has existence problems. Data from a Maria Albina Puente analyze a problem
real software project are used to illustrate the in system reliability in their article, “On
hypothesis-testing procedures. the Reliability of the Self-Dual k-out-of-n
Yili Hong, William Q. Meeker, and Systems.” They consider a class of k-out-of-
Luis A. Escobar examine the use of normal n systems and study the variation of their
approximation confidence intervals (NACIs) reliability under different assumptions. It
in “Avoiding Problems with Normal is proved that a way to achieve highly reli-
Approximation Confidence Intervals for able self-dual k-out-of-n systems consists in
Probabilities.” They note that such intervals maintaining the reliability of the compo-
are widely used, especially with censored nents at a level of performance above 1/2.
data. The performance of normal approxi- They derive asymptotic results and extend
mation intervals can be poor when the sam- them to systems with multiple self-dual
ple size is not large or when there is heavy k-out-of-n subsystems. ■
12 AMSTAT NEWS FEBRUARY 2008
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