DETERMINATION OF CHANGE POINTS OF
NON MONOTONIC FAILURE RATES
Ramesh C. Gupta
Robin Warren
University of Maine, Department of Mathematits
5752 Neville Hall
Orono, ME 04469-5752 U.S.A.
R,~upta~tnaine.nlaine.edu
and Statistics
Survival and failure time date are frequently modeled by increasing or decreasing
failure rate distributions. This may be inappropriate when the course of the disease is
such that mortalny reaches a peak after some finite period and than declines slowly. For
example, in a study of curability of breast cancer, Langlands et. al. (1979) found that the
peak mortality occurred after about three years. Bennett (1983) analyzed the data from
Veterans Administration lung cancer presented by Prentice (1973) and showed that the
empirital rates for both low PS and high PS groups are non-monotonic.
Therefore, it is
important to analyze such data sets with appropriate models like log normal, inverse
Gaussian log-logistic and Burr type X II, having non-monotonic failure rates.
Sinte most of the failure rates have complex expressions because of the integral
in the denominator, the determination of the monotonicity is not straightforward.
To
alleviate this difficulty Glaser (1980) presented a method to determine the monotonicity
of the failure rate having one turning point. Glaser’s method uses the density function
rather than the failure rate, which, in many cases, is much simpler-. Glaser’s method can
be used in many complicated examples such as quadratic exponential family (Pham-Gia
1994), having generalized Rayleigh, half normal, gamma, Maxwell Boltzman, classical
Rayleigh and Chi-square us special cases. Other examples include inverse Gaussian, log
normal and weighted inverse Gaussian distributions, see Gupta and Akman (1995). In
addition to these, Glaser’s method can be used to determine the monotonicity in certain
mixture models having two components, see Gutland and Sethuraman (1994, 1995) and
Al-Hussaini and Abd - El. Hakim (1989).
This paper has been motivated
by pooling two gamma distributions.
the conjecture presented by Glaser is
Glaser to models having two or more
procedure helps us to identify failure
by an interesting example of a failure rate obtained
For certain values of the parameters it is found that
not correct. This lead us to extend the results of
turning points of the failure rate. The extended
rates of more complex form,
REFERENCES
Al-Hussaini, EK. and Abd-El-Hakim, N.S. (1989). Failure rate of the inverse GaussianWeibull mixture model. Annals of the Institute of Statistical Mathematits 4 l(3 ),
617622.
Bennett, S. (1983). Log-logistic
32(2), 165-171.
regression models for survival data. Applied Statistics
Glaser. R.E. (1980). Bathtub and related failure rate characterizations.
American Statistical Association ‘75, 667-672.
Journal of the
Gupta R.C. and Akman, 0. (1995). On the reliability studies of a weighted inverse
Gaussian model. Journal of Statistical Planning and Inference, 48, 69-83.
Gurland, J. and Sethuraman, J. (1994). Reversal of increasing failure rates when pooling
failure data. Technometrics 36(4), 416-4 18.
Gurland, J. and Sethuraman, J. (1995). How pooling failure data may reverse increasing
failure rate. Journal of the American Statistical Association, 90(432), 14 16 1423.
Langlands, A.O. Pocock, SJ., Kerr, G.R. and Gore, S.M. (1979). Long term survival of
patients with breast cancer: a study of curability of the disease. British Medieal
Journal 2, 1247- 125 1.
Pham-Gia, T. ( 1996). The hazard rate of the power-quadratic exponential
distributions. Statistics and Probability Letters 20, 375-382.
Prentice, R.L. ( 1973). Exponential
Biometrits 60, 279-288.
family of
survivals with censoring and explanatory
variables.
ABSTRACT
This paper was motivated by the problem of the determination of the change
points of the failure rate of a mixture of two gamma distributions. For certain values of
the parameters the existing methods are not applicable sinte, in this case, there are two
turning points of the failure rate. Thus, we extend the results to models having two or
more turning points of the failure fates. The extended procedure helps us to identify
failure rates of more complex forms. Finally, the mixture gamma case is completely
resolved employing theoretical, graphical and numerical techniques wherever necessary.