Failure rate and mean residual life with trend changes

Asia - Pacific Journal of Operational Research, Nov 1998 by Guess, Frank, Nam, Kyung H, Park, Dong Ho

In this paper we consider certain parametric models that show the trend change in either its failure rate or its mean residual life. The conditions that characterize the relations between failure rate and mean residual life are derived in general and we apply these results to the Hjorth model with three parameters to explore such relations in details. We also present some graphical comparisons for the change points of both failure rate and mean residual life for selected combinations of three parameter values for the Hjorth model.

Keywords. Mean residual life, failure rate, Hjorth model, trend change.

1. Introduction

Let X be a nonnegative random variable with the cumulative distribution function F and the survival function F = 1- F. The mean residual life (mrl) function is defined as

if F(t) > 0 and the probability density function, f (t), exists. The interpretation of r(t)dt is that it is the conditional probability of failure within the interval (t, t dt), given that there has been no failures up to age t.

The classes of life distributions which exhibit the trend change in their aging properties, such as failure rate, mrl, etc., have attracted a great deal of interests among the reliability analysts and engineers. Such classes of life distributions include IDFR, DIFR, DIMRL and IDMRL, among others.

Many authors have studied the trend changes either in failure rate or in mrl. Glaser (1980) discusses the relation between the density function and trend change in its failure rate. Aarset (1987), Park (1988) and Xie (1989) propose new procedures for testing exponentiality against the trend change in failure rate under somewhat different situations. Regarding the trend change in mrl, Guess, Hollander and Proschan (1986) define the DIMRL (IDMRL) class and propose testing procedures for a constant mrl against a trend change in mrl. Ebrahimi (1991) estimates the change point of mrl function and Lim and Park (1995) propose a new testing procedure for a trend change in mrl.

Our main objective of this paper is to study the relations between failure rate and mrl, especially in terms of locations of their change points. Recently, Mi (1995) shows that the change point of failure rate for DIFR life distribution is greater than or equal to the point at which IDMRL distribution changes its mrl. Gupta and Akman (1995) show similar results under some regularity conditions and apply them to the length biased inverse Gaussian distribution. Chang and Tang (1993) consider the Birnbaum-Saunders distribution to derive the point and interval estimates for the change point of failure rate (which is referred to as `critical time' in their paper) when the distribution shows IDFR.

In this paper, we study the conditions on the failure rate function under which DIFR (IDFR) implies IDMRL (DIMRL) each having a nonnegative change point. We are also interested in deriving the conditions under which DIFR (IDFR) implies DMRL (IMRL). The results of Mi (1995) and Gupta and Akman (1995) do not provide such conditions. Although Mi (1995) proves that the trend change of mrl function occurs before the failure rate changes its trend in general, the results do not discuss any specific conditions under which DIFR (IDFR) implies DMRL (IMRL). This paper also generalizes Mi's results by consideration of IDFR and DIMRL as well.

3. Comparison of Failure Rate and MRL for Hjorth Model

To compute the failure rate and mrl for the Hjorth model with 0 1/theta and the Hjorth model shows DIFR and IDMRL. We also note that the trend change of failure rate occurs later than the trend change of mrl for each case.

Table 1 shows that for fixed delta and beta, as the value of theta increases the percentiles of change points for both r(t) and m(t) increase. On the other hand, for fixed delta and beta, as delta increases the trend changes for both r(t) and m(t) occur earlier.

Acknowledgement

Dong Ho Park's work was supported in part by the Basic Science Research Institute Programs, Ministry of Education, 1998, Project No. BSRI98-1403, KOREA. We are grateful to the anonymous referee and the managing editor for helpful comments.

References

Aarset, M. V. (1987), How to identify a bathtub hazard rate, IEEE Transactions on Reliability 36, 106-108.

Chang, D. S. and L. C. Tang (1993), Reliability bounds and critical time for Birnbaum-Saunders distribution, IEEE Transactions on Reliability 42, 464-469.

Ebrahimi, N. (1991), On estimating change point in a mean residual life function, Sankhya: The Indian Journal of Statistics 53, Series A, 206219.

Glaser, R. E. (1980), Bathtub and related failure rate characterizations, Journal of American Statistical Association 75, 667-672.

Guess, F., M. Hollander and F. Proschan (1986), Testing exponentiality versus a trend change in mean residual life, The Annals of Statistics 75, 1388-1398.

Gupta, R. C. and H. O. Akman (1995), On the reliability studies of a weighted inverse Gaussian model, Journal of Statistical Planning and Inferences 48, 69-83.

Hjorth, U. (1980), A reliability distribution with increasing, decreasing, constant and bathtub shaped failure rates, Technometrics 22 (1), 99-104. Lim, J. H. and D. H. Park (1995), Trend change in mean residual life, IEEE Transactions on Reliability 44, 291-296.