# Inconsistency of the Maximum Likelihood Estimator of a Distribution Having Increasing Failure Rate Average

@article{Boyles1985InconsistencyOT, title={Inconsistency of the Maximum Likelihood Estimator of a Distribution Having Increasing Failure Rate Average}, author={Russell A. Boyles and A. W. Marshall and Frank Proschan}, journal={Annals of Statistics}, year={1985}, volume={13}, pages={413-417} }

Abstract : Marshall and Proschan showed that the MLE for a life distribution with increasing failure rate is strongly consistent (Ann. Math. Statist., 1965, 36, 69-77). In this note the author shows that the MLE for a life distribution with increasing failure rate average is not consistent; in fact, maximum likelihood estimation in the IFRA case yields estimators of the average failure rate and of the distribution function which, in general, coverage a.s. to values other than the true values…

## 23 Citations

### Estimation of a distribution function with increasing failure rate average

- MathematicsJournal of Statistical Planning and Inference
- 2021

### On Estimating a Survival Curve Subject to a Uniform Stochastic Ordering Constraint

- Mathematics
- 1993

Abstract If F and G are cumulative distribution functions on [0, ∞) governing the lifetimes of specific systems under study, and if F and G are their corresponding survival functions, then F is said…

### Estimating a distribution function subject to a stochastic order restriction: a comparative study

- Mathematics, Economics
- 2012

In this article, we compare four nonparametric estimators of a distribution function (DF), estimated under a stochastic order restriction. The estimators are compared by simulation using four…

### On nonparametric maximum likelihood estimation of a distribution uniformly stochastically smaller than a standard

- Mathematics
- 1991

### Inconsistency of the MLE for the Joint Distribution of Interval‐Censored Survival Times and Continuous Marks

- MathematicsScandinavian journal of statistics, theory and applications
- 2008

This paper provides a new explicit formula and uses this formula and the mark-specific cumulative hazard function of Huang & Louis (1998) to obtain the almost sure limit of the MLE, which implies that the Mle is inconsistent in general.

### Further moments inequalities of life distributions with hypothesis testing applications: the IFRA, NBUC and DMRL classes

- Mathematics
- 2004

### Nonparametric quantile estimation under order constraints

- Mathematics
- 1995

The problmeof the estimation of a quantile function G is considered when G is moredispersed than a distribution function F.In the one-sample problem {F is know}the esitimators presented here render…

### Uniform Strong Consistent Estimation of an Ifra Distribution Function

- Mathematics
- 1994

Let F?n be an estimator of an IFRA survival function F? and let A be such that 0 < F?(A) < 1. The main result constructs an IFRA estimator by splicing the smallest increasing failure rate on the…

### Consistent likelihood-based estimation of a star-shaped distribution

- Mathematics
- 2009

We consider the classical problem of nonparametrically estimating a star-shaped distribution, i.e., a distribution function F on [0,∞) with the property that F(u)/u is nondecreasing on the set {u :…

## References

SHOWING 1-9 OF 9 REFERENCES

### Maximum Likelihood Estimation for Distributions with Monotone Failure Rate

- Mathematics
- 1965

Abstract : Using the idea of maximum likelihood, we derive an estimator for a distribution function possessing an increasing (decreasing) failure rate and also obtain corresponding estimators for the…

### CONSISTENCY OF THE MAXIMUM LIKELIHOOD ESTIMATOR IN THE PRESENCE OF INFINITELY MANY INCIDENTAL PARAMETERS

- Mathematics
- 1956

0 and ai. The parameter 0, upon which all the distributions depend, is called "structural"; the parameters {aiI} are called "incidental". Throughout this paper we shall assume that the Xi, are…

### Families of components, and systems, exposed to a compound poisson damage process

- Mathematics
- 1973

Abstract : A fairly common failure model in a wide variety of contexts is a cumulative damage process, in which shocks occur randomly in time and associated with each shock there is a random amount…

### Shock models with underlying birth process

- MathematicsJournal of Applied Probability
- 1975

This paper extends results of Esary, Marshall and Proschan (1973) and A-Hameed and Proschan (1973). We consider the life distribution of a device subject to a sequence of shocks occurring randomly in…

### A Stochastic Characterization of Wear-out for Components and Systems

- Mathematics
- 1966

nential life distributions are accepted as characterizing the phenomenon of no wear. The problem of finding a class of life distributions which would similarly reflect the phenomenon of wear-out has…