Increasing Failure Rate Average Research Articles

A New Generalization of the Class of NBU Distributions

It is well-known that the increasing-failure-rate-average (IFRA), new-better-than-used (NBU), and new-better-than-used-in-expectation (NBUE) classes of life distributions are successive generalizations of the class of increasing-failure-rate (IFR) distributions. Further, each of the IFR, IFRA and NBU classes can be characterized in terms of the exponential distribution and certain partial orderings on the family of all distributions. It is not known, however, if the same holds for the NBUE class. This paper introduces a new generalization (called NBAFR) of the NBU class that allows a characterization in terms of the exponential distribution and a (new) partial ordering. The NBAFR class neither contains nor is contained in the NBUE class. Like the latter, the NBAFR class is preserved under the reliability operation of addition of life lengths. But unlike the NBUE, the NBAFR class is also preserved under formation of s-coherent systems - just like the IFR, IFRA, and NBU classes.

The class of mifra lifetimes and its relation to other classes

AbstractIn a previous paper, the authors have introduced a class of multivariate lifetimes (MIFRA) which generalize the univariate lifetimes with increasing failure rate average (IFRA). They have also shown that this class satisfies many fundamental properties. In this paper it is shown that other concepts of multivariate IFRA do not satisfy all of these properties. Relationships between MIFRA and these other concepts are given. Finally, positive dependence implications with respect to these classes are also discussed.

Multivariate Classes of Life Distributions in Reliability Theory

Four classes of life distributions which have been useful in describing aging where systems are assumed to have independent univariate component lifetimes are: (1) the increasing failure rate (IFR) class; (2) the increasing failure rate average (IFRA) class; (3) the new better than used (NBU) class; and (4) the new better than used in expectation (NBUE) class. These classes together with multivariate analogs of the IFR and IFRA classes are reviewed and discussed. An up to date survey of the literature on this topic is included. New multivariate definitions of the NBU and NBUE classes are introduced.

Multivariate Increasing Failure Rate Average Distributions

A class of multivariate distributions which is an extension of the class of univariate distributions with increasing failure rate averages is introduced. Properties of this class are studied and examples of distributions which are members of this class are given.

A catalog of failure distributions

In the following, several useful distributions are classified as members of Increasing failure rate (IFR) , Decreasing failure rate (DFR), Increasing failure rate average (IFRA), Decreasing failure rate average (DFRA) or Exponential (EXP) family.

Minimax Results for IFRA Scale Alternatives

Recent results by Birnbaum, Esary and Marshall (1966), Barlow and Proschan (1967) and others suggest that the exponential models used be Epstein and Sobel (1953) and others for life testing problems should be extended to models in which the lifetimes have increasing failure rate average (IFRA) distributions. In this paper, IFRA scale models are dealt with. Consider two independent random samples $X_1, \cdots, X_m$ and $Y_1, \cdots, Y_n$ from populations with distributions $F(x)$ and $G(y) = F(\Delta y)$ respectively, where $F$ is a continuous, unknown, IFRA distribution. The null hypothesis $H_0:\Delta \leqq 1$ is to be tested against the alternative $H_1:\Delta > 1$. Since $F$ is unknown, it is not possible to maximize the power $E\lbrack \phi\mid F(\cdot), F(\Delta\cdot) \rbrack, \Delta > 1$. However, it is shown (Theorems 2.1, 2.2, 3.1, 3.2) that the tests that maximize the power for the exponential alternative $F(\Delta y) = 1 - \exp (- \Delta y)$ actually maximize the minimum power $\inf_{F(\cdot)} E\lbrack \phi\mid F(\cdot), F(\Delta\cdot) \rbrack$. Thus these tests are minimax. They have been computed by Lehmann (1953), Savage (1956), and Rao, Savage and Sobel (1960). The results indicate that in the case of uncensored samples, one should use one of the statistics \begin{align*}L = {\prod}^n_{i=1} (N + i - s_{n+1-i}), \text{or} \\ S = \sum^m_{i=1} J_0(r_i), \text{with} J_0(k) = \sum^N_{j=N+1-k} 1/j,\end{align*} where $N = m + n$ and $r_1, \cdots, r_m (s_1, \cdots, s_n)$ are the ordered ranks of the $X$'s ($Y$'s) in the combined sample of $X$'s and $Y$'s. $L$. is minimax for $\Delta$ in an interval about two, and $S$ is minimax for $\Delta$ in an interval $(1, \delta)$ to the right of one. The minimax statistics in the case of censored samples are more complicated (see (3.1) and (3.2)) and one might use one of the approximations suggested by Gastwirth (1965) or Basu (1967) (see (3.3)). Only finite sample size properties are dealt with. Asymptotic results are given in [6].

A Note on Tests for Monotone Failure Rate Based on Incomplete Data

Abstract : Tests for exponential versus IFRA (increasing failure rate average) distributions based on incomplete data are defined and shown to be unbiased. The tests are motivated by a class of tests considered in detail by Bickel and Doksum. Tests for exponential versus IFR distributions based on the ranks of total time on test statistics are also considered.