A test for the increasing log-odds rate family
Introduction
The hazard rate (HR), which is also referred to as the failure rate, is probably the most widely-used tool for the characterisation of lifetime distributions under nonparametric assumptions. The mathematical properties of the HR give rise to several classes of distribution, among which the increasing hazard rate (IHR) family is probably the most popular thanks to their desirable features and wide range of applications (Barlow et al., 1963, Marshall and Olkin, 2007, Shaked and Shanthikumar, 2007). Various methods have been proposed to test the ageing properties of distributions, with particular reference to the IHR condition (Barlow and Proschan, 1969, Tenga and Santner, 1984, Bickel, 1969, Bickel and Doksum, 1969, Proschan and Pyke, 1967, Sahoo and Sengupta, 2017).
This paper deals with an alternative and closely related function, namely the log-odds rate (LOR), and with the families of distributions that are characterised by its behaviour. The LOR can be expressed in terms of time or log-time and it represents an alternative model for the hazard of ageing, which is comparable to the model based on the HR. In particular, distributions with monotone LOR are relevant in reliability and survival analysis (Zimmer et al., 1998, Wang et al., 2003, Wang et al., 2005, Banerjee et al., 2007, Sunoj et al., 2007, Navarro et al., 2008, Khorashadizadeh et al., 2013, Lando et al., 2020).
The objective of the present study is to test the goodness-of-fit to the increasing log-odds rate (ILOR) family w.r.t. time or log-time. This paper draws inspiration from the work of Tenga and Santner (1984) (see also Sengupta and Paul (2005)) and proposes a Kolmogorov–Smirnov type test that is based on the distance between a special ageing function, which is defined in Section 3, and its greatest convex minorant (GCM) (Barlow et al., 1971). The distribution of the test statistic can be computed through simulation: approximate critical values and -values are given in Section 4. The performance of the test is validated through a simulation study. Some possible applications are discussed in Section 5.
Section snippets
Ageing functions and related families
Throughout this paper, “increasing” signifies “non-decreasing” and “decreasing” signifies “non-increasing”.
Let be an absolutely continuous random variable (RV) with support , cumulative distribution (CDF) and probability density function (PDF) .
Barlow et al. (1971) define the generalised hazard rate of as where is an absolutely continuous CDF. Correspondingly, the generalised hazard function is defined as where is a
Goodness-of-fit test for ILOR properties
Consider the following two null hypotheses The test proposed in this paper is inspired from the approach of Tenga and Santner (1984). Taking into account that is IHR iff the corresponding hazard function, , is convex, Tenga and Santner (1984) developed a method to test the convexity of , based on the distance between an estimator of and its GCM. Their work provides a useful framework to check the convexity of a general composition , where is a
Simulations
The performance of the test is analysed through a simulation study.3 Three choices of weights were considered in (16): (1) , (2) , (3) . Choices (1) and (2) have two opposite defects, namely, (1) places more weights on larger values of , whereas (2) places more weights on the smaller ones. Overall, choice (3) is well balanced between (1) and (2), and it provides the best performance in
Applications
The ILOR property has many applications in reliability (Zimmer et al., 1998, Wang et al., 2003, Wang et al., 2005, Banerjee et al., 2007, Sunoj et al., 2007, Navarro et al., 2008, Khorashadizadeh et al., 2013, Lando et al., 2020). For instance, Zimmer et al. (1998) determine bounds for the reliability function, , under the assumption that is log-ILOR. Lando et al. (2020) establish second-order stochastic comparisons of -out-of- systems under the assumption that the component lifetimes
Funding
This research was supported by the Czech Science Foundation (GACR) under project 20-16764S and moreover by SP2020/11, an SGS research project of VŠB-TU Ostrava
References (20)
- et al.
Testing the hypothesis of increasing hazard ratio in two samples
Comput. Statist. Data Anal.
(2017) - et al.
Bayesian analysis of generalized odds-rate hazards models for survival data
Lifetime Data Anal.
(2007) - et al.
Statistical Inference Under Order Restrictions
(1971) - et al.
Properties of probability distributions with monotone hazard rate
Ann. Math. Stat.
(1963) - et al.
A note on tests for monotone failure rate based on incomplete data
Ann. Math. Stat.
(1969) Tests for monotone failure rate II
Ann. Math. Stat.
(1969)- et al.
Tests for monotone failure rate based on normalized spacings
Ann. Math. Stat.
(1969) - et al.
Characterization of life distributions using log-odds rate in discrete aging
Comm. Statist. Theory Methods
(2013) - et al.
Second-order stochastic comparisons of order statistics
(2020) - et al.
Life Distributions, Vol. 13
(2007)
Cited by (3)
Properties of increasing odds rate distributions with a statistical application
2022, Journal of Statistical Planning and InferenceCitation Excerpt :In the literature, various methods have been proposed to test ageing properties of distributions. For example, Barlow and Proschan (1969), Bickel and Doksum (1969), Bickel (1969) and Proschan and Pyke (1967) test exponentiality against the IHR alternative, Tenga and Santner (1984), Hall and Van Keilegom (2005) and Groeneboom and Jongbloed (2012) test the IHR null hypothesis against non-IHR alternatives, Sengupta and Paul (2005) test log-concavity against non-log-concave alternatives, Lando (2020) tests the null hypothesis of increasingness of the log-odds rate. Another stochastic order, reminded below, which has important applications in reliability, is the hazard rate order.
Testing convexity of the generalised hazard function
2022, Statistical PapersSecond-order stochastic comparisons of order statistics
2021, Statistics