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A new family of tests for DMTTF alternatives under complete and censored samples
Abstract
A family of nonparametric tests is proposed for detecting DMTTF property. The exact distributions of the test statistics are derived under the null hypothesis of exponentiality and critical values are calculated. The proposed test statistics are shown to be asymptotically normal and the consistency of the tests is established. Pitman asymptotic efficacy values are calculated. A large sample test under random censoring is also proposed and the relevant asymptotics are worked out. A Monte Carlo simulation study is carried out in order to assess the merit of the proposed tests. Finally, the test is applied to some real data sets for illustration.
... In recent times, the literature in reliability has focused quite heavily on these functions and the ageing classes derived from them namely, DMTTF (IMTTF), IMIT and DRHR. Various probabilistic aspects have been explored in [32], [42], [22], [33], [63], [60], [43], [62], [8], [72] etc. while inferential issues have been dealt with in [11], [15], [14], [13], [41], [57], [54], [12], [24], [58], [37]. However, the problem of obtaining appropriate conditions in presence of which the first-passage times of a Markov process possess such ageing properties remain unexplored. ...
While modelling deterioration or ageing of devices, first-passage times of Markov processes play a significant role especially when the devices are subject to shocks and wear during their operation. In view of this, obtaining sufficient conditions for first-passage times to belong to specific ageing families constitute an important problem. There exists a rich literature dealing with this class of problems, see, for example [17], [35], [61], [10]. We address the same problem in the context of some new ageing classes such as DMTTF (IMTTF), IMIT and DRHR. We also rectify some erroneous results in Belzunce et al. (Adv. Appl. Prob. 34, 2002) which explore connections between such sufficient conditions pertaining to different reliability classes. We further strengthen a certain result of Belzunce et al. (2002).
In this paper, we propose a new non-parametric test for testing mean time to failure order. The asymptotic properties of the proposed test statistic are studied. We also develop a jackknife empirical likelihood (JEL) ratio test for testing the mean time to failure order. Using the Monte Carlo simulation study, we establish that the JEL ratio test has good power under various alternatives. Finally, we illustrate the proposed test procedure using two real data sets.
The mission duration plays a very vital role in maintaining engineering systems in reality. This paper considers the mission duration into the age replacement policies with multiple missions for multi-state systems (MSSs). We first introduce the MSS whose degradation process follows a non-homogenous continuous time Markov process. Then, the basic replacement policy is given, and the optimum number of missions is derived analytically and numerically. To address the trade-off between the cost and maintainability, we consider the replacement time in maintenance and propose two policies, i.e., replacement first and replacement last. The optimal policies of each replacement are discussed analytically. Numerical examples are proposed to verify the applicability of the methodology derived by this paper. Finally, the suggested replacement policies are illustrated in maintaining the coal transportation system in a power station based on the assumed data.
The performance and effectiveness of an age replacement policy can be assessed by its mean time to failure (MTTF) function. In this paper we propose a class of tests to detect trend change in MTTF function. We develop test statistics utilising a measure of deviation based on a weighted integral approach. We derive the exact and asymptotic distributions of our test statistics exploiting L-statistic theory and also establish the consistency of the test as a consequence of our results. A Monte Carlo study is conducted to evaluate the performance of the proposed test. Finally, we apply our test to some real life data sets for illustrative purposes.
In this paper, a new class of life distribution is proposed, named renewal new better than used in Laplace transform order (RNBUL). The test statistic for testing exponentiality versus (RNBUL) based on U-statistic is proposed. Pitman’s asymptotic efficiencies of the test and Pitman’s asymptotic relative efficiencies (PARE) are calculated. The percentiles of this test statistic are tabulated, and the powers of this test are estimated for some famous alternatives distributions in reliability such as Weibull, linear failure rate (LFR), and Gamma distributions. The problem regarding the right-censored data is also handled. Finally, some applications to elucidate the usefulness of the proposed test in reliability analysis are discussed.
In this paper, we develop a simple non-parametric test for testing exponentiality against decreasing mean time to failure class alternatives. We derive the exact null distribution of the test statistic and find the critical values for different sample sizes. Asymptotic properties of the test statistics are studied. The test is compared with some other test by evaluating Pitman’s asymptotic efficacy. We also discuss how the proposed method takes the censoring information into consideration. Finally, some numerical results are presented and the test procedure is illustrated using a real data.
The purpose of this paper is to enhance the study of the mean time to failure (MTTF) order in an age replacement model, and the decreasing mean time to failure (DMTTF) class of life distributions. We present new characterizations of the MTTF order in terms of the well-known hazard rate and reversed hazard rate orders. Preservation properties of this order under some reliability operations such as monotonic transformations, mixture, and weighted distributions are investigated. In this context, we provide new results related to the DMTTF class. Finally, the problem of testing exponentiality against such class is discussed, and some numerical results are presented.
In a recent paper, Basu and Mitra (20021.
Basu , S. and
Mitra , M. (2002). Testing exponentiality against Laplace order dominance. Statistics, 36: 223–229. View all references) introduced a class of tests for exponentiality against the non-parametric class ℒ of life distributions. The test statistics are fractional sample moments. In a simulation study, the tests show a remarkable behavior: when performed on a nominal level of 5%, each of the tests always accepted the null hypothesis of exponentiality if the underlying data came from an exponential distribution. Furthermore, power of the tests was very low compared to other procedures designed for the same testing situation. It is the aim of this paper to show that the reasons for this behavior are an incorrectly stated asymptotic distribution, a slow convergence of the finite sample distributions of the test statistics to their limit distribution, and, to a minor extent, the dependence of the proposed tests on the choice of parameter values.To this end, we derive the limit distributions of the test statistics in case of a general underlying distribution and the local approximate Bahadur efficiency of the tests against several parametric families of alternatives to exponentiality. Additionally, the finite sample behavior of the tests is examined by means of a simulation study.Further, we enlarge the proposed class of tests by extending some characterization of exponentiality within the ℒ-class.
Let F be a distribution with density f, and let be the failure rate of F. Tests for constant versus monotone increasing failure rate based on the ranks of the normalized spacings between the ordered observations have been considered by Proschan and Pyke (1967). They show that these statistics are asymptotically normally distributed for fixed alternatives F and compute the ratios of the efficacies of one of their rank tests to the best statistics for Weibull and Gamma alternatives. In this paper, it is shown that asymptotic normality holds also for sequences of alternatives that approach the distribution as ; and that the above mentioned ratios of efficacies are in fact Pitman efficiencies. Let be the ranks of the normalized spacings, and . Then is asymptotically equivalent to the Proschan Pyke statistic. It is shown that the Pitman efficiency satisfies \begin{equation*} \tag{1.1} e(T_1, T_2) \equiv \frac{3}{4}\end{equation*} for all sequences of alternatives and thus is asymptotically inadmissible. Statistics that are linear in the normalized spacings and asymptotically most powerful for parametric alternatives if the scale parameter is known, are derived, and it is shown that the rank statistics that are asymptotically most powerful in the class of linear rank tests, are nowhere most powerful in the class of all tests, when is known. If is unknown, studentizing of the linear normalized spacing tests which are asymptotically most powerful for known leads to procedures which have only the same asymptotic power as the most powerful linear rank tests. Unbiasedness is shown for tests that are monotone in the normalized spacings, and Monte Carlo power estimates are used to compare the various statistics with the likelihood ratio tests considered by Barlow (1967).
Mean time to failure (MTTF) function plays an important role in connection with age replacement policy. In this paper, a goodness of fit test is proposed for MTTF function utilizing a weighted integral as a measure of deviation. The test statistic is shown to be asymptotically normal and the consistency of the test is also established. The proposed test is compared with some parametric tests of exponentiality in the sense of Pitman Asymptotic Relative Efficiency (PARE). The performance of the test is assessed by means of a simulation study. Finally, the test is applied to a real-life data set for illustrative purposes and the results are shown to be in accordance with those obtained by applying the one-sample Kolmogorov-Smirnov test.
The performance and effectiveness of an age replacement policy can be assessed by its mean time to failure (MTTF) function. We develop shock model theory in different scenarios for classes of life distributions based on the MTTF function where the probabilities of surviving the first k shocks are assumed to have discrete DMTTF, IMTTF and IDMTTF properties. The cumulative damage model of A-Hameed and Proschan [1] is studied in this context and analogous results are established. Weak convergence and moment convergence issues within the IDMTTF class of life distributions are explored. The preservation of the IDMTTF property under some basic reliability operations is also investigated. Finally we show that the intersection of IDMRL and IDMTTF classes contains the BFR family and establish results outlining the positions of various non-monotonic ageing classes in the hierarchy.
A two-sample nonparametric test based on two independent samples is proposed for comparing mean time to failure (MTTF) functions of two life distributions. The proposed test has the flexibility of handling unequal sample sizes and crossings of mean time to failure functions. The test is shown to be consistent and a bound on asymptotic size has been derived. A computational technique is devised for evaluating the proposed test statistic. A simulation study is carried out for assessment of the merit of the proposed test procedure. Finally, the test is applied to real-life data sets for illustration.
A consistent test for detecting DMTTF property is proposed. The exact null distribution of the test statistic as well as its asymptotic distribution have been derived. Efficacy values are calculated, and a simulation study is carried out to assess the performance of the test. Finally, the test is applied to some real-life data sets.
The mean time to failure (MTTF) function plays a vital role in the theory of age replacement policies. The point at which the MTTF function changes trend has important implications in the context of cost optimization in such policies. We develop a general methodology for change point estimation in this scenario and also establish the strong consistency of the proposed estimator. We also examine the performance of our estimator by applying it to simulated and real life data sets.
Mean time to failure in age replacement evaluates the performance and effectiveness of the age replacement policy. In this paper, we propose a test for exponentiality against a trend change in mean time to failure in age replacement. We derive the asymptotic distribution of the test statistics under the null hypothesis to approximate the critical values. We conduct a simulation study to investigate the performance of the proposed test. To illustrate the application, we apply our test on a real data set.
In this article, we propose a family of non-parametric tests for testing exponentiality against decreasing mean time to failure class alternatives. We obtain the exact null distribution of the test statistic and calculate the critical values for different sample sizes. We also extend our test to incorporate right censored observations. We discuss the efficiency loss due to censoring and calculate it for some choice of censoring distribution. A simulation result is presented to show the performance of the proposed test. Finally, the test procedure is illustrated using a real data.
The mean time to failure (MTTF) function in age replacement is used to evaluate the performance and effectiveness of the age replacement policy. In this paper, based on the MTTF function, we introduce two new nonparametric classes of lifetime distributions with nonmonotonic mean time to failure in age replacement; increasing then decreasing MTTF (IDMTTF) and decreasing then increasing MTTF (DIMTTF). The implications between these classes of distributions and some existing classes of nonmonotonic ageing classes are studied. The characterizations of IDMTTF and DIMTTF in terms of the scaled total time on test transform are also obtained.
We consider the problem of testing exponentiality against IFR alternatives. A measure of deviation from exponentiality is developed and a class of test statistics is constructed on the basis of this measure. It is shown that the test statistic is an L-statistic. The asymptotic as well as the exact distributions of the test statistics are obtained and the test statistics are proved to be consistent. The Pitman efficiency has also been studied.
In this article some properties of the mean time to failure in an age replacement model is presented by examining the relationship it has with hazard (reversed hazard) rate and mean (reversed mean) residual life functions. An ordering based on mean time to failure is used to examine its implications with other stochastic orders.
In this paper, we investigate an ageing class of life distributions. We show that the family of life distributions is closed under formation of parallel and series systems. This class is situated between IFR and NBUE.
Pooled data on the times of successive failures of the air conditioning system of a fleet of jet airplanes seemed to indicate that the life distribution had a decreasing failure rate. More refined analysis showed that the failure distribution for each airplane separately was exponential, but with a different failure rate. Using the theorem that a mixture of distributions each having a non-increasing failure rate (e.g., a mixture of exponential distributions) itself has a non-increasing failure rate, the apparent decreasing failure rate of the pooled air-conditioning life distribution was satisfactorily explained. The present study has implications in other areas where an observed decreasing failure rate may well be the result of mixing exponential distributions having different parameters.
The exact null distribution is obtained for the test statistic V* proposed by Hollander & Proschan (1975) for testing exponentiality versus decreasing mean residual life alternatives. Selected
critical values are tabulated for sample sizes 2 to 60.
Let F be a life distribution with survival function [Fbar] = 1 - F and finite mean The scaled total time on test transform was introduced by Barlow and Campo (1975) as a tool in the statistical analysis of life data. The aging properties IFR, IFRA, NBUE and DMRL can be translated to properties of ϕF(t). Guided by these properties of ϕF(t) we suggest some test statistics for testing exponentiality against IFR, IFRA, NBUE and DMRL, respectively. The IFR and IFRA tests are new; the NBUE and DMRL tests have already been proposed and stu- died by Hollander and Proschan (1975) . The exact and asymptotic distributions of the statistics are derived and the asymptotic efficiencies of the tests are studied. The power of some tests is estimated by simulation for some alternatives when the sample size is n = 10 or n = 20.
We present a statistical procedure to test that a life distribution belongs to the class of exponential distributions against that it belongs to a class of alternatives based on the Laplace transform. The test has been shown to be consistent and the asymptotic distribution of the test statistic has been obtained. The performance of the test against various classes of alternatives has been studied by means of Monte Carlo simulation. An interesting characterization theorem for exponentials, which motivates our test procedure, has been proved.
Total time on test (TTT) plots provide a useful graphical method for tentative identification of failure distribution models. Identification is based on properties of the TTT transform. New properties of the TTT transform distribution are obtained. These results are useful to the user of TTT plots. Although IFR (DFR) distributions are particularly easy to identify from TTT plots, the user must exercise caution relative to identification of IFR A (DFRA) distributions.
In Knopik (2005) the ageing class MTFR (Mean Time to Failure or Repair) of lifetime distribution was introduced. In this paper, we show that the family MTFR is closed under weak convergence of distribution and convolution. We prove that the dual family MTFR D (in a particular case) is closed under mixtures.
This is the first of two papers describing a study of the effect of departures from assumptions, other than normality, on the null-distribution of the F-statistic in the analysis of variance. In this paper, certain theorems required in the study and concerning the distribution of quadratic forms in multi-normally distributed variables are first enunciated and simple approximations tested numerically. The results are then applied to determine the effect of group-to-group inequality of variance in the one-way classification. It appears that if the groups are equal, moderate inequality of variance does not seriously affect the test. However, with unequal groups, much larger discrepancies appear. In a second paper, similar methods are used to determine the effect of inequality of variance and serial correlation between errors in the two-way classification.
This paper provides an empirical Bayes approach to the problem of nonparametric estimation of a distribution (or survival) function when the observations are censored on the right. The results use the notion of a Dirichlet process prior introduced by Ferguson. The paper presents a generalization to the case of right censored observations of the rate result of an empirical Bayes nonparametric estimator of a distribution function of Korwar and Hollander in the uncensored case. The rate of asymptotic convergence of optimality is shown to be the best obtainable for the problem considered.
This paper considers linear functions of order statistics of the form . The main results are that is asymptotically normal if the second moment of the population is finite and J is bounded and continuous a.e. , and that this first result continues to hold even if the unordered observations are not identically distributed. The moment condition can be discarded if J trims the extremes. In addition, asymptotic formulas for the mean and variance of are given for both the identically and non-identically distributed cases. All of the theorems of this paper apply to discrete populations, continuous populations, and grouped data, and the conditions on J are easily checked (and are satisfied by most robust statistics of the form ). Finally, a number of applications are given, including the trimmed mean and Gini's mean difference, and an example is presented which shows that may not be asymptotically normal if J is discontinuous.
This paper deals with preservation of the reversed hazard rate order between equilibrium random variables under formations
of some reliability structures. We further investigate a new aging notion based upon the reversed hazard rate order between
a random life and its equilibrium version. A nonparametric method is developed to test the exponentiality against such a strict
aging property, some numerical results are presented as well.
In this paper we develop a new family of tests for the dilation order based in a characterization of the dilation order. This
family of tests statistics can be used for testing the exponentiality against HNBUE (HNWUE) alternatives. Asymptotic distributional
results are given for both families of tests. For the HNBUE (HNWUE) we also derive the exact distribution under the null hypothesis.
At any age the mean residual life function gives the expected remaining life at that age. Reliabilists and biometricians have found it useful to categorize failure distributions by the monotonicity properties of the mean residual life function. Hollander and Proschan (1975, Biometrika 62, 585-593) have derived tests of the null hypothesis that the underlying failure distribution is exponential, versus the alternative that it has a monotone mean residual life function. These tests are based on a complete sample. Often, however, data are incomplete because of withdrawals from the study and because of survivors at the time the data are analyzed. In this paper we generalize the Hollander-Proschan tests to accommodate randomly censored data. The efficiency loss due to the presence of censoring is also investigated.
Inferences on the parameters of the Birnbaum-Saunders fatigue life distribution based on maximum likelihood estimation
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Asymptotic normality of L-statistics with randomly censored data
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Linear functions of order statistics with smooth weight functions
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