Harmonic New Better Than Used In Expectation Research Articles

A new non-parametric hypothesis testing with reliability analysis applications to model some real data

Due to the essential necessities of current technology and contemporary sciences, a methodical approach to investigating every occurrence or action occurring in the world was required. As a result, it makes sense that such an approach would be required in the investigation of the reliability of technological systems and products. In actuality, there are instances when system parts gradually deteriorate over time t, at which point the system needs to be updated by replacing the spare parts. Although renewal tries to improve system usability, it is unable to restore the system to its prior state. We discovered that an entirely novel domain of reliability known as classes of life distributions has emerged as a result of the growth of criteria for determining the success or failure of dependability over the previous few decades. Moreover, the other nonparametric tests of life distribution lack test efficiency and have low test power. On the other hand, based on that approach, the non-parametric statistical test utilized in this study evaluates different types of treatment by examining failure behavior in the survival data that was collected. As a result, in this ivestigation, we have developed a novel test of life distribution in this study that considers the power and effectiveness of the test. In order to compare data belonging to the harmonic new better than used in expectation (HNBUE) class with data that follow an exponential distribution, this article suggests nonparametric tests for both censored and uncensored data. The higher percentile points of the test statistics are collected, and the suggested test's normality is verified. The Pitman asymptotic relative efficiency (PARE) and powers of proposed test are calculated a few well-known alternative asymmetric models. Three asymmetric real-world data sets are examined to demonstrate the paper's findings.

A new test for exponentiality against HNBUE alternatives

Two families of test statistics for testing the null hypothesis of exponentiality against Harmonic New Better than Used in Expectation (HNBUE) alternatives are proposed. Asymptotic distributions of the test statistics are derived under the null and alternative hypotheses and the consistency of the tests established. Comparison with competing tests are made in terms of Pitman Asymptotic Relative Efficiency (PARE). Simulation studies have been carried out to assess the performance of the tests. Finally, the test has been applied to three real life data sets described in Proschan, Susarla and Van Ryzin and Engelhardt, Bain and Wright.

A Test for Convex Dominance With Respect to the Exponential Class Based on an <formula formulatype="inline"><tex Notation="TeX">$L^1$</tex></formula> Distance

We consider the problem of testing if a non-negative random variable is dominated, in the convex order, by the exponential class. Under the null hypothesis, the variable is harmonic new better than used in expectation (HNBUE), a well-known class of ageing distributions in reliability theory. As a test statistic, we propose the L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> norm of a suitable distance between the empirical and the exponential distributions, and we completely determine its asymptotic properties. The practical performance of our proposal is illustrated with simulation studies, which show that the asymptotic test has a good behavior and power, even for small sample sizes. Finally, three real data sets are analyzed.

Modeling of Discrete HNBUE Class with Hypothesis Testing Application

A new model is proposed to test whether an equipment whose life is measured in discrete time units fails due to uncontrollable chance factors alone against the alternatives that it degrades over time in the sense of discrete HNBUE (harmonic new better than used in expectation) aging. The test is shown to be consistent and the asymptotic distribution of the test statistic has been obtained. The critical points and the power of the test against various alternatives from discrete HNBUE class have been studied by means of simulation. Finally, the test is applied to a discrete reliability data to examine the performance.

Characterizations of exponentiality within the HNBUE class and related tests

A harmonic new better than used in expectation (HNBUE) variable is a random variable which is dominated by an exponential distribution in the convex stochastic order. We use a recently obtained condition on stochastic equality under convex domination to derive characterizations of the exponential distribution and bounds for HNBUE variables based on the mean values of the order statistics of the variable. We apply the results to generate discrepancy measures to test if a random variable is exponential against the alternative that is HNBUE, but not exponential.

Reliability bounds on HNBUE life distributions with known first two moments

In this paper, we study upper and lower bounds for the reliability function in harmonic new better than used in expectation (HNBUE) life distribution class with known first two moments. Here we say a life distribution has HNBUE property if the integral harmonic mean value of the residual life in any interval [0, t] is no more than its mean. By a constructive proof, we determine the lower and upper reliability bounds analytically and show that these bounds are all sharp.

Testing for Harmonic New Better than used in Expectation

A statistic for testing the null hypothesis that F is the exponential distribution against the alternative of harmonic new better than used in expectation (HNBUE) is proposed. The asymptotic distribution theory for this statistic is derived under the null hypothesis and asymptotic relative efficiency (ARE) with respect to other competing tests for HNBUE is evaluated. This test is applied to the leukemia data described in Bryson and Siddiqui (1969).

An exact test for hnbue class of life distributions

This paper introduces an exact test for testing H 0: F is exponential against H 1: F is harmonic new better than used in expectation (HNBUE) or F is harmonic worse than used in expectation (HNWUE) on complete data. Selected critical values are tabulated for sample size n = 5(1)60. An asymptotic test is made available for large values of n. The standard normal table can be used for the above testing problem. Power of the test is estimeted by simulation. The problem when right censored data is available is also handled. An example of 81 patients of melanoma disease demonstrates the practical application of the proposed test in medical Sciences.

Structures of systems with exponential life and HNBUE components

Structures of systems with exponential life and HNBUE (harmonic new better than used in expectation) components are studied. If a monotonic system with independent HNBUE components has exponential life then it is essentially a series system with exponential components. Thus the surmise of Ahmed and Alzaid (1988) is correct. Further, without the assumption of independence, the author derives a similar result for parallel systems with HNBUE components.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Multivariate harmonic new better than used in expectation distributions

The definition of multivariate harmonic new better than used in expectation (MHNBUE) life distributions is introduced. The definition is multivariate generalization of the useful univariate aging property, harmonic new better than used in expectation (HNBUE), which is weaker than new better than used in expectation (NBUE). Various closure properties are proved. Unlike the previous extensions we also prove the very important result of closure under convolution. Some examples are given to illustrate the relationship between the proposed MHNBUE and MHNBUE properties presented by Basu, Ebrahimi, and Klefsjö (1983). We also study the dual property of multivariate harmonic new worse than used in expectation (MHNWUE).

Testing whether survival function is harmonic new better than used in expectation

Statistical procedures to test that a life distribution is exponential against the alternative that it is harmonic new better than used in expectation (HNBUE) are considered.

Inequalities involving the lifetime of series and parallel systems

AbstractLet {Xi} be independent HNBUE (Harmonic New Better Than Used in Expectation) random variables and let {Yi} be independent exponential random variables such that E{Xi}=E{Yi} It is shown that \documentclass{article}\pagestyle{empty}\begin{document}$ E\left[{u\left({\mathop {\min \,X_i}\limits_{l \le i \le n}} \right)} \right] \ge E\left[{u\left({\mathop {\min \,Y_i}\limits_{l \le i \le n}} \right)} \right] $\end{document} for all increasing and concave u. This generalizes a result of Kubat. When comparing two series systems with components of equal cost, one with lifetimes {Xi} and the other with lifetimes {Yi}, it is shown that a risk‐averse decision‐maker will prefer the HNBUE system. Similar results are obtained for parallel systems.