scholarly journals Residual Probability Function, Associated Orderings, and Related Aging Classes

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
M. Kayid ◽  
S. Izadkhah ◽  
S. Alshami
Keyword(s):  
Order Statistics ◽  
Probability Function ◽  
Equilibrium Distribution ◽  
Real Data ◽  
Life Testing ◽  
Monotone Transformation ◽  
Weighted Distributions ◽  
Original Distribution ◽  
Life Distributions

The concept of residual probability plays an important role in reliability and life testing. In this investigation, we study further the residual probability order and its related aging classes. Several characterizations and preservation properties of this order under some statistical and reliability operations of monotone transformation, mixture, weighted distributions, and order statistics are discussed. In addition, by comparing the original distribution with its associated equilibrium distribution with respect to the residual probability order, new aging classes of life distributions are proposed and studied. Finally, a test of exponentiality against such classes is derived and sets of real data are used as examples to elucidate the use of the proposed test for practical problems.

scholarly journals Form-Invariance of the Non-Regular Exponential Family of Distributions

2018 ◽  
Vol 41 (2) ◽  
pp. 157-172
Author(s):  
Samereh Ghorbanpour ◽  
Rahim Chinipardaz ◽  
Seyed Mohammad Reza Alavi
Keyword(s):  
Exponential Family ◽  
Information Matrix ◽  
Likelihood Estimation ◽  
Weight Functions ◽  
Weighted Distribution ◽  
Weighted Distributions ◽  
Original Distribution ◽  
Special Cases ◽  
Exponential Family Of Distributions ◽  
Regular Family

The weighted distributions are used when the sampling mechanism records observations according to a nonnegative weight function. Sometimes the form of the weighted distribution is the same as the original distribution except possibly for a change in the parameters that is called the form-invariant weighted distribution. In this paper, by identifying a general class of weight functions, we introduce an extended class of form-invariant weighted distributions belonging to the non-regular exponential family which included two common families of distribution: exponential family and non-regular family as special cases. Some properties of this class of distributions such as the sufficient and minimal sufficient statistics, maximum likelihood estimation and the Fisher information matrix are studied.

scholarly journals On Bayesian estimation of step stress accelerated life testing for exponentiated Lomax distribution based on censored samples

2020 ◽  
pp. 239-248
Author(s):  
Refah Mohamed Alotaibi ◽  
Hoda Ragab Rezk
Keyword(s):  
Numerical Study ◽  
Real Data ◽  
Operating Conditions ◽  
Stress Factors ◽  
Cumulative Exposure ◽  
Exposure Model ◽  
Life Testing ◽  
Unknown Parameters ◽  
Accelerated Life ◽  
Credible Intervals

In reliability analysis and life-testing experiments, the researcher is often interested in the effects of changing stress factors such as “temperature”, “voltage” and “load” on the lifetimes of the units. Step-stress (SS) test, which is a special class from the well-known accelerated life-tests, allows the experimenter to increase the stress levels at some constant times to obtain information on the unknown parameters of the life models more speedily than under usual operating conditions. In this paper, a simple SS model from the exponentiated Lomax (ExpLx) distribution when there is time limitation on the duration of the experiment is considered. Bayesian estimates of the parameters assuming a cumulative exposure model with lifetimes being ExpLx distribution are resultant using Markov chain Monte Carlo (M.C.M.C) procedures. Also, the credible intervals and predicted values of the scale parameter, reliability and hazard are derived. Finally, the numerical study and real data are presented to illustrate the proposed study

scholarly journals A Bayes Inference for Step-Stress Accelerated Life Testing

2017 ◽  
Vol 6 (6) ◽  
pp. 1
Author(s):  
Naijun Sha ◽  
Hao Yang Teng
Keyword(s):  
Real Data ◽  
Accelerated Life Testing ◽  
Cumulative Exposure ◽  
Data Sets ◽  
Life Testing ◽  
Bayes Inference ◽  
Monte Carlo Algorithms ◽  
Mathematical Properties ◽  
Posterior Inference ◽  
Accelerated Life

In this article, we present a Bayesian analysis with convex tent priors for step-stress accelerated life testing (SSALT) using a proportional hazard (PH) model. As flexible as the cumulative exposure (CE) model in fitting step-stress data and its attractive mathematical properties, the PH model makes Bayesian inference much more accessible than the CE model. Two sampling methods through Markov chain Monte Carlo algorithms are employed for posterior inference of parameters. The performance of the methodology is investigated using both simulated and real data sets.

Reliability Bounds for Exponential Mixtures

1984 ◽  
Vol 33 (1-2) ◽  
pp. 3-16 ◽  
Author(s):  
Manish C. Bhattacharjee
Keyword(s):  
Upper Bound ◽  
Renewal Process ◽  
Survival Probability ◽  
Equilibrium Distribution ◽  
Sufficient Conditions ◽  
Main Thrust ◽  
Reliability Bounds ◽  
Exponential Mixture ◽  
Life Distributions

We obtain some new reliability bounds for the class of life distributions which are exponential mixtures. It is shown that our bounds often improve on the usual DFR upper bound as well as those contained in the recent results of Shaked, Heyde and Leslie, Hall , and Brown. Simple sufficient conditions, under which such is the case, are developed. The main thrust of our findings is that our bounds are tighter for moderate to heavy departures from exponentiality. Among other results, it is shown that every DFR survival probability with a finite mean is stochastically strictly dominated by an exponential mixture. An application of our methods yields a new bound on the tail of the equilibrium distribution of a DFR renewal process, which can be tighter than the corresponding results of Brown.

New exact Bayesian prediction of the range for the exponential lifetime based on fixed and random sample sizes

2017 ◽  
Vol 6 (1-2) ◽  
pp. 169
Author(s):  
A. H. Abd Ellah
Keyword(s):  
Sample Size ◽  
Random Sample ◽  
Real Data ◽  
Predictive Distribution ◽  
Data Sets ◽  
Life Testing ◽  
Random Sample Size ◽  
Fixed Sample Size ◽  
Fixed Sample ◽  
Special Case

We consider the problem of predictive interval for the range of the future observations from an exponential distribution. Two cases are considered, (1) Fixed sample size (FSS). (2) Random sample size (RSS). Further, I derive the predictive function for both FSS and RSS in closely forms. Random sample size is appeared in many application of life testing. Fixed sample size is a special case from the case of random sample size. Illustrative examples are given. Factors of the predictive distribution are given. A comparison in savings is made with the above method. To show the applications of our results, we present some simulation experiments. Finally, we apply our results to some real data sets in life testing.

scholarly journals Characterizations of the Beta Kumaraswamy Exponential Distribution

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 23
Author(s):  
Zakeia A. Al-saiary ◽  
Rana A. Bakoban ◽  
Areej A. Al-zahrani
Keyword(s):  
Order Statistics ◽  
Exponential Distribution ◽  
Hazard Function ◽  
Real Data ◽  
Quantile Function ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Data Set ◽  
Skewness And Kurtosis

In this article, the five-parameter beta Kumaraswamy exponential distribution (BKw-E) is introduced, and some characterizations of this distribution are obtained. The shape of the hazard function and some other important properties—such as median, mode, quantile function, and mean—are studied. In addition, the moments, skewness, and kurtosis are found. Furthermore, important measures such as Rényi entropy and order statistics are obtained; these have applications in many fields. An example of a real data set is discussed.

Some Characterizations of the Exponential and Related Life Distributions

1986 ◽  
Vol 35 (3-4) ◽  
pp. 189-198 ◽  
Author(s):  
S. P. Mukherjee ◽  
Dilip Roy
Keyword(s):  
Order Statistics ◽  
Failure Rate ◽  
Residual Life ◽  
Mean Residual Life ◽  
Finite Range ◽  
Conditional Expectations ◽  
Coefficients Of Variation ◽  
Constant Coefficients ◽  
Life Distributions ◽  
Pearsonian Type

Characterizations of the exponential, the Pearsonian Type XI and a finite range distributions have been obtained in terms of different constant coefficients of variation of residual life. These results have also been stated in terms of conditional expectations of suitably chosen functions of order statistics. A few earlier results have been generalised. Some related results in IMRL and DMRL. classes of distributions have been presented . Also characterizations involving the product of failure rate and mean residual life are aiven.

scholarly journals Bayesian Prediction for Exponentiated Generalized Xgamma Distribution Based on Dual Generalized Order Statistics with Application to Poverty and COVID-19 Mortality Rates

2021 ◽  
pp. 30-53
Author(s):  
R. E. Abd EL-Kader ◽  
A. M. Abd AL-Fattah ◽  
G. R. AL-Dayian ◽  
A. A. EL-Helbawy
Keyword(s):  
Order Statistics ◽  
Real Data ◽  
Generalized Order Statistics ◽  
Statistical Prediction ◽  
Data Sets ◽  
Interval Prediction ◽  
Potential Applications ◽  
Dual Generalized Order Statistics ◽  
Lower Records ◽  
Generalized Order

Statistical prediction is one of the most important problems in life testing; it has been applied in medicine, engineering, business and other areas as well. In this paper, the exponentiated generalized xgamma distribution is introduced as an application on the exponentiated generalized general class of distributions. Bayesian point and interval prediction of exponentiated generalized xgamma distribution based on dual generalized order statistics are considered. All results are specialized to lower records. The results are verified using simulation study as well as applications to real data sets to demonstrate the flexibility and potential applications of the distribution.

scholarly journals Probability Functions of Order Statistics from Discrete Uniform Distribution

2019 ◽  
Vol 7 (1) ◽  
Author(s):  
Ayse Metin KarakaÅŸ ◽  
S. Çalik
Keyword(s):  
Order Statistics ◽  
Uniform Distribution ◽  
Probability Function ◽  
Probability Functions ◽  
Discrete Uniform Distribution ◽  
Discrete Uniform

In this paper, we firstly give basic definitions and theorems for order statistics. Later, we show that r. probability function of order statistics from discrete uniform distribution can be obtained in another form.

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