THE MAXIMUM FAILURE TIME DISTRIBUTION

2009 ◽  
Vol 09 (02) ◽  
pp. 369-381
Author(s):  
SARALEES NADARAJAH ◽  
SAMUEL KOTZ
Keyword(s):  
Failure Time ◽  
Moment Generating Function ◽  
Rate Function ◽  
Cumulative Distribution ◽  
Mean Deviation ◽  
Exact Probability ◽  
Failure Time Distribution ◽  
Failure Times ◽  
Exact Probability Distribution ◽  
Methods Of Moments

For systems with parallel components, the variable of primary importance is the maximum of the failure times of the different components. In this paper, we study the exact probability distribution of the maximum failure time. Explicit expressions are derived for the cumulative distribution function, probability density function, hazard rate function, moment-generating function, nth moment, variance, skewness, kurtosis, mean deviation, Shannon entropy, and the order statistics. Estimation procedures are derived by the methods of moments and maximum likelihood. We expect that these results could be useful for performance assessment of parallel systems.

Optimal Preventive Maintenance Under Decision Dependent Uncertainty

2006 ◽  
Author(s):  
Alexander Galenko ◽  
Elmira Popova ◽  
Ernie Kee ◽  
Rick Grantom
Keyword(s):  
Preventive Maintenance ◽  
Time Distribution ◽  
Failure Time ◽  
South Texas ◽  
Maintenance Cost ◽  
Failure Time Distribution ◽  
Failure Times ◽  
Decision Dependent Uncertainty ◽  
Dependent Failure ◽  
Operating Company

We analyze a system of N components with dependent failure times. The goal is to obtain the optimal block replacement interval (different for each component) over a finite horizon that minimizes the expected total maintenance cost. In addition, we allow each preventive maintenance action to change the future joint failure time distribution. We illustrate our methodology with an example from South Texas Project Nuclear Operating Company.

scholarly journals Study of a unit power-logarithmic distribution

2021 ◽  
Vol 5 (1) ◽  
pp. 218-235
Author(s):  
Christophe Chesneau ◽  
Keyword(s):  
Stochastic Order ◽  
Quantile Function ◽  
Moment Generating Function ◽  
Rate Function ◽  
Cumulative Distribution ◽  
Functional Capabilities ◽  
Logarithmic Distribution ◽  
Logarithmic Functions ◽  
Mean Variance ◽  
Selection Of

This article proposes a new unit distribution based on the power-logarithmic scheme. The corresponding cumulative distribution function is defined by a special ratio of power and logarithmic functions that is dependent on one parameter. We show that this function benefits from great flexibility characterized by a large selection of convex and concave shapes. The other key functions are determined and studied. In particular, we show that the probability density function may take on different decreasing or U shapes, and the hazard rate function has a wide panel of U shapes. These functional capabilities are rare for a one-parameter unit distribution. In addition, we prove certain stochastic order results, provide the expression of the quantile function via the Lambert function, some interesting distributional results, and give simple expressions for the ordinary moments, mean, variance, skewness, kurtosis, moment generating function and incomplete moments. Subsequently, a basic statistical approach is described, to show how the new distribution can be applied in a data analysis scenario. Finally, complementary mathematical findings are presented, yielding new integrals linked to the Euler constant via some well-known moments properties.

Random wear models in reliability theory

1971 ◽  
Vol 3 (02) ◽  
pp. 229-248 ◽  
Author(s):  
David S. Reynolds ◽  
I. Richard Savage
Keyword(s):  
Maximum Likelihood ◽  
Failure Time ◽  
Maximum Likelihood Estimates ◽  
Time To Failure ◽  
Failure Time Distribution ◽  
Wear Process ◽  
Joint Distributions ◽  
Failure Times ◽  
Random Failure ◽  
Interesting Class

Gaver (1963) and Antelman and Savage (1965) have proposed models for the distribution of the time to failure of a simple device exposed to a randomly varying environment. Each model represents cumulative wear as a specified function of a non-negative stochastic process with independent increments, and assumes that the reliability of the device is conditioned upon realizations of this process. From these models are derived the corresponding unconditional joint distributions for the random failure time vector of n independent, identical devices exposed to the same realization of the wear process. It is shown that the identical failure time distribution for one component can arise from each model. In the Gaver model simultaneous failure times occur with positive probability. The probabilities of specific tie configurations are developed. For an interesting class of Gaver models involving a time scale parameter, the maximum likelihood estimates from several devices in one environment are examined. In that case the tie configuration probability does not depend on the parameter. For the corresponding Antelman-Savage models a consistent sequence of estimators is obtained; the maximum likelihood theory did not appear tractable.

scholarly journals The Hamza Distribution with Statistical Properties and Applications

2020 ◽  
pp. 28-42
Author(s):  
Ahmad Aijaz ◽  
Muzamil Jallal ◽  
S. Qurat Ul Ain ◽  
Rajnee Tripathi
Keyword(s):  
Cumulative Distribution Function ◽  
Likelihood Estimation ◽  
Moment Generating Function ◽  
Rate Function ◽  
Cumulative Distribution ◽  
Parameter Distribution ◽  
Residual Function ◽  
Lorenz Curves ◽  
Method Of Maximum Likelihood ◽  
Two Parameter

This paper suggested a new two parameter distribution named as Hamza distribution. A detailed description about the properties of a suggested distribution including moments, moment generating function, deviations about mean and median, stochastic orderings, Bonferroni and Lorenz curves, Renyi entropy, order statistics, hazard rate function and mean residual function has been discussed. The behavior of a probability density function (p.d.f) and cumulative distribution function (c.d.f) have been depicted through graphs. The parameters of the distribution are estimated by the known method of maximum likelihood estimation. The performance of the established distribution have been illustrated through applications, by which we conclude that the established distribution provide better fit.

scholarly journals Poisson Transmuted-G Family of Distributions: Its Properties and Applications

2021 ◽  
pp. 309-332 ◽  
Author(s):  
Laba Handique ◽  
Subrata Chakraborty ◽  
M.S. Eliwa ◽  
Dr. G.G. Hamedani
Keyword(s):  
Maximum Likelihood ◽  
Hazard Function ◽  
Failure Time ◽  
Quantile Function ◽  
Moment Generating Function ◽  
Residual Lifetime ◽  
Mean Deviation ◽  
Time Data ◽  
Family Based ◽  
Family Of Distributions

In this article, an extension of the transmuted-G family is proposed, in the so-called Poison transmuted-G family of distributions. Some of its statistical properties including quantile function, moment generating function, order statistics, probability weighted moment, stress-strength reliability, residual lifetime, reversed residual lifetime, Rényi entropy and mean deviation are derived. A few important special models of the proposed family are listed. Stochastic characterizations of the proposed family based on truncated moments, hazard function and reverse hazard function, are also studied. The family parameters are estimated via the maximum likelihood approach. A simulation study is carried out to examine the bias and mean square error of the maximum likelihood estimators. The advantage of the proposed family in data fitting is illustrated by means of two applications to failure time data sets.

scholarly journals One-sided Variations on Tries: Path Imbalance, Climbing, and Key Sampling

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Costas A. Christophi ◽  
Hosam M. Mahmoud
Keyword(s):  
Probability Distribution ◽  
Generating Function ◽  
Path Length ◽  
Mellin Transform ◽  
Moment Generating Function ◽  
Exact Probability ◽  
Digital Trees ◽  
Exact Probability Distribution ◽  
International Audience ◽  
The Mean

International audience One-sided variations on path length in a trie (a sort of digital trees) are investigated: They include imbalance factors, climbing under different strategies, and key sampling. For the imbalance factor accurate asymptotics for the mean are derived for a randomly chosen key in the trie via poissonization and the Mellin transform, and the inverse of the two operations. It is also shown from an analysis of the moving poles of the Mellin transform of the poissonized moment generating function that the imbalance factor (under appropriate centering and scaling) follows a Gaussian limit law. The method extends to several variations of sampling keys from a trie and we sketch results of climbing under different strategies. The exact probability distribution is computed in one case, to demonstrate that such calculations can be done, at least in principle.

Random wear models in reliability theory

1971 ◽  
Vol 3 (2) ◽  
pp. 229-248 ◽  
Author(s):  
David S. Reynolds ◽  
I. Richard Savage
Keyword(s):  
Maximum Likelihood ◽  
Failure Time ◽  
Maximum Likelihood Estimates ◽  
Time To Failure ◽  
Failure Time Distribution ◽  
Wear Process ◽  
Joint Distributions ◽  
Failure Times ◽  
Random Failure ◽  
Interesting Class

Gaver (1963) and Antelman and Savage (1965) have proposed models for the distribution of the time to failure of a simple device exposed to a randomly varying environment. Each model represents cumulative wear as a specified function of a non-negative stochastic process with independent increments, and assumes that the reliability of the device is conditioned upon realizations of this process. From these models are derived the corresponding unconditional joint distributions for the random failure time vector of n independent, identical devices exposed to the same realization of the wear process. It is shown that the identical failure time distribution for one component can arise from each model. In the Gaver model simultaneous failure times occur with positive probability. The probabilities of specific tie configurations are developed.For an interesting class of Gaver models involving a time scale parameter, the maximum likelihood estimates from several devices in one environment are examined. In that case the tie configuration probability does not depend on the parameter. For the corresponding Antelman-Savage models a consistent sequence of estimators is obtained; the maximum likelihood theory did not appear tractable.

scholarly journals Blinking statistics and molecular counting in direct stochastic reconstruction microscopy (dSTORM)

2021 ◽  
Author(s):  
Lekha Patel ◽  
David Williamson ◽  
Dylan M Owen ◽  
Edward A K Cohen
Keyword(s):  
Probability Distribution ◽  
Single Molecule ◽  
Immunological Synapse ◽  
Training Data ◽  
Supplementary Information ◽  
Cellular Structures ◽  
Exact Probability ◽  
Stochastic Optical Reconstruction Microscopy ◽  
Exact Probability Distribution ◽  
Optical Reconstruction

Abstract Motivation Many recent advancements in single-molecule localization microscopy exploit the stochastic photoswitching of fluorophores to reveal complex cellular structures beyond the classical diffraction limit. However, this same stochasticity makes counting the number of molecules to high precision extremely challenging, preventing key insight into the cellular structures and processes under observation. Results Modelling the photoswitching behaviour of a fluorophore as an unobserved continuous time Markov process transitioning between a single fluorescent and multiple dark states, and fully mitigating for missed blinks and false positives, we present a method for computing the exact probability distribution for the number of observed localizations from a single photoswitching fluorophore. This is then extended to provide the probability distribution for the number of localizations in a direct stochastic optical reconstruction microscopy experiment involving an arbitrary number of molecules. We demonstrate that when training data are available to estimate photoswitching rates, the unknown number of molecules can be accurately recovered from the posterior mode of the number of molecules given the number of localizations. Finally, we demonstrate the method on experimental data by quantifying the number of adapter protein linker for activation of T cells on the cell surface of the T-cell immunological synapse. Availability and implementation Software and data available at https://github.com/lp1611/mol_count_dstorm. Supplementary information Supplementary data are available at Bioinformatics online.

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