Residual Life Function Research Articles

Reliability function of k-out-of-n system equipped with two cold standby components

In this article, we focus on reliability properties of a k-out-of-n system equipped with two cold standby components. We describe the lifetime of the system and compute the reliability function of the system, which will allow us to investigate the risk of system failure. Further, three different mean residual life functions of the system are derived. We also provide some stochastic ordering results. The theoretical results established in this article addresses a problem proposed by Eryilmaz and Wang. We have presented a few numerical results to illustrate the importance of our analysis.

A zero truncated discrete distribution: Theory and applications to count data

The analysis and modeling of zero truncated count data is of primary interest in many elds such as engineering, public health, sociology, psychology, epidemiology. Therefore, in this article we have proposed a new and simple structure model, named a zero truncated discrete Lindley distribution. Thedistribution contains some submodels and represents a two-component mixture of a zero truncated geometric distribution and a zero truncated negative binomial distribution with certain parameters. Several properties of the distribution are obtained such as mean residual life function, probability generating function, factorial moments, negative moments, moments of residual life function, Bonferroni and Lorenz curves, estimation of parameters, Shannon and Renyi entropies, order statistics with the asymptotic distribution of their extremes and range, a characterization, stochastic ordering and stress-strength parameter. Moreover, the collective risk model is discussed by considering theproposed distribution as primary distribution and exponential and Erlang distributions as secondary ones. Test and evaluation statistics as well as three real data applications are considered to assess the peformance of the distribution among the most frequently zero truncated discrete probability models.

A weighted transmuted exponential distributions with environmental applications

In this paper, we introduce a new three-parameter distribution. It is based on the combination of a re-parametrization of a general family of distributions (known as the EGNB2 distribution) and the so-called quadratic rank transmutation map defined with the exponential distribution as baseline. We explore some mathematical properties of this distribution including the hazard rate function, moments, the moment generating function, the quantile function, various entropy measures and (reversed) residual life functions. A statistical study investigates estimation of the parameters using the method of maximum likelihood. The distribution along with other existing distributions are fitted to two environmental data sets and its superior performance is assessed by using some goodness-of-fit tests. As a result, some environmental measures associated with these data are obtained such as the return level and mean deviation about this level.

Transmuted Gumbel univariate exponential distribution

A functional composition of the distribution function of one prob ability distribution with the inverse distribution function of another is called the transmutation map. The present paper is purported to show how the transmuted distribution can be obtained by using the convex combination of failure probability of two-component systems. The transmuted Gumbel uni variate exponential distribution is presented by changing convex combination parameter. This new distribution is dened and studied. Some mathematical properties of this distribution including the generating function and ordinary moments are derived. The survival, hazard rate and mean residual life func tions are discussed. Finally, three applications to real data are presented.

Testing ageing notions through percentiles of the residual life

ABSTRACTIn applications, it is common to classify life distributions according to different ageing properties based on different reliability measures. Those based on the failure rate function or the mean residual life are very popular in the literature. However, the increasing or decreasing failure rate ageing notion is very restrictive and, on the other hand, the mean residual life function has some drawbacks which make convenient to prevent its use: it may not exist and, even when it does, sometimes the empirical mean residual life function cannot be calculated or a single long-term survivor can have a marked effect upon it which will tend to be unstable due to its strong dependence on very long durations. The median, or other percentiles, of the residual life of a random variable, are, in general, useful alternatives to the mean residual life. In this paper, we propose a test statistic to check the ageing notion decreasing percentile residual life. This estimator is based on the difference of the empirical percentile residual life function and the monotone estimator. We study the asymptotic and the bootstrap distributions of the proposed statistic. A Monte Carlo simulation study has been carried out to see the performance of the statistic in finite sample situations. Finally, we apply our procedure to a real data set.

Exponentiated Rayleigh Poisson Distribution: Model, Properties and Applications

In this research paper, a new class of life-time distribution is introduced by compounding A new generalization of Rayleigh distribution; properties and applications and The Exponentiated G Poisson model, the so-called Exponentiated Rayleigh Poisson distribution. Main aim of this research article is to enhance the flexibility of Exponentiated G. Poisson distribution by power transformation technique. The probability density function, the survival function and the hazard function of the new proposed model in graphical form are illustrated. We study the properties of this new distribution with special emphasis on its quantile function, mode, skewness, kurtosis and moments. We have discussed residual life function, the probability-weighted moments, order statistics, R'enyi and entropies. We also discussed parameter estimation considering the maximum likelihood estimation approach. We have calculated the value of log-likelihood, Akaike's information criteria, Bayesian information criteria, corrected Akaike's information criteria and Hennan-Quinn information criteria of Generalized Rayleigh distribution, Exponentiated Chen distribution, Exponentiated Exponential distribution, Exponentiated Inverted Weibull distribution, Compound Rayleigh distribution and newly proposed Exponentiated Rayleigh Poisson distribution and found that the newly proposed model has smaller values in comparison to other. We have studied the P-P plot, Q-Q plot Kolmogorov Smirnov test and TTT plot of the proposed distribution for model validation. We compared the empirical distribution CDF and estimated distributed function CDF of the proposed model with five other models. A real dataset is analyzed for illustrative purposes. The importance and flexibility of the new family is illustrated by applying different techniques and tools. A final conclusion concludes the paper.

The extended gamma distribution with regression model and applications

<abstract> <p>This paper introduces a new extension of the gamma distribution, named as a <italic>new extended gamma</italic> distribution, via mixture representation of xgamma and gamma distributions. The statistical properties of the proposed distribution are derived such as moment generating and characteristic functions, variance, skewness, and kurtosis measures, Lorenz curve, and mean residual life function. The maximum likelihood, parametric bootstrap, method of moments, least squares, and weighted least squares estimation methods are considered to obtain the unknown model parameters. The finite sample performance of estimation methods is discussed via a simulation study. Using the proposed distribution, we propose a new regression model for the right-skewed response variable as an alternative to the gamma regression model. Two real data sets are analyzed to convince the readers for the usefulness of the proposed model.</p> </abstract>

A Weighted Pranav Distribution and its Application to Survival Data

A weighted Pranav distribution which includes one parameter Pranav distribution as a special case has been proposed for modelling lifetime data. Properties including moments and moment-related measures, hazard rate function and mean residual life function have been discussed. The estimation of its parameters has been discussed using maximum likelihood estimation. The goodness of fit of the proposed distribution has been discussed with a real lifetime dataset and the fit has been found satisfactory over several lifetime distributions.

A Univariate, Bivariate and Multivariate Extensions for the Inverse Rayleigh Model with Properties and Applications to the Univariate Version

A new univariate extension of the Inverse Rayleigh distribution is proposed and studied. Some of its fundamental statistical properties such as some stochastic properties, ordinary and incomplete moments, moments generating functions, residual life and reversed residual life functions, order statistics, quantile spread ordering, Rényi, Shannon and q-entropies are derived. A simple type Copula based construction via Morgenstern family and via Claytoncopula is employed to derive many bivariate and multivariate extensions of the new model. We assessed the performance of the maximum likelihood estimators using a simulation study. The importance of the new model is shown via two applications to real data sets.

A Two-Parameter Pranav Distribution with Properties and Its Application.

Introduction: Lifetime distribution has drawn so much attention in recent research, and this has lead to the development of new lifetime distribution. Addition of parameters to the existing distribution makes the distribution more flexible and reliable and applicable model has become the focus of the recent search. This paper proposed a two-parameter Pranav distribution which has its base from a one-parameter Pranav and Ishita distribution. Methods Two parameter Pranav distribution was proposed. Mathematical and statistical properties of the distribution which includes; moments, coefficient of variation, skewness, kurtosis, index of dispersion, hazard rate function, mean residual life function, stochastic ordering, mean deviation, Bonferroni and Lorenz curves were developed. Other lifetime distributions such as Ishita, Akash, Sujatha, Shanker, Lindley, and Exponential distributions were considered in the study. Results: This new distribution was compared with two-parameter Akash, Lindley, one parameter Pranav, Ishita, Akash, Sujatha, Shanker, Lindley, and Exponential distributions to determine the efficiency of the new model. The estimation of parameters has been X-rayed using the method of moments and maximum likelihood. Also, AIC and BIC were used to test for the goodness of fit of the model which was applied to a real-life data of hypertensive patients. The results show that the new two-parameter Pranav distribution has the lowest value of AIC and BIC Conclusion: Based on the AIC and BIC values we can conclude that the two-parameter Pranav is more efficient than the other distribution for modeling survival of hypertensive patients. Hence two-parameter Pranav can be seen as an important distribution in modeling lifetime data.

Weighted Quasi Lindley Distribution with Properties and Applications

In this paper a three-parameter weighted quasi Lindley distribution which includes two-parameter quasi Lindley distribution, weighted Lindley distribution and gamma distribution and one parameter Lindley and exponential distributions as particular cases has been proposed and its statistical properties including moments and moments based measures, hazard rate function, and mean residual life function have been discussed. Maximum likelihood estimation has been discussed for estimating its parameters. Applications of the distribution have explained with two real failure time data and the goodness of fit shows quite satisfactory fit over other lifetime distributions.

Exponential intervened Poisson distribution

We introduce a new family of distributions using intervened power series distribution. It is a rich family containing distributions such as Marshall–Olkin extended families of distributions, families of distributions generated through truncated negative binomial, and families of distributions generated through truncated binomial distribution. Also, the proposed distribution is a generalization of the families of distributions generated through zero truncated Poisson distribution. As a particular case, we study exponential intervened Poisson (EIP) distribution. This is a generalization of exponential-Poisson distribution. The shape properties of the pdf and hazard rate are proved. The model identifiability is established. The expression for moments, mean deviation, distribution of order statistics, and residual life function of EIP distribution are derived. The unknown parameters of the distribution are estimated using the method of maximum likelihood. The existence and uniqueness of maximum likelihood estimates (MLEs) are proved. Simulation study is carried out to show the performance of MLEs. EIP distribution is fitted to two real data sets and it is shown that the distribution is more appropriate than other competitive models. The adequacy of the EIP model for the data set is established using parametric bootstrap approach.

Rayleigh-half normal distribution for modeling tooth movement data

ABSTRACTClear aligners provide a recent technique to induce orthodontic tooth movement. In this paper, a new one-parameter model, called, Rayleigh-Half Normal distribution (RHN) is proposed for modeling tooth movement data. Various statistical and reliability properties including moments and its associated measures, mean residual life, survival and hazard functions are derived and studied. Moreover, the model parameter is estimated via the method of moments and maximum likelihood. A simulation study was performed to examine the bias and mean square error of the estimated parameter. Finally, the flexibility of the proposed model is illustrated by utilizing a real dental data. The results showed that it could give a better fit than other well-known models.

Remaining useful life prediction of lubricating oil with dynamic principal component analysis and proportional hazards model

Lubricating oil contains a lot of tribological information of the machine and plays an important role in machine health. Oil degrades with serving time and causes severe wear afterwards, which is a complex dynamic process, and difficult to be accurately described by a single property. Therefore, the main purpose of deterioration prediction is to estimate the remaining useful life that the oil can still fulfill its functions by analyzing oil condition monitoring data. With a large amount of oil condition monitoring data collected, a vector autoregressive model is applied to the original oil data to describe the dynamic deterioration process. Then dynamic principal component analysis, an effective dimensionality reduction method, is employed to obtain the principal components capturing the most information of the oil data. The proportional hazards model is then built to calculate the failure risk of the lubricating oil based on the condition monitoring information, where its baseline function represents the aging process assuming to follow the Weibull distribution and its positive link function represents the influence of covariates (the principal components) on the failure risk. Finally, the remaining useful life prediction of lubricating oil can be obtained by explicit formulas of the characteristics such as the conditional reliability function and the mean residual life function. This work provides an approach to assess the health of lubricating oil, and a guidance for oil maintenance strategy.

The log-weighted exponential regression model: alternative to the beta regression model

In this paper, we introduce a new two-parameter distribution, called log-weighted exponential (log-WE) distribution, defined on the unit interval. Some of the statistical properties of the proposed distribution including moments, incomplete moments and mean residual life function are obtained. The maximum likelihood, method of moments, least squares and weighted least squares estimation methods are used to estimate unknown parameters of the proposed model and finite sample performance of estimation methods are evaluated through Monte-Carlo simulation study. Moreover, a new regression model as an alternative to the beta regression model is introduced.

Semiparametric model of mean residual life with biased sampling data

The mean residual life function is an important and attractive alternative to the survival function or the hazard function of survival time in practice. It describes the remaining life expectancy of a subject surviving up to time t. To study the relationship between the mean residual life and its associated covariates, a broad class of mean residual life models under general biased sampling data is proposed in this paper, thereby extending the reach of this flexible and powerful tool. The unknown parameters are estimated by using inverse probability weighting method. An easily used model diagnostic method is also presented to assess the adequacy of the model. Both asymptotic properties and finite sample performances of the proposed estimators are established. Finally the practical appeal of the estimator is shown via two real applications using the Channing House data and the Canadian Study of Health and Aging (CSHA) dementia data.

A Two-parameter Rama Distribution

In this paper, a two-parameter Rama distribution is proposed. This is coined from Lindley distribution and Rama distribution. Its mathematical and statistical properties which include its shapes, moment, coefficient of variation, skewness, kurtosis, index of dispersion, hazard rate function, mean residual life function, stochastic ordering, mean deviation; Bonferroni and Lorenz curves are also discussed. The estimation of parameters has been X-rayed using methods of moment and maximum likelihood. Also AIC and BIC are used to test for the goodness of fit of the model which is applied to a real life data of hepatitis B patients. This new distribution is compared with Rama, 2-parameter Akash, 2-parameter Lindley, Akash, Shanker, Ishita, Lindley and Exponential distributions in order to determine the efficiency of the new model.

A Two-Parameter Weighted Rama Distribution with Properties and Application

In this paper, a two-parameter weighted Rama distribution which includes one parameter Rama distribution introduced by Shanker [1] has been proposed for modelling real lifetime data. Statistical properties of the distribution including shapes of a probability density function, moments and moment related measures, hazard rate function, mean residual life function, and stochastic orderings have been discussed. The estimation of its parameters has been discussed using the method of maximum likelihood. Application of the proposed distribution has been discussed.

Alpha-Power Pareto distribution: Its properties and applications.

In Statistical theory, inclusion of an additional parameter to standard distributions is a usual practice. In this study, a new distribution referred to as Alpha-Power Pareto distribution is introduced by including an extra parameter. Several properties of the proposed distribution, including moment generating function, mode, quantiles, entropies, mean residual life function, stochastic orders and order statistics are obtained. Parameters of the proposed distribution have been estimated using maximum likelihood estimation technique. Two real datasets have been considered to examine the usefulness of the proposed distribution. It has been observed that the proposed distribution outperforms different variants of Pareto distribution on the basis of model selection criteria.

Odoma Distribution and Its Application

In this study, a new continuous one parameter lifetime distribution is proposed. Its mathematical properties such as moments, order statistics, entropy, survival function, hazard rate function and mean residual life function are derived. The new distribution is applied to real-life data from engineering and the method of maximum likelihood is used to estimate the parameter. The goodness-of-fit of the new distribution shows its better fit to the data than some competing distributions.