Goodness-of-fit test for exponentiality based on spacings for general progressive Type-II censored data

There have been numerous tests proposed to determine whether or not the exponential model is suitable for a given data set. In this article, we propose a new test statistic based on spacings to test whether the general progressive Type-II censored samples are from exponential distribution. The null distribution of the test statistic is discussed and it could be approximated by the standard normal distribution. Meanwhile, we propose an approximate method for calculating the expectation and variance of samples under null hypothesis and corresponding power function is also given. Then, a simulation study is conducted. We calculate the approximation of the power based on normality and compare the results with those obtained by Monte Carlo simulation under different alternatives with distinct types of hazard function. Results of simulation study disclose that the power properties of this statistic by using Monte Carlo simulation are better for the alternatives with monotone increasing hazard function, and otherwise, normal approximation simulation results are relatively better. Finally, two illustrative examples are presented.     

Testing the validity of the exponential model for hybrid Type-I censored data

The exponential distribution has been used in life-testing and reliability studies. In this article, we first express the entropy of Type-I hybrid censoring scheme in terms of hazard function and provide an estimate of the entropy of Type-I hybrid censored data. Then, we construct a goodness-of-fit test statistic based on Kullback–Leibler information for Type-I hybrid censored data. The test statistic is used to test for exponentiality. A Monte Carlo simulation is conducted to obtain the power of the proposed test against various alternatives. Finally, a data example is presented for illustrative purpose.     

Moments of nonparametric probability density functions of entropy estimators applied to testing the inverse Gaussian distribution

The inverse Gaussian (IG) distribution is widely used to model data and then it is important to develop efficient goodness of fit tests for this distribution. In this article, we introduce some new test statistics for examining the IG goodness of fit based on correcting moments of nonparametric probability density functions of entropy estimators. These tests are consistent against all alternatives. Critical points and power of the tests are explored by simulation. We show that the proposed tests are more powerful than competitor tests. Finally, the proposed tests are illustrated by real data examples.     

Smooth tests of fit for censored gmma samples

In this paper properties of two estimators of Cpm are investigated in terms of changes in the process mean and variance. The bias and mean squared error of these estimators are derived. It can be shown that the estimate of Cpm proposed by Chan, Cheng and Spiring (1988) has smaller bias than the one proposed by Boyles (1991) and also has a smaller mean squared error under certain conditions. Various approximate confidence intervals for Cpm are obtained and are compared in terms of coverage probabilities, missed rate and average interval width.     

Interval estimation for the Pareto distribution based on the progressive Type II censored sample

For the complete sample and the right Type II censored sample, Chen [Joint confidence region for the parameters of Pareto distribution. Metrika 44 (1996), pp. 191–197] proposed the interval estimation of the parameter θ and the joint confidence region of the two parameters of Pareto distribution. This paper proposed two methods to construct the confidence region of the two parameters of the Pareto distribution for the progressive Type II censored sample. A simulation study comparing the performance of the two methods is done and concludes that Method 1 is superior to Method 2 by obtaining a smaller confidence area. The interval estimation of parameter ν is also given under progressive Type II censoring. In addition, the predictive intervals of the future observation and the ratio of the two future consecutive failure times based on the progressive Type II censored sample are also proposed. Finally, one example is given to illustrate all interval estimations in this paper.     

On the goodness-of-fit test based on the ratio of cumulative hazard functions with the Type II censored data

For comparing two cumulative hazard functions, we consider an extension of the Kullback–Leibler information to the cumulative hazard function, which is concerning the ratio of cumulative hazard functions. Then we consider its estimate as a goodness-of-fit test with the Type II censored data. For an exponential null distribution, the proposed test statistic is shown to outperform other test statistics based on the empirical distribution function in the heavy censoring case against the increasing hazard alternatives.     

Likelihood inference for the component lifetime distribution based on progressively censored parallel systems data

Point and interval estimators for the scale parameter of the component lifetime distribution of a k-component parallel system are obtained when the component lifetimes are assumed to be independently and identically exponentially distributed. We prove that the maximum likelihood estimator of the scale parameter based on progressively Type-II censored system lifetimes is unique and can be obtained by a fixed-point iteration procedure. In particular, we illustrate that the Newton–Raphson method does not converge for any initial value. Furthermore, exact confidence intervals are constructed by a transformation using normalized spacings and other component lifetime distributions including Weibull distribution are discussed.     

Testing exponentiality based on Kullback—Leibler information for progressively Type II censored data

In many life-testing and reliability experiments, data are often censored in order to reduce the cost and time associated with testing and since the conventional Type-I and Type-II censoring schemes are not flexible enough, progressive censoring is developed by researchers. In this article, we develop a general goodness of fit test by using a new estimate of Kullback–Leibler information based on progressively Type-II censored data. Consistency and other properties of the proposed test are shown. Then, we use the proposed test statistic to test for exponentiality based on progressively Type-II censored data. The power values of the proposed test under different progressively Type-II censoring schemes are computed, through Monte Carlo simulations. It is observed that the proposed test is quite powerful in compared with the test proposed by Balakrishnan et al. (2007 Balakrishnan, N., Habibi Rad, A., and Arghami, N. R. (2007). Testing exponentiality based on Kullback–Leibler information with progressively type-II censored data. IEEE Transactions on Reliability 56:301–307. [Google Scholar]). Two real datasets from progressive censoring literature are finally presented for illustrative purpose.     

Likelihood estimation for a general class of inverse exponentiated distributions based on complete and progressively censored data

In this paper, we establish the existence and uniqueness of the maximum-likelihood estimates of the parameters of a general class of inverse exponentiated distributions based on complete as well as progressively Type-I and Type-II censored data.     

Progressively Type-II censored competing risks data for exponential distributions based on sequential order statistics

We consider the progressively Type-II censored competing risks model based on sequential order statistics. It is assumed that the latent failure times are independent and the failure of each unit influences the lifetime distributions of the latent failure times of surviving units. We provide explicit expressions for the likelihood function of the available data under the conditional proportional hazard rate (CPHR) and the power trend conditional proportional hazard rate (PTCPHR) models. Under CPHR and PTCPHR models and assumption that the baseline distributions of the latent failure times are exponential, classical and Bayesian estimates of the unknown parameters are provided. Monte Carlo simulations are then performed for illustrative purposes. Finally, two datasets are analyzed.     

An extensive power evaluation of some tests for the inverse Gaussian distribution

Many applications of the Inverse Gaussian distribution, including numerous reliability and life testing results are presented in statistical literature. The paper studies the problem of using entropy tests to examine the goodness of fit of an Inverse Gaussian distribution with unknown parameters. Some entropy tests based on different entropy estimates are proposed. Critical values of the test statistics for various sample sizes are obtained by Monte Carlo simulations. Type I error of the tests is investigated and then power values of the tests are compared with the competing tests against various alternatives. Finally, recommendations for the application of the tests in practice are presented.     

Goodness of fit tests for Nakagami distribution based on smooth tests

ABSTRACT

Nakagami distribution is one of the most common distributions used to model positive valued and right skewed data. In this study, we interest goodness of fit problem for Nakagami distribution. Thus, we propose smooth tests for Nakagami distribution based on orthonormal functions. We also compare these tests with some classical goodness of fit tests such as Cramer–von Mises, Anderson–Darling, and Kolmogorov–Smirnov tests in respect to type-I error rates and powers of tests. Simulation study indicates that smooth tests give better results than these classical tests give in respect to almost all cases considered.     

A consistent method of estimation for the three-parameter lognormal distribution based on Type-II right censored data

ABSTRACT

In this paper, we propose a parameter estimation method for the three-parameter lognormal distribution based on Type-II right censored data. In the proposed method, under mild conditions, the estimates always exist uniquely in the entire parameter space, and the estimators also have consistency over the entire parameter space. Through Monte Carlo simulations, we further show that the proposed method performs very well compared to a prominent method of estimation in terms of bias and root mean squared error (RMSE) in small-sample situations. Finally, two examples based on real data sets are presented for illustrating the proposed method.     

Testing exponentiality based on Type-I censored data

The problem of goodness-of-fit for the exponential distribution when the available data are subject to Type-I censoring is discussed here. A test procedure is proposed in this case that exhibits more power as compared to existing methods. The power of the proposed test is assessed for several alternative distributions by means of Monte Carlo simulations. Finally, the proposed test is illustrated with a real data set.     

Inference for the scale parameter of lifetime distribution of k-unit parallel system based on progressively censored data

In this paper, inference for the scale parameter of lifetime distribution of a k-unit parallel system is provided. Lifetime distribution of each unit of the system is assumed to be a member of a scale family of distributions. Maximum likelihood estimator (MLE) and confidence intervals for the scale parameter based on progressively Type-II censored sample are obtained. A β-expectation tolerance interval for the lifetime of the system is obtained. As a member of the scale family, half-logistic distribution is considered and the performance of the MLE, confidence intervals and tolerance intervals are studied using simulation.     

Testing goodness-of-fit for some lifetime distributions with conventional Type-I censoring

A goodness-of-fit test procedure is proposed for some lifetime distributions when the available data are subject to Type-I censoring. The proposed method extends the test procedure of Pakyari and Balakrishnan to other lifetime distributions. The extension to Weibull and log-normal models is studied in details. The new test recovers the nominal level of significance and exhibits more power in comparison to the existing tests for several alternative distributions by means of Monte Carlo simulations. Finally, a real dataset is considered for illustrative purposes.     

Point and Interval Estimation for a Generalized Logistic Distribution Under Progressive Type II Censoring

The likelihood equations based on a progressively Type II censored sample from a Type I generalized logistic distribution do not provide explicit solutions for the location and scale parameters. We present a simple method of deriving explicit estimators by approximating the likelihood equations appropriately. We examine numerically the bias and variance of these estimators and show that these estimators are as efficient as the maximum likelihood estimators (MLEs). The probability coverages of the pivotal quantities (for location and scale parameters) based on asymptotic normality are shown to be unsatisfactory, especially when the effective sample size is small. Therefore we suggest using unconditional simulated percentage points of these pivotal quantities for the construction of confidence intervals. A wide range of sample sizes and progressive censoring schemes have been considered in this study. Finally, we present a numerical example to illustrate the methods of inference developed here.     

Shapiro–Wilk test for skew normal distributions based on data transformations

A probability property that connects the skew normal (SN) distribution with the normal distribution is used for proposing a goodness-of-fit test for the composite null hypothesis that a random sample follows an SN distribution with unknown parameters. The random sample is transformed to approximately normal random variables, and then the Shapiro–Wilk test is used for testing normality. The implementation of this test does not require neither parametric bootstrap nor the use of tables for different values of the slant parameter. An additional test for the same problem, based on a property that relates the gamma and SN distributions, is also introduced. The results of a power study conducted by the Monte Carlo simulation show some good properties of the proposed tests in comparison to existing tests for the same problem.     

Existence,uniqueness and consistency of estimation of life characteristics of three-parameter Weibull distribution based on Type-II right censored data

In this paper, we propose a new method of estimation for the parameters and quantiles of the three-parameter Weibull distribution based on Type-II right censored data. The method, based on a data transformation, overcomes the problem of unbounded likelihood. In the proposed method, under mild conditions, the estimates always exist uniquely, and the estimators are also consistent over the entire parameter space. Through Monte Carlo simulations, we further show that the proposed method of estimation performs well compared to some prominent methods in terms of bias and root mean squared error in small-sample situations. Finally, two real data sets are used to illustrate the proposed method of estimation.