This research aims to analyze data coming from step stress life testing experiments where the str... more This research aims to analyze data coming from step stress life testing experiments where the stress level is incremented at a preset time to obtain failure data faster. To analyze step stress data, a model that extrapolates the information attained from the accelerated tests to normal conditions needs to be fit to the life test data. We used the Cumulative Exposure Model (CEM) to model simple step stress lognormal life test data where hybrid censoring is present and applied the maximum likelihood estimation method to find the point and interval estimates of the parameters. Bootstrap intervals (bootstrap-t intervals and percentile intervals) were also constructed. We then performed a simulation study to assess the proposed methods of estimation under different hybrid censoring schemes. The Bias and MSE of the maximum likelihood estimators (MLEs) along with the coverage probability and average lengths of the corresponding confidence intervals were investigated. Finally, an illustrative example has been used to demonstrate the application of the methods discussed in this paper.
We consider nonparametric interval estimation for the population quantiles based on unbalanced ra... more We consider nonparametric interval estimation for the population quantiles based on unbalanced ranked set samples. We derived the large sample distribution of the empirical log likelihood ratio statistic for the quantiles. Approximate intervals for quantiles are obtained by inverting the likelihood ratio statistic. The performance of the empirical likelihood interval is investigated and compared with the performance of the intervals based on the ranked set sample order statistics.
International Journal of Statistics and Economics, Jun 20, 2008
We consider the problem of setting confidence intervals for the stress-strength probability using... more We consider the problem of setting confidence intervals for the stress-strength probability using empirical likelihood techniques with paired samples. Some modifications to the usual empirical likelihood technique like bootstrapping and smoothing are considered. The intervals are studied by simulation in terms of their coverage probabilities. Comparisons with other known intervals are also undertaken. Our simulation studies of the properties of the intervals indicate that some modifications are capable of improving the performance of the empirical likelihood based intervals. Moreover they appear to compete well with other known intervals in the literature.
The problem of interval estimation for the regression parameter of the model of parallel systems ... more The problem of interval estimation for the regression parameter of the model of parallel systems with covariates is investigated. Intervals based on inverted likelihood ratio tests were considered. Two corrections to the likelihood ratio statistic were examined, the Bartlett correction and the mean and variance correction (Dicicco [4]). A simulation study is conducted to investigate the coverage probability and the symmetry of upper and lower error probabilities for various degrees of time censoring. It is found that the uncorrected intervals perform equally well as that of the corrected ones, except when the sample size is small and the censoring is heavy. In this case, intervals based on the mean and variance correction perform very well.
Computational Statistics & Data Analysis, Mar 1, 2008
We consider the likelihood and Bayesian estimation of the stress-strength reliability based on lo... more We consider the likelihood and Bayesian estimation of the stress-strength reliability based on lower record values from the generalized exponential distribution. The estimators are derived and their properties are studied. Confidence intervals, exact and approximate, as well as the Bayesian credible sets for the stress-strength reliability are obtained. A simulation study is conducted to investigate and compare the performance of the intervals.
Journal of Statistical Computation and Simulation, Jul 8, 2013
We consider the problem of setting confidence intervals for the stress-strength probability using... more We consider the problem of setting confidence intervals for the stress-strength probability using empirical likelihood techniques with paired samples. Some modifications to the usual empirical likelihood technique like bootstrapping and smoothing are considered. The intervals are studied by simulation in terms of their coverage probabilities. Comparisons with other known intervals are also undertaken. Our simulation studies of the properties of the intervals indicate that some modifications are capable of improving the performance of the empirical likelihood based intervals. Moreover they appear to compete well with other known intervals in the literature.
Communications in Statistics - Simulation and Computation, Nov 4, 2014
We consider confidence intervals for the stress–strength reliability Pr(X< Y) in the two-param... more We consider confidence intervals for the stress–strength reliability Pr(X< Y) in the two-parameter exponential distribution. We have derived the Bayesian highest posterior density interval using non-informative prior distributions. We have compared its performance with the intervals based on the generalized pivot variable intervals in terms of their coverage probabilities and expected lengths. Our simulation study shows that the Bayesian interval performs better according to the criteria used, especially when the sample sizes are very small. An example is given.
Australian & New Zealand Journal of Statistics, Oct 1, 2007
A distribution-free test is proposed for the symmetry of a continuous distribution about a specif... more A distribution-free test is proposed for the symmetry of a continuous distribution about a specified median. The test is based on a longest run statistic on the upper portion of the sequence of ordered 'centred' observations in magnitude. The probability distribution of the longest run statistic is derived, and a computationally simple and accurate approximation of the right-tail probabilities of this statistic is given. This approximation is based on a partial fraction expansion of the corresponding generating function and is derived for use with large samples. The powers of the proposed test, some variations of the test, and other rival tests are investigated under a wide variety of asymmetric alternatives. Simulations indicate that the proposed test is competitive with other tests in terms of power performance.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
In this paper, we propose a modified version of ranked set sample which allows for the incorporat... more In this paper, we propose a modified version of ranked set sample which allows for the incorporation of more information in the inference procedure at almost no cost. This procedure is studied for the location-scale family using a likelihood framework. Our results show that the modified procedure produce more accurate inferences than the original ranked set sampling scheme.
Journal of Modern Applied Statistical Methods, May 1, 2006
Point estimation of the parameters of the lognormal distribution with censored data is considered... more Point estimation of the parameters of the lognormal distribution with censored data is considered. The often employed maximum likelihood estimator does not exist in closed form and iterative methods that require very good starting points are needed. In this article, some techniques of finding closed form estimators to this situation are presented and extended. An extensive simulation study is carried out to investigate and compare the performance of these techniques. The results show that some of them are highly efficient as compared with the maximum likelihood estimator.
Journal of Information and Optimization Sciences, Sep 1, 1999
Abstract Test statistics having scores which emphasize extreme observations and hence are effecti... more Abstract Test statistics having scores which emphasize extreme observations and hence are effective in detecting a shift in location in light tailed distributions were proposed by Randles and Hogg (1973). While for heavy tailed distributions, statistics that downweight extreme observations are developed by Pollicello (1978). A Modified Wilcoxon signed rank statistic for shift in location is proposed. It is based on modifying the Wilcoxon scores according to the distributions’s tail lengths. The extreme scores are determined according to the distribution’s tail lengths estimated from the sample. For light tailed distributions, it emphasizes extrema observations by pushing up their scores. While for heavy tailed distributions it downweights the scores of extreme observations. In this sense it is a continuously adaptive test. The proposed test is compared, through a simulation study, with the Wilcoxon signed rank test and the sign test. The results show that the proposed test has a better power performance t...
This paper considers the preliminary test estimator of the unknown parameter of the exponential d... more This paper considers the preliminary test estimator of the unknown parameter of the exponential distribution based on progressively Type-II censored samples. The mean squared error of the proposed estimator is derived and it is shown that this estimator dominates the corresponding usual estimator in the neighborhood of null hypothesis. The minimax regret estimator and shrinkage estimator are also discussed. A numerical example is given to illustrate the proposed estimators. Nova Science Publishers, Inc.Scopu
A goodness of fit test procedure is proposed for the log-logistic distribution when the available... more A goodness of fit test procedure is proposed for the log-logistic distribution when the available data are subject to Type I censoring. The proposed test is based on transforming type 1 censored data into complete data from a suitably truncated distribution. A Monte Carlo power study is conducted to evaluate and compare the performance of the proposed method with the existing classical methods. An application based on a real dataset is considered for illustrative purposes.
Australian & New Zealand Journal of Statistics, Jun 1, 2005
This paper considers the problem of testing for shift in location when the symmetry of the underl... more This paper considers the problem of testing for shift in location when the symmetry of the underlying distribution is in doubt. Various adaptive test procedures have been suggested in the literature; they are mainly based on a preliminary test or measure of asymmetry, and then choosing between the sign or the Wilcoxon tests accordingly. However, as this paper demonstrates, there are some disadvantages with such procedures. This paper develops a test that does not suffer from such disadvantages. The proposed test is based on modifying the Wilcoxon scores according to the evidence of asymmetry of the distribution present in the data as indicated by the magnitude of the P-value from a preliminary test of symmetry. A simulation study investigates and compares the performance of the proposed test and other known adaptive procedures in terms of power and attainment of the nominal size. The performance of a suitable bootstrap procedure for the situation under consideration is also studied. In most cases under consideration, the proposed test is found to be superior to the other tests.
Given a prior guess of the unknown parameter h, the preliminary test estimator based on the maxim... more Given a prior guess of the unknown parameter h, the preliminary test estimator based on the maximum likelihood estimator for the exponential scale parameter is developed. The optimal significance levels based on the minimax regret criterion and the corresponding critical values are obtained.
We consider goodness of fit tests for the Rayleigh distribution with grouped data. New Kolmogrov-... more We consider goodness of fit tests for the Rayleigh distribution with grouped data. New Kolmogrov-Smirnov type tests are suggested and compared with the traditional chi-square and likelihood ratio tests. The results show that some of the suggested tests have a good power performance as compared with the traditional ones.
We consider maximum likelihood estimation for the parameters and certain functions of the paramet... more We consider maximum likelihood estimation for the parameters and certain functions of the parameters in the Inverse Weibull (IW) distribution based on type II censored data. The functions under consideration are the Mean Residual Life (MRL), which is very important in reliability studies, and Tail Value at Risk (TVaR), which is an important measure of risk in actuarial studies. We investigated the performance of the MLE of the parameters and derived functions under various experimental conditions using simulation techniques. The performance criteria are the bias and the mean squared error of the estimators. Recommendations on the use of the MLE in this model are given. We found that the parameter estimators are almost unbiased, while the MRL and TVaR estimators are asymptotically unbiased. Moreover, the mean squared error of all estimators decreased for larger sample sizes and it increased when the censoring proportion is increased for a fixed sample size. The conclusion is that the maximum likelihood method of estimation works well for the parameters and the derived functions of the parameter like the MRL and TVaR. Two examples on a real data set are presented to illustrate the application of the methods used in this paper. The first one is on survival time of pigs while the other is on fire losses.
This research aims to analyze data coming from step stress life testing experiments where the str... more This research aims to analyze data coming from step stress life testing experiments where the stress level is incremented at a preset time to obtain failure data faster. To analyze step stress data, a model that extrapolates the information attained from the accelerated tests to normal conditions needs to be fit to the life test data. We used the Cumulative Exposure Model (CEM) to model simple step stress lognormal life test data where hybrid censoring is present and applied the maximum likelihood estimation method to find the point and interval estimates of the parameters. Bootstrap intervals (bootstrap-t intervals and percentile intervals) were also constructed. We then performed a simulation study to assess the proposed methods of estimation under different hybrid censoring schemes. The Bias and MSE of the maximum likelihood estimators (MLEs) along with the coverage probability and average lengths of the corresponding confidence intervals were investigated. Finally, an illustrative example has been used to demonstrate the application of the methods discussed in this paper.
We consider nonparametric interval estimation for the population quantiles based on unbalanced ra... more We consider nonparametric interval estimation for the population quantiles based on unbalanced ranked set samples. We derived the large sample distribution of the empirical log likelihood ratio statistic for the quantiles. Approximate intervals for quantiles are obtained by inverting the likelihood ratio statistic. The performance of the empirical likelihood interval is investigated and compared with the performance of the intervals based on the ranked set sample order statistics.
International Journal of Statistics and Economics, Jun 20, 2008
We consider the problem of setting confidence intervals for the stress-strength probability using... more We consider the problem of setting confidence intervals for the stress-strength probability using empirical likelihood techniques with paired samples. Some modifications to the usual empirical likelihood technique like bootstrapping and smoothing are considered. The intervals are studied by simulation in terms of their coverage probabilities. Comparisons with other known intervals are also undertaken. Our simulation studies of the properties of the intervals indicate that some modifications are capable of improving the performance of the empirical likelihood based intervals. Moreover they appear to compete well with other known intervals in the literature.
The problem of interval estimation for the regression parameter of the model of parallel systems ... more The problem of interval estimation for the regression parameter of the model of parallel systems with covariates is investigated. Intervals based on inverted likelihood ratio tests were considered. Two corrections to the likelihood ratio statistic were examined, the Bartlett correction and the mean and variance correction (Dicicco [4]). A simulation study is conducted to investigate the coverage probability and the symmetry of upper and lower error probabilities for various degrees of time censoring. It is found that the uncorrected intervals perform equally well as that of the corrected ones, except when the sample size is small and the censoring is heavy. In this case, intervals based on the mean and variance correction perform very well.
Computational Statistics & Data Analysis, Mar 1, 2008
We consider the likelihood and Bayesian estimation of the stress-strength reliability based on lo... more We consider the likelihood and Bayesian estimation of the stress-strength reliability based on lower record values from the generalized exponential distribution. The estimators are derived and their properties are studied. Confidence intervals, exact and approximate, as well as the Bayesian credible sets for the stress-strength reliability are obtained. A simulation study is conducted to investigate and compare the performance of the intervals.
Journal of Statistical Computation and Simulation, Jul 8, 2013
We consider the problem of setting confidence intervals for the stress-strength probability using... more We consider the problem of setting confidence intervals for the stress-strength probability using empirical likelihood techniques with paired samples. Some modifications to the usual empirical likelihood technique like bootstrapping and smoothing are considered. The intervals are studied by simulation in terms of their coverage probabilities. Comparisons with other known intervals are also undertaken. Our simulation studies of the properties of the intervals indicate that some modifications are capable of improving the performance of the empirical likelihood based intervals. Moreover they appear to compete well with other known intervals in the literature.
Communications in Statistics - Simulation and Computation, Nov 4, 2014
We consider confidence intervals for the stress–strength reliability Pr(X< Y) in the two-param... more We consider confidence intervals for the stress–strength reliability Pr(X< Y) in the two-parameter exponential distribution. We have derived the Bayesian highest posterior density interval using non-informative prior distributions. We have compared its performance with the intervals based on the generalized pivot variable intervals in terms of their coverage probabilities and expected lengths. Our simulation study shows that the Bayesian interval performs better according to the criteria used, especially when the sample sizes are very small. An example is given.
Australian & New Zealand Journal of Statistics, Oct 1, 2007
A distribution-free test is proposed for the symmetry of a continuous distribution about a specif... more A distribution-free test is proposed for the symmetry of a continuous distribution about a specified median. The test is based on a longest run statistic on the upper portion of the sequence of ordered 'centred' observations in magnitude. The probability distribution of the longest run statistic is derived, and a computationally simple and accurate approximation of the right-tail probabilities of this statistic is given. This approximation is based on a partial fraction expansion of the corresponding generating function and is derived for use with large samples. The powers of the proposed test, some variations of the test, and other rival tests are investigated under a wide variety of asymmetric alternatives. Simulations indicate that the proposed test is competitive with other tests in terms of power performance.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
In this paper, we propose a modified version of ranked set sample which allows for the incorporat... more In this paper, we propose a modified version of ranked set sample which allows for the incorporation of more information in the inference procedure at almost no cost. This procedure is studied for the location-scale family using a likelihood framework. Our results show that the modified procedure produce more accurate inferences than the original ranked set sampling scheme.
Journal of Modern Applied Statistical Methods, May 1, 2006
Point estimation of the parameters of the lognormal distribution with censored data is considered... more Point estimation of the parameters of the lognormal distribution with censored data is considered. The often employed maximum likelihood estimator does not exist in closed form and iterative methods that require very good starting points are needed. In this article, some techniques of finding closed form estimators to this situation are presented and extended. An extensive simulation study is carried out to investigate and compare the performance of these techniques. The results show that some of them are highly efficient as compared with the maximum likelihood estimator.
Journal of Information and Optimization Sciences, Sep 1, 1999
Abstract Test statistics having scores which emphasize extreme observations and hence are effecti... more Abstract Test statistics having scores which emphasize extreme observations and hence are effective in detecting a shift in location in light tailed distributions were proposed by Randles and Hogg (1973). While for heavy tailed distributions, statistics that downweight extreme observations are developed by Pollicello (1978). A Modified Wilcoxon signed rank statistic for shift in location is proposed. It is based on modifying the Wilcoxon scores according to the distributions’s tail lengths. The extreme scores are determined according to the distribution’s tail lengths estimated from the sample. For light tailed distributions, it emphasizes extrema observations by pushing up their scores. While for heavy tailed distributions it downweights the scores of extreme observations. In this sense it is a continuously adaptive test. The proposed test is compared, through a simulation study, with the Wilcoxon signed rank test and the sign test. The results show that the proposed test has a better power performance t...
This paper considers the preliminary test estimator of the unknown parameter of the exponential d... more This paper considers the preliminary test estimator of the unknown parameter of the exponential distribution based on progressively Type-II censored samples. The mean squared error of the proposed estimator is derived and it is shown that this estimator dominates the corresponding usual estimator in the neighborhood of null hypothesis. The minimax regret estimator and shrinkage estimator are also discussed. A numerical example is given to illustrate the proposed estimators. Nova Science Publishers, Inc.Scopu
A goodness of fit test procedure is proposed for the log-logistic distribution when the available... more A goodness of fit test procedure is proposed for the log-logistic distribution when the available data are subject to Type I censoring. The proposed test is based on transforming type 1 censored data into complete data from a suitably truncated distribution. A Monte Carlo power study is conducted to evaluate and compare the performance of the proposed method with the existing classical methods. An application based on a real dataset is considered for illustrative purposes.
Australian & New Zealand Journal of Statistics, Jun 1, 2005
This paper considers the problem of testing for shift in location when the symmetry of the underl... more This paper considers the problem of testing for shift in location when the symmetry of the underlying distribution is in doubt. Various adaptive test procedures have been suggested in the literature; they are mainly based on a preliminary test or measure of asymmetry, and then choosing between the sign or the Wilcoxon tests accordingly. However, as this paper demonstrates, there are some disadvantages with such procedures. This paper develops a test that does not suffer from such disadvantages. The proposed test is based on modifying the Wilcoxon scores according to the evidence of asymmetry of the distribution present in the data as indicated by the magnitude of the P-value from a preliminary test of symmetry. A simulation study investigates and compares the performance of the proposed test and other known adaptive procedures in terms of power and attainment of the nominal size. The performance of a suitable bootstrap procedure for the situation under consideration is also studied. In most cases under consideration, the proposed test is found to be superior to the other tests.
Given a prior guess of the unknown parameter h, the preliminary test estimator based on the maxim... more Given a prior guess of the unknown parameter h, the preliminary test estimator based on the maximum likelihood estimator for the exponential scale parameter is developed. The optimal significance levels based on the minimax regret criterion and the corresponding critical values are obtained.
We consider goodness of fit tests for the Rayleigh distribution with grouped data. New Kolmogrov-... more We consider goodness of fit tests for the Rayleigh distribution with grouped data. New Kolmogrov-Smirnov type tests are suggested and compared with the traditional chi-square and likelihood ratio tests. The results show that some of the suggested tests have a good power performance as compared with the traditional ones.
We consider maximum likelihood estimation for the parameters and certain functions of the paramet... more We consider maximum likelihood estimation for the parameters and certain functions of the parameters in the Inverse Weibull (IW) distribution based on type II censored data. The functions under consideration are the Mean Residual Life (MRL), which is very important in reliability studies, and Tail Value at Risk (TVaR), which is an important measure of risk in actuarial studies. We investigated the performance of the MLE of the parameters and derived functions under various experimental conditions using simulation techniques. The performance criteria are the bias and the mean squared error of the estimators. Recommendations on the use of the MLE in this model are given. We found that the parameter estimators are almost unbiased, while the MRL and TVaR estimators are asymptotically unbiased. Moreover, the mean squared error of all estimators decreased for larger sample sizes and it increased when the censoring proportion is increased for a fixed sample size. The conclusion is that the maximum likelihood method of estimation works well for the parameters and the derived functions of the parameter like the MRL and TVaR. Two examples on a real data set are presented to illustrate the application of the methods used in this paper. The first one is on survival time of pigs while the other is on fire losses.
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