Bootstrap confidence intervals for the mode of the hazard function (original) (raw)
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Log-logistic and Weibull distributions have both accelerated survival time property. The log-logistic distribution has also proportional odds property. Log-logistic distribution has unimodal hazard curve which changes direction. Link [6,7] presented a confidence interval estimate of survival function using Cox's proportional hazard model with covariates. Her idea more recently extended by [1] to the exponential distribution and [2] to exponential proportional hazard model, respectively. The same idea has been extended to the Weibull proportional hazard regression model by [3]. In this study, it is formed on confidence interval for log-logistic distribution survival function for any values of the time provided that the survival times have a log-logistic distributed random variable. It is also extended the same results to the proportional odds regression. A Real time data and a simulation data examples are also considered in the study for illustration the discussed confidence interval. Journa l o f B io m etrics & B io s ta tistics
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Journal of Statistical Computation and Simulation, 2014
Interval censored data arises when a failure time say, T can not be observed directly but can only be determined to lie in an interval obtained from a series of inspection times. The frequentist approach for analysing interval-censored data has been developed for some time now. It is very common due to unavailability of software in the field of biological, medical and reliability studies to simplify the interval censoring structure of the data into that of a more standard right censoring situation by imputing the midpoints of the censoring intervals. In this research paper, we apply the Bayesian approach by employing Lindley's 1980, and Tierney and Kadane 1986 numerical approximation procedures when the survival data under consideration are interval-censored. The Bayesian approach to interval censored data has barely been discussed in literature. The essence of this study is to explore and promote the Bayesian methods when the survival data been considered or investigated is interval-censored. We have considered only a parametric approach by assuming that the survival data follows a loglogistic distribution model. We illustrate the proposed methods with two real datasets. A simulation study is also carried out to compare the performances of the methods.
2015
The log-normal distribution is often used to model lifetime data due to its non-monotonic hazard rate. However, with left-truncated data the normal approximation fails due to the increased skewness in this distribution. This sometimes results in the poor performance of the confidence interval estimation based on the asymptotic normality of the maximum likelihood estimates, especially when the sample sizes are small. The purpose of this research is to compare and analyze the performance of the Wald, likelihood ratio and jackknife confidence intervals based on the widths of the intervals for the parameters of the log-normal model with fixed covariates through a coverage probability study. A lifetime data is therefore simulated under six different settings; model 1 (no truncation with exact observations), model 2 (low truncation with exact observations), model 3 (high truncation with exact observations), model 4 (no truncation with low censoring), model 5 (low truncation with low censo...
Statistics & Probability Letters, 2014
Two requirements for pivoting a cumulative distribution function (CDF) in order to construct exact confidence intervals or bounds for a real-valued parameter θ are the monotonicity of this CDF with respect to θ and the existence of solutions of some pertinent equations for θ. The second requirement is not fulfilled by the CDF of the maximum likelihood estimator of the exponential scale parameter when the data come from some life-testing scenarios such as type-I censoring, hybrid type-I censoring, and progressive type-I censoring that are subject to time constraints. However, the method has been used in these cases probably because the non-existence of the solution usually happens only with small probability. Here, we illustrate the problem by giving formal details in the case of type-I censoring and by providing some further examples. We also present a suitable extension of the basic pivoting method which is applicable in situations wherein the considered equations have no solution.
International Journal of Statistics and Probability, 2015
We consider statistical inference of the unknown parameters of a two-parameter bathtub-shaped distribution (Chen, 2000) [Stat. & Prob. Letters 49 (2000) 155-161]. The inference will be conducted for Type-II censored and progressively Type-II censored data using the maximum likelihood and Bayes techniques. There are no explicit expressions for the estimators of the parameters. In the case of the maximum likelihood estimator (MLE), we propose a simple fixed point algorithm to compute the MLE and construct different confidence intervals and confidence regions of the unknown parameters. Bayes analyses of the unknown parameters are also discussed under fairly general priors for the unknown parameters. We propose to use the Markov Chain Monte Carlo (MCMC) and simulation-based technique to compute the Bayes estimates and the two-sided Bayesian probability intervals of the parameters. Also, we use the rejection sampling algorithm to produce the exact Bayes estimates. The methods developed will be applied in the analyses of two real data sets and a simulated data set. A Monte Carlo simulation is used to compare the results from the MLE and Bayes techniques.
Classical and Bayesian studies for a new lifetime model in presence of type-II censoring
Communications for Statistical Applications and Methods, 2019
This paper proposes a new class of distribution using the concept of exponentiated of distribution function that provides a more flexible model to the baseline model. It also proposes a new lifetime distribution with different types of hazard rates such as decreasing, increasing and bathtub. After studying some basic statistical properties and parameter estimation procedure in case of complete sample observation, we have studied point and interval estimation procedures in presence of type-II censored samples under a classical as well as Bayesian paradigm. In the Bayesian paradigm, we considered a Gibbs sampler under Metropolis-Hasting for estimation under two different loss functions. After simulation studies, three different real datasets having various nature are considered for showing the suitability of the proposed model.
Mathematics, 2020
After defining a new log-logistic model and studying its properties, some new bivariate type versions using “Farlie-Gumbel-Morgenstern Copula”, “modified Farlie-Gumbel-Morgenstern Copula”, “Clayton Copula”, and “Renyi’s entropy Copula” are derived. Then, using the Bagdonavicius-Nikulin goodness-of-fit (BN-GOF) test for validation, we proposed a goodness-of-fit test for a new log-logistic model. The modified test is applied for the “right censored” real dataset of survival times. All elements of the modified test are explicitly derived and given. Three real data applications are presented for measuring the flexibility and the importance of the new model under the uncensored scheme. Two other real datasets are analyzed for censored validation.
Asymptotic Properties of Bootstrap Likelihood Ratio Statistics for Time Censored Data
2003
Much research has been done on the asymptotic distributions of likelihood ratio statistics for complete data. In this paper we consider the situation in which the data are time censored and the distribution of the likelihood ratio statistic is a mixture of continuous and discrete distributions. We show that the distribution of a signed square root likelihood ratio statistic can be approximated by its bootstrap distribution up to second order accuracy. Similar results are shown to hold for likelihood ratio statistics with or without a Bartlett correction. The main tool used is a continuous Edgeworth expansion for the likelihood-based statistics, which may be of some independent interest. Further, we use a simulation study to investigate the adequacy of the approximation provided by the theoretical result by comparing the finite-sample coverage probability of several competing confidence interval (CI) procedures based on the two parameter Weibull model. Our simulation results show that, in finite samples, the methods based on the bootstrap signed square root likelihood ratio statistic outperform the bootstrap-t and BCa methods in constructing one-sided confidence bounds (CBs) when the data are Type I censored.
Estimation Strategies for Censored Lifetimes with a Lexis-Diagram Type Model
Scandinavian Journal of Statistics, 2008
ABSTRACT When collecting right-censored lifetime data, a sampling bias occurs in many situations. The present authors offer a new sampling scheme description based on the Lexis diagram, which includes time-window and cohort studies. This implies a selection bias for the data collected. We also assume that censoring may occur. Hence, we provide a model selection strategy to estimate the hazard and density functions in this model. Adaptive projection estimators on trigonometric bases are developed by contrast penalization and optimal non-parametric rates of convergence are given. Both estimators are practically studied through simulation experiments as well as real data in the case of the time-window study. Copyright (c) Board of the Foundation of the Scandinavian Journal of Statistics 2008.