Conservative confidence intervals for a single parameter

The method of constructing confidence intervals from hypothesis tests is studied in the case in which there is a single unknown parameter and is proved to provide confidence intervals with coverage probability that is at least the nominal level. The confidence intervals obtained by the method in several different contexts are seen to compare favorably with confidence intervals obtained by traditional methods. The traditional intervals are seen to have coverage probability less than the nominal level in several instances, This method can be applied to all confidence interval problems and reduces to the traditional method when an exact pivotal statistic is known.     

Multiple comparisons with a control for exponential location parameters under heteroscedasticity

In this paper, a new design-oriented two-stage two-sided simultaneous confidence intervals, for comparing several exponential populations with control population in terms of location parameters under heteroscedasticity, are proposed. If there is a prior information that the location parameter of k exponential populations are not less than the location parameter of control population, one-sided simultaneous confidence intervals provide more inferential sensitivity than two-sided simultaneous confidence intervals. But the two-sided simultaneous confidence intervals have advantages over the one-sided simultaneous confidence intervals as they provide both lower and upper bounds for the parameters of interest. The proposed design-oriented two-stage two-sided simultaneous confidence intervals provide the benefits of both the two-stage one-sided and two-sided simultaneous confidence intervals. When the additional sample at the second stage may not be available due to the experimental budget shortage or other factors in an experiment, one-stage two-sided confidence intervals are proposed, which combine the advantages of one-stage one-sided and two-sided simultaneous confidence intervals. The critical constants are obtained using the techniques given in Lam [9,10]. These critical constant are compared with the critical constants obtained by Bonferroni inequality techniques and found that critical constant obtained by Lam [9,10] are less conservative than critical constants computed from the Bonferroni inequality technique. Implementation of the proposed simultaneous confidence intervals is demonstrated by a numerical example.     

Confidence and prediction intervals based on interpolated records

In several statistical problems, nonparametric confidence intervals for population quantiles can be constructed and their coverage probabilities can be computed exactly, but cannot in general be rendered equal to a pre-determined level. The same difficulty arises for coverage probabilities of nonparametric prediction intervals for future observations. One solution to this difficulty is to interpolate between intervals which have the closest coverage probability from above and below to the pre-determined level. In this paper, confidence intervals for population quantiles are constructed based on interpolated upper and lower records. Subsequently, prediction intervals are obtained for future upper records based on interpolated upper records. Additionally, we derive upper bounds for the coverage error of these confidence and prediction intervals. Finally, our results are applied to some real data sets. Also, a comparison via a simulation study is done with similar classical intervals obtained before.     

Editorial collaborators

Confidence intervals obtained by bootstrap methods and normal approximation are compared, based on output data from terminating and steady-state simulations. Bootstrap intervals are equal or better than normal approximation intervals in actual probability coverages. Furthermore, bootstrap methods capture the skewness in the distribution of outputs and, therefore, are more desirable than normal approximation.     

Confidence intervals for treatment effect from restricted maximum likelihood

An explicit form of confidence intervals for the treatment effect in random effects meta-analysis model obtained from Harville–Jeske–Kenward–Roger approach is given. These restricted likelihood based intervals are compared to alternative procedures commonly used in collaborative studies when the number of participants is small and study-specific variances are heterogeneous. Monte Carlo simulation experiments show that the former intervals have quite conservative coverage probabilities and favor the latter intervals.     

CONDITIONAL INFERENCE PROCEDURES FOR THE PARETO AND POWER FUNCTION DISTRIBUTIONS BASED ON TYPE-II RIGHT CENSORED SAMPLES

In this paper we consider conditional inference procedures for the Pareto and power function distributions. We develop procedures for obtaining confidence intervals for the location and scale parameters as well as upper and lower n probability tolerance intervals for a proportion g, given a Type-II right censored sample from the corresponding distribution. The intervals are exact, and are obtained by conditioning on the observed values of the ancillary statistics. Since, for each distribution, the procedures assume that a shape parameter x is known, a sensitivity analysis is also carried out to see how the procedures are affected by changes in x.     

Reliability estimation in multicomponent stress–strength model for Topp-Leone distribution

In this paper, we consider the estimation reliability in multicomponent stress-strength (MSS) model when both the stress and strengths are drawn from Topp-Leone (TL) distribution. The maximum likelihood (ML) and Bayesian methods are used in the estimation procedure. Bayesian estimates are obtained by using Lindley’s approximation and Gibbs sampling methods, since they cannot be obtained in explicit form in the context of TL. The asymptotic confidence intervals are constructed based on the ML estimators. The Bayesian credible intervals are also constructed using Gibbs sampling. The reliability estimates are compared via an extensive Monte-Carlo simulation study. Finally, a real data set is analysed for illustrative purposes.     

Distribution-free prediction intervals for the future current record statistics

Prediction of records plays an important role in many applications, such as, meteorology, hydrology, industrial stress testing and athletic events. In this paper, based on the observed current records of an iid sequence sample drawn from an arbitrary unknown distribution, we develop distribution-free prediction intervals as well as prediction upper and lower bounds for current records from another iid sequence. We also present sharp upper bounds for the expected lengths of the so obtained prediction intervals. Numerical computations of the coverage probabilities are presented for choosing the appropriate limits of the prediction intervals.      

Usual and Shortest Confidence Intervals on Odds Ratios from Logistic Regression

Based on the large-sample normal distribution of the sample log odds ratio and its asymptotic variance from maximum likelihood logistic regression, shortest 95% confidence intervals for the odds ratio are developed. Although the usual confidence interval on the odds ratio is unbiased, the shortest interval is not. That is, while covering the true odds ratio with the stated probability, the shortest interval covers some values below the true odds ratio with higher probability. The upper and lower limits of the shortest interval are shifted to the left of those of the usual interval, with greater shifts in the upper limits. With the log odds model γ + Xβ, in which X is binary, simulation studies showed that the approximate average percent difference in length is 7.4% for n (sample size) = 100, and 3.8% for n = 200. Precise estimates of the covering probabilities of the two types of intervals were obtained from simulation studies, and are compared graphically. For odds ratio estimates greater (less) than one, shortest intervals are more (less) likely to include one than are the usual intervals. The usual intervals are likelihood-based and the shortest intervals are not. The usual intervals have minimum expected length among the class of unbiased intervals. Shortest intervals do not provide important advantages over the usual intervals, which we recommend for practical use.     

Exact Likelihood Inference for k Exponential Populations Under Joint Type-II Censoring

In this paper, when a jointly Type-II censored sample arising from k independent exponential populations is available, the conditional MLEs of the k exponential mean parameters are derived. The moment generating functions and the exact densities of these MLEs are obtained using which exact confidence intervals are developed for the parameters. Moreover, approximate confidence intervals based on the asymptotic normality of the MLEs and credible confidence regions from a Bayesian viewpoint are also discussed. An empirical comparison of the exact, approximate, bootstrap, and Bayesian intervals is also made in terms of coverage probabilities. Finally, an example is presented in order to illustrate all the methods of inference developed here.     

Asymptotic Confidence Intervals for a New Inequality Measure

This work aims at assessing, by simulation methods, the performance of asymptotic confidence intervals for Zenga's new inequality measure. The results are compared with those obtained on Gini's measure, perhaps the most widely used index for measuring inequality in income and wealth distributions. Our findings show that the coverage accuracy and the size of the confidence intervals for the two measures are very similar in samples from economic size distributions.     

Exact Likelihood Inference for k Exponential Populations Under Joint Progressive Type-II Censoring

Comparative lifetime experiments are of great importance when the interest is in ascertaining the relative merits of k competing products with regard to their reliability. In this paper, when a joint progressively Type-II censored sample arising from k independent exponential populations is available, the conditional MLEs of the k exponential mean parameters are derived. Their conditional moment generating functions and exact densities are obtained, using which exact confidence intervals are developed for the parameters. Moreover, approximate confidence intervals based on the asymptotic normality of the MLEs and credible confidence regions from a Bayesian viewpoint are discussed. An empirical evaluation of the exact, approximate, bootstrap, and Bayesian intervals is also made in terms of coverage probabilities and average widths. Finally, an example is presented in order to illustrate all the methods of inference developed here.     

New Confidence Intervals for the Proportion of Interest in One-Sample Correlated Binary Data

Correlated binary data is obtained in many fields of biomedical research. When constructing a confidence interval for the proportion of interest, asymptotic confidence intervals have already been developed. However, such asymptotic confidence intervals are unreliable in small samples. To improve the performance of asymptotic confidence intervals in small samples, we obtain the Edgeworth expansion of the distribution of the studentized mean of beta-binomial random variables. Then, we propose new asymptotic confidence intervals by correcting the skewness in the Edgeworth expansion in one direct and two indirect ways. New confidence intervals are compared with the existing confidence intervals in simulation studies.     

On the simple step-stress model for two-parameter exponential distribution

In this paper, we consider the simple step-stress model for a two-parameter exponential distribution, when both the parameters are unknown and the data are Type-II censored. It is assumed that under two different stress levels, the scale parameter only changes but the location parameter remains unchanged. It is observed that the maximum likelihood estimators do not always exist. We obtain the maximum likelihood estimates of the unknown parameters whenever they exist. We provide the exact conditional distributions of the maximum likelihood estimators of the scale parameters. Since the construction of the exact confidence intervals is very difficult from the conditional distributions, we propose to use the observed Fisher Information matrix for this purpose. We have suggested to use the bootstrap method for constructing confidence intervals. Bayes estimates and associated credible intervals are obtained using the importance sampling technique. Extensive simulations are performed to compare the performances of the different confidence and credible intervals in terms of their coverage percentages and average lengths. The performances of the bootstrap confidence intervals are quite satisfactory even for small sample sizes.     

Point and Interval Estimation for Bivariate Normal Distribution Based on Progressively Type-II Censored Data

ABSTRACT

The maximum likelihood estimates (MLEs) of parameters of a bivariate normal distribution are derived based on progressively Type-II censored data. The asymptotic variances and covariances of the MLEs are derived from the Fisher information matrix. Using the asymptotic normality of MLEs and the asymptotic variances and covariances derived from the Fisher information matrix, interval estimation of the parameters is discussed and the probability coverages of the 90% and 95% confidence intervals for all the parameters are then evaluated by means of Monte Carlo simulations. To improve the probability coverages of the confidence intervals, especially for the correlation coefficient, sample-based Monte Carlo percentage points are determined and the probability coverages of the 90% and 95% confidence intervals obtained using these percentage points are evaluated and shown to be quite satisfactory. Finally, an illustrative example is presented.     

Distribution-free confidence bounds for pr y< when fx and gy = fx-6 are continuous and symmetric

For and continuous and symmetric and differing at most by a shift parameter, distribution-free confidence intervals for are obtained by means of the Chebyshev inequality and an upper bound for the variance of the Mann-Whitney statistic. The (two-sided) intervals are reliable for small samples and about 20 to 30 per cent shorter than those obtained by Ury for and completely unknown for equal sample sizes, with larger savings otherwise. They are also shorter than the upper bounds obtained by Birnbaum and McCarty (1958) when the confidence coefficient does not exceed 0.95.     

Reliability of stress–strength model for exponentiated Pareto distributions

In this paper, the reliability of a system is discussed when the strength of the system and the stress imposed on it are independent, non-identical exponentiated Pareto distributed random variables. Different point estimations and interval estimations are proposed. The point estimators obtained are maximum likelihood, uniformly minimum variance unbiased and Bayesian estimators. The interval estimations obtained are approximate, exact, bootstrap-p and bootstrap-t confidence intervals and Bayesian credible interval. Different methods and the corresponding confidence intervals are compared using Monte-carlo simulations.     

Data-based interval estimation of classification error rates

Leave-one-out and 632 bootstrap are popular data-based methods of estimating the true error rate of a classification rule, but practical applications almost exclusively quote only point estimates. Interval estimation would provide better assessment of the future performance of the rule, but little has been published on this topic. We first review general-purpose jackknife and bootstrap methodology that can be used in conjunction with leave-one-out estimates to provide prediction intervals for true error rates of classification rules. Monte Carlo simulation is then used to investigate coverage rates of the resulting intervals for normal data, but the results are disappointing; standard intervals show considerable overinclusion, intervals based on Edgeworth approximations or random weighting do not perform well, and while a bootstrap approach provides intervals with coverage rates closer to the nominal ones there is still marked underinclusion. We then turn to intervals constructed from 632 bootstrap estimates, and show that much better results are obtained. Although there is now some overinclusion, particularly for large training samples, the actual coverage rates are sufficiently close to the nominal rates for the method to be recommended. An application to real data illustrates the considerable variability that can arise in practical estimation of error rates.     

Some Further Issues Concerning Likelihood Inference for Left Truncated and Right Censored Lognormal Data

The maximum likelihood estimates (MLEs) of the parameters of a two-parameter lognormal distribution with left truncation and right censoring are developed through the Expectation Maximization (EM) algorithm. For comparative purpose, the MLEs are also obtained by the Newton–Raphson method. The asymptotic variance-covariance matrix of the MLEs is obtained by using the missing information principle, under the EM framework. Then, using asymptotic normality of the MLEs, asymptotic confidence intervals for the parameters are constructed. Asymptotic confidence intervals are also obtained using the estimated variance of the MLEs by the observed information matrix, and by using parametric bootstrap technique. Different confidence intervals are then compared in terms of coverage probabilities, through a Monte Carlo simulation study. A prediction problem concerning the future lifetime of a right censored unit is also considered. A numerical example is given to illustrate all the inferential methods developed here.     

Estimation and prediction for Chen distribution with bathtub shape under progressive censoring

We consider estimation of the unknown parameters of Chen distribution [Chen Z. A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Statist Probab Lett. 2000;49:155–161] with bathtub shape using progressive-censored samples. We obtain maximum likelihood estimates by making use of an expectation–maximization algorithm. Different Bayes estimates are derived under squared error and balanced squared error loss functions. It is observed that the associated posterior distribution appears in an intractable form. So we have used an approximation method to compute these estimates. A Metropolis–Hasting algorithm is also proposed and some more approximate Bayes estimates are obtained. Asymptotic confidence interval is constructed using observed Fisher information matrix. Bootstrap intervals are proposed as well. Sample generated from MH algorithm are further used in the construction of HPD intervals. Finally, we have obtained prediction intervals and estimates for future observations in one- and two-sample situations. A numerical study is conducted to compare the performance of proposed methods using simulations. Finally, we analyse real data sets for illustration purposes.