Generalized Confidence Interval for the Scale Parameter of the Power-Law Process

The power-law process is widely used in the analysis of repairable system reliability. In this article, interval estimation for the scale parameter is investigated under some general conditions. A procedure to derive a generalized confidence interval for the scale parameter is presented. We also study the accuracy of the generalized confidence interval by Monte Carlo simulation. Finally, two examples are shown to illustrate the proposed procedure.     

Confidence intervals for the slope of a linear functional relationship

Creasy (1956) obtained confidence limits for the slope of a bivariate linear structural relationship. This paper shows that her results are also applicable to the linear functional relationship.     

Random Weighting Estimation of Confidence Intervals for Quantiles

This paper presents a new random weighting method for confidence interval estimation for the sample ‐quantile. A theory is established to extend ordinary random weighting estimation from a non‐smoothed function to a smoothed function, such as a kernel function. Based on this theory, a confidence interval is derived using the concept of backward critical points. The resultant confidence interval has the same length as that derived by ordinary random weighting estimation, but is distribution‐free, and thus it is much more suitable for practical applications. Simulation results demonstrate that the proposed random weighting method has higher accuracy than the Bootstrap method for confidence interval estimation.     

Confidence Intervals for a Binomial Parameter after Observing No Successes

The easily computed, one-sided confidence interval for the binomial parameter provides the basis for an interesting classroom example of scientific thinking and its relationship to confidence intervals. The upper limit can be represented as the sample proportion from a number of “successes” in a future experiment of the same sample size. The upper limit reported by most people corresponds closely to that producing a 95 percent classical confidence interval and has a Bayesian interpretation.     

Confidence Intervals for the Hyperparameters in Structural Models

This article deals with the bootstrap as an alternative method to construct confidence intervals for the hyperparameters of structural models. The bootstrap procedure considered is the classical nonparametric bootstrap in the residuals of the fitted model using a well-known approach. The performance of this procedure is empirically obtained through Monte Carlo simulations implemented in Ox. Asymptotic and percentile bootstrap confidence intervals for the hyperparameters are built and compared by means of the coverage percentages. The results are similar but the bootstrap procedure is better for small sample sizes. The methods are applied to a real time series and confidence intervals are built for the hyperparameters.     

Robust Confidence Intervals for the Bernoulli Parameter

Despite the simplicity of the Bernoulli process, developing good confidence interval procedures for its parameter—the probability of success p—is deceptively difficult. The binary data yield a discrete number of successes from a discrete number of trials, n. This discreteness results in actual coverage probabilities that oscillate with the n for fixed values of p (and with p for fixed n). Moreover, this oscillation necessitates a large sample size to guarantee a good coverage probability when p is close to 0 or 1.

It is well known that the Wilson procedure is superior to many existing procedures because it is less sensitive to p than any other procedures, therefore it is less costly. The procedures proposed in this article work as well as the Wilson procedure when 0.1 ≤p ≤ 0.9, and are even less sensitive (i.e., more robust) than the Wilson procedure when p is close to 0 or 1. Specifically, when the nominal coverage probability is 0.95, the Wilson procedure requires a sample size 1, 021 to guarantee that the coverage probabilities stay above 0.92 for any 0.001 ≤ min {p, 1 ?p} <0.01. By contrast, our procedures guarantee the same coverage probabilities but only need a sample size 177 without increasing either the expected interval width or the standard deviation of the interval width.     

Confidence Intervals for the Current Status Model

We discuss a new way of constructing pointwise confidence intervals for the distribution function in the current status model. The confidence intervals are based on the smoothed maximum likelihood estimator, using local smooth functional theory and normal limit distributions. Bootstrap methods for constructing these intervals are considered. Other methods to construct confidence intervals, using the non‐standard limit distribution of the (restricted) maximum likelihood estimator, are compared with our approach via simulations and real data applications.     

Confidence intervals based on the deviance statistic for the hyperparameters in state space models

The main objective of this work is to evaluate the performance of confidence intervals, built using the deviance statistic, for the hyperparameters of state space models. The first procedure is a marginal approximation to confidence regions, based on the likelihood test, and the second one is based on the signed root deviance profile. Those methods are computationally efficient and are not affected by problems such as intervals with limits outside the parameter space, which can be the case when the focus is on the variances of the errors. The procedures are compared to the usual approaches existing in the literature, which includes the method based on the asymptotic distribution of the maximum likelihood estimator, as well as bootstrap confidence intervals. The comparison is performed via a Monte Carlo study, in order to establish empirically the advantages and disadvantages of each method. The results show that the methods based on the deviance statistic possess a better coverage rate than the asymptotic and bootstrap procedures.     

A Wavelet Method for Confidence Intervals in the Density Model

Abstract.  We present a wavelet procedure for defining confidence intervals for f ( x ₀), where x ₀ is a given point and f is an unknown density from which there are independent observations. We use an undersmoothing method which is shown to be near optimal (up to a logarithmic term) in a first order sense. We propose a second order correction using the Edgeworth expansion. The adaptation with respect to the unknown regularity of f is given via a Lepskii type algorithm and has the advantage to be well located. The theoretical results are proved under weak assumptions and concern very irregular or oscillating functions. An empirical study gives some hints for choosing the constant of the threshold level. The results are very encouraging for the length of the intervals as well as for the coverage accuracy.     

Confidence Intervals for Current Status Data

Abstract.  The likelihood ratio statistic for testing pointwise hypotheses about the survival time distribution in the current status model can be inverted to yield confidence intervals (CIs). One advantage of this procedure is that CIs can be formed without estimating the unknown parameters that figure in the asymptotic distribution of the maximum likelihood estimator (MLE) of the distribution function. We discuss the likelihood ratio-based CIs for the distribution function and the quantile function and compare these intervals to several different intervals based on the MLE. The quantiles of the limiting distribution of the MLE are estimated using various methods including parametric fitting, kernel smoothing and subsampling techniques. Comparisons are carried out both for simulated data and on a data set involving time to immunization against rubella. The comparisons indicate that the likelihood ratio-based intervals are preferable from several perspectives.     

Revisiting Beal's Confidence Intervals for the Difference of Two Binomial Proportions

Confidence interval construction for the difference of two independent binomial proportions is a well-known problem with a full panoply of proposed solutions. In this paper, we focus largely on the family of intervals proposed by Beal (1987 Beal , S. ( 1987 ). Asymptotic confidence intervals for the difference between two binomial parameters for use with small samples . Biometrics 43 : 941 – 950 . [CSA] [CROSSREF] [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). This family, which includes the Haldane and Jeffreys–Perks intervals as special cases, assumes a symmetric prior distribution for the population proportions p ₁ and p ₂. We propose new methods that allow the currently observed data to set the prior distribution by taking a parametric empirical-Bayes approach; in addition, we also provide an investigation of the new interval' behaviors in small-sample situations. Unlike other solutions, our intervals can be used adaptively for experiments conducted in multiple stages over time. We illustrate this notion using data from an Argentinean study involving the Mal Rio Cuarto virus and its transmission to susceptible maize crops.     

Some Small-Sample Properties of the Student t Confidence Intervals

For the two-sided Student t confidence interval for the mean of a normal distribution there is, for any sample size, a sufficiently large confidence level that ensures that the interval covers all the observations; there are also sufficiently small confidence levels guaranteeing, respectively, that (a) the interval does not cover all the observations and (b) the interval lies within the extreme observations. Necessary and sufficient conditions are also obtained for the width of the confidence interval to always exceed the sample range, as well as for the reverse inequality. Some implications of the results are discussed.     

关于捕获再捕获抽样的置信区间

捕获再捕获抽样是一种应用广泛的抽样方法。运用随机模拟,研究捕获再捕获抽样的三种估计量的均值、方差、偏度、峰度、利用近似正态分布构造的置信区间的统计性质。改进了估计量的样本方差计算公式,使得利用近似正态分布构造的置信区间更优。     

Simultaneous confidence bands and regions for log-location-scale distributions with censored data

In many areas of application, especially life testing and reliability, it is often of interest to estimate an unknown cumulative distribution (cdf). A simultaneous confidence band (SCB) of the cdf can be used to assess the statistical uncertainty of the estimated cdf over the entire range of the distribution. Cheng and Iles [1983. Confidence bands for cumulative distribution functions of continuous random variables. Technometrics 25 (1), 77–86] presented an approach to construct an SCB for the cdf of a continuous random variable. For the log-location-scale family of distributions, they gave explicit forms for the upper and lower boundaries of the SCB based on expected information. In this article, we extend the work of Cheng and Iles [1983. Confidence bands for cumulative distribution functions of continuous random variables. Technometrics 25 (1), 77–86] in several directions. We study the SCBs based on local information, expected information, and estimated expected information for both the “cdf method” and the “quantile method.” We also study the effects of exceptional cases where a simple SCB does not exist. We describe calibration of the bands to provide exact coverage for complete data and type II censoring and better approximate coverage for other kinds of censoring. We also discuss how to extend these procedures to regression analysis.     

Approximate Bayesianity of Frequentist Confidence Intervals for a Binomial Proportion

The well-known Wilson and Agresti–Coull confidence intervals for a binomial proportion p are centered around a Bayesian estimator. Using this as a starting point, similarities between frequentist confidence intervals for proportions and Bayesian credible intervals based on low-informative priors are studied using asymptotic expansions. A Bayesian motivation for a large class of frequentist confidence intervals is provided. It is shown that the likelihood ratio interval for p approximates a Bayesian credible interval based on Kerman’s neutral noninformative conjugate prior up to O(n^{? 1}) in the confidence bounds. For the significance level α ? 0.317, the Bayesian interval based on the Jeffreys’ prior is then shown to be a compromise between the likelihood ratio and Wilson intervals. Supplementary materials for this article are available online.     

Asymptotic properties of mean cumulative function estimators from window-observation recurrence data

A variety of nonparametric and parametric methods have been used to estimate the mean cumulative function (MCF) for the recurrence data collected from the counting process. When the recurrence histories of some units are available in disconnected observation windows with gaps in between, Zuo et al. (2008) showed that both the nonparametric and parametric methods can be extended to estimate the MCF. In this article, we establish some asymptotic properties of the MCF estimators for the window-observation recurrence data.     

Empirical Likelihood for Compound Poisson Processes

Let {N(t), t > 0} be a Poisson process with rate λ > 0, independent of the independent and identically distributed random variables with mean μ and variance . The stochastic process is then called a compound Poisson process and has a wide range of applications in, for example, physics, mining, finance and risk management. Among these applications, the average number of objects, which is defined to be λμ, is an important quantity. Although many papers have been devoted to the estimation of λμ in the literature, in this paper, we use the well‐known empirical likelihood method to construct confidence intervals. The simulation results show that the empirical likelihood method often outperforms the normal approximation and Edgeworth expansion approaches in terms of coverage probabilities. A real data set concerning coal‐mining disasters is analyzed using these methods.     

Confidence intervals for the comparison of two poisson distributions

Confidence interval construction the difference in mean event rates for two Index independent , Poisson samples is discussed. Intervals are derived by considering Bayes estimates of the mean event rates using a family of noninformative priors. The coverage probabilities of the proposed are compared to those of the standard Wald interval for of observed events. A compromise method of constructing interval based on the data is suggested and its properties are evaluated. The method is illustrated in several examples.     

Exact Non-parametric Confidence Intervals for Quantiles with Progressive Type-II Censoring

It is shown how various exact non-parametric inferences based on order statistics in one or two random samples can be generalized to situations with progressive type-II censoring, which is a kind of evolutionary right censoring. Ordinary type-II right censoring is a special case of such progressive censoring. These inferences include confidence intervals for a given parent quantile, prediction intervals for a given order statistic of a future sample, and related two-sample inferences based on exceedance probabilities. The proposed inferences are valid for any parent distribution with continuous distribution function. The key result is that each observable uncensored order statistic that becomes available with progressive type-II censoring can be represented as a mixture with known weights of underlying ordinary order statistics. The importance of this mixture representation lies in that various properties of such observable order statistics can be deduced immediately from well-known properties of ordinary order statistics.     

Lower Confidence Limits on the Generalized Exponential Distribution Percentiles Under Progressive Type-I Interval Censoring

In industrial life test and survival analysis, the percentile estimation is always a practical issue with lower confidence bound required for maintenance purpose. Sampling distributions for the maximum likelihood estimators of percentiles are usually unknown. Bootstrap procedures are common ways to estimate the unknown sampling distributions. Five parametric bootstrap procedures are proposed to estimate the confidence lower bounds on maximum likelihood estimators for the generalized exponential (GE) distribution percentiles under progressive type-I interval censoring. An intensive simulation is conducted to evaluate the performances of proposed procedures. Finally, an example of 112 patients with plasma cell myeloma is given for illustration.