Construction of fixed width confidence intervals for a Bernoulli success probability using sequential sampling: a simulation study

This article considers the construction of level 1?α fixed width 2d confidence intervals for a Bernoulli success probability p, assuming no prior knowledge about p and so p can be anywhere in the interval [0, 1]. It is shown that some fixed width 2d confidence intervals that combine sequential sampling of Hall [Asymptotic theory of triple sampling for sequential estimation of a mean, Ann. Stat. 9 (1981), pp. 1229–1238] and fixed-sample-size confidence intervals of Agresti and Coull [Approximate is better than ‘exact’ for interval estimation of binomial proportions, Am. Stat. 52 (1998), pp. 119–126], Wilson [Probable inference, the law of succession, and statistical inference, J. Am. Stat. Assoc. 22 (1927), pp. 209–212] and Brown et al. [Interval estimation for binomial proportion (with discussion), Stat. Sci. 16 (2001), pp. 101–133] have close to 1?α confidence level. These sequential confidence intervals require a much smaller sample size than a fixed-sample-size confidence interval. For the coin jamming example considered, a fixed-sample-size confidence interval requires a sample size of 9457, while a sequential confidence interval requires a sample size that rarely exceeds 2042.     

Estimating Process Capability Index C PM Using a Bootstrap Sequential Sampling Procedure

Construction of a confidence interval for process capability index C _PM is often based on a normal approximation with fixed sample size. In this article, we describe a different approach in constructing a fixed-width confidence interval for process capability index C _PM with a preassigned accuracy by using a combination of bootstrap and sequential sampling schemes. The optimal sample size required to achieve a preassigned confidence level is obtained using both two-stage and modified two-stage sequential procedures. The procedure developed is also validated using an extensive simulation study.     

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.     

Fixed-width sequential confidence interval for the correlation coefficient of a bivariate normal distribution

A sequential confidence interval of fixed width 2d d > 0, is constructed for the correlation coefficient of a bivariate normal distribution. It is shown that the coverage probability is approximately equal to a preassigned number γ, 0 < γ < as d → 0.     

Confidence sets based on the positive part James–Stein estimator with the asymptotically constant coverage probability

The asymptotic expansions for the coverage probability of a confidence set centred at the James–Stein estimator presented in our previous publications show that this probability depends on the non-centrality parameter τ² (the sum of the squares of the means of normal distributions). In this paper we establish how these expansions can be used for a construction of confidence region with constant confidence level, which is asymptotically (the same formula for both case τ→0 and τ→∞) equal to some fixed value 1?α. We establish the shrinkage rate for the confidence region according to the growth of the dimension p and also the value of τ for which we observe quick decreasing of the coverage probability to the nominal level 1?α. When p→∞ this value of τ increases as O(p^1/4). The accuracy of the results obtained is shown by the Monte-Carlo statistical simulations.     

Sequential Fixed-Width Confidence Interval for a Function of the Parameters

Let X₁…, X_m and Y₁…, Y_n be two independent sequences of i.i.d. random variables with distribution functions F_x(.|θ) and F_y(. | φ) respectively. Let g(θ, φ) be a real-valued function of the unknown parameters θ and φ. The purpose of this paper is to suggest a sequential procedure which gives a fixed-width confidence interval for g(θ, φ) so that the coverage probability is approximately α (preas-signed). Certain asymptotic optimality properties of the sequential procedure are established. A Monte Carlo study is presented.     

An alternative to confidence intervals constructed after a Hausman pretest in panel data

Consider panel data modelled by a linear random intercept model that includes a time‐varying covariate. Suppose that our aim is to construct a confidence interval for the slope parameter. Commonly, a Hausman pretest is used to decide whether this confidence interval is constructed using the random effects model or the fixed effects model. This post‐model‐selection confidence interval has the attractive features that it (a) is relatively short when the random effects model is correct and (b) reduces to the confidence interval based on the fixed effects model when the data and the random effects model are highly discordant. However, this confidence interval has the drawbacks that (i) its endpoints are discontinuous functions of the data and (ii) its minimum coverage can be far below its nominal coverage probability. We construct a new confidence interval that possesses these attractive features, but does not suffer from these drawbacks. This new confidence interval provides an intermediate between the post‐model‐selection confidence interval and the confidence interval obtained by always using the fixed effects model. The endpoints of the new confidence interval are smooth functions of the Hausman test statistic, whereas the endpoints of the post‐model‐selection confidence interval are discontinuous functions of this statistic.     

SOME PROPERTIES OF PROFILE BOOTSTRAP CONFIDENCE INTERVALS

We consider the problem of finding an equi-tailed confidence interval, with coverage probability (1-α), for a scalar parameter θ₀ in the presence of a (possibly infinite dimensional) nuisance parameter ψ₀. It is supposed that the value taken by θ₀ does not restrict the value that ψ₀ may take and vice-versa. Given a sensible estimate ψ_n of ψ₀, profile bootstrap confidence interval for θ₀ is defined to be the exact equi-tailed confidence interval with coverage probability (1-α) assuming that ψ₀=ψ_n. We compare the properties of the profile bootstrap confidence interval and the ordinary bootstrap confidence interval when they are based on studentised and unstudentised quantities. Under mild regularity conditions the profile bootstrap confidence interval is always a subset of the set of allowable values of θ₀ and is transformation-respecting when based on either an unstudentised quantity or a studentised quantity satisfying certain restrictions. As a confidence interval for the autoregressive parameter of an AR(1) process, the profile bootstrap confidence interval has important advantages over the ordinary bootstrap confidence interval based on a studentised quantity.     

Expansions of the coverage probabilities of prediction region based on a shrinkage estimator

An expansion formula for the coverage probability of prediction region based on a shrinkage estimator proposed by Joshi [Joshi, V. M. (1967). Inadmissibility of the usual confidence sets for the mean of a multivariate normal population. Ann. Math. Statist., 38, 1868–1875.] is obtained. Its error bound is evaluated in terms of a function of an unknown parameter. Applying this result, three types of asymptotic expansions are derived. These expansions show inadmissibility of the usual prediction region.     

Point estimation after early stopping in a repeated measures trial

When the individual measurements are statistically independent, the maximum likelihood estimator calculated at the end of a sequential procedure overestimates the underlying effect. There are many clinical trials in which we are interested in comparing changes in responses between two treatment groups sequentially. Lee and DeMets (1991, JASA 86, 757–762) proposed a group sequential method for comparing rates of change when a response variable is measured for eaeh patient at successive follow-up visits. They assumed that the response follows the linear mixed effects model and derived the asymptotic joint distribution of the sequentially computed statistics. In this article, we consider the maximum likelihood estimator (MLE), the median unbiased estimator (MUE) and the midpoint of a 100(1-α)% confidence interval as point estimators for the rate of change in the linear mixed effects model, and investigate their properties by Monte Carlo simulation.     

UNIFORM ASYMPTOTIC EXPANSIONS APPLIED TO SEQUENTIAL t - and t2-TESTS

Using some uniform asymptotic expansions for parabolic cylinder functions recently developed by Olver (1959), various integrals associated with the sequential t- and t² -tests are evaluated asymptotically in terms of the sample size. Then the continuation region inequalities for these tests are inverted and expressed in terms of well known test criteria. It should be pointed out that the inversion of the continuation regions in terms of the well known statistics yields forms for the sequential tests that are more easily applicable by the practitioner than the forms yielded by the method of Rushton. Furthermore, using these inequalities and the asymptotic normality of the test criteria, finite sure termination of sequential t- and t²-test procedures readily follow. Based on simulation studies, power comparisons of the two approximations are also made.     

Confidence Intervals for Quantiles of a Two-parameter Exponential Distribution under Progressive Type-II Censoring

Confidence intervals for the pth-quantile Q of a two-parameter exponential distribution provide useful information on the plausible range of Q, and only inefficient equal-tail confidence intervals have been discussed in the statistical literature so far. In this article, the construction of the shortest possible confidence interval within a family of two-sided confidence intervals is addressed. This shortest confidence interval is always shorter, and can be substantially shorter, than the corresponding equal-tail confidence interval. Furthermore, the computational intensity of both methodologies is similar, and therefore it is advantageous to use the shortest confidence interval. It is shown how the results provided in this paper can apply to data obtained from progressive Type II censoring, with standard Type II censoring as a special case. The applications of more complex confidence interval constructions through acceptance set inversions that can employ prior information are also discussed.     

ASYMPTOTIC EXPANSIONS APPLIED TO F-TEST CRITERIA

Using asymptotic expansions of the Kummer hypergeometric function, the sequential. F-test criterion is evaluated asymptotically in terms of the sample size. The continuation region inequalities for the test are inverted and expressed in terms of known test criteria. A rapidly converging algorithm for carrying out the sequential procedure is provided. This makes the F-test easier for the practitioner to use. Almost sure finite termination of the sequential. F-test is asserted by appealing to the continuation inequalities and a heuristic asymptotic expansion of the test criterion. Average stopping times of the sequential procedure for a variety of population means and population number configurations are tabulated. The computer symbolic manipulation program MAPLE was used to derive some formulae.     

Approximate Marginal Posterior Distributions Using Asymptotic Modes

This article gives asymptotic expansions for marginal posterior distributions with asymptotic modes of order n ^?2, and shows their validity. In addition, by using the asymptotic expansion, an approximate central posterior credible interval is derived.     

New bootstrap confidence intervals for means of positively skewed distributions

ABSTRACT

In this paper, we consider the problem of constructing non parametric confidence intervals for the mean of a positively skewed distribution. We suggest calibrated, smoothed bootstrap upper and lower percentile confidence intervals. For the theoretical properties, we show that the proposed one-sided confidence intervals have coverage probability α + O(n^{? 3/2}). This is an improvement upon the traditional bootstrap confidence intervals in terms of coverage probability. A version smoothed approach is also considered for constructing a two-sided confidence interval and its theoretical properties are also studied. A simulation study is performed to illustrate the performance of our confidence interval methods. We then apply the methods to a real data set.     

Confidence Interval for Shrinkage Parameters in Ridge Regression

In ridge regression, the estimation of ridge parameter k is an important problem. There are several methods available in the literature to do this job some what efficiently. However, no attempts were made to suggest a confidence interval for the ridge parameter using the knwoledge from the data. In this article, we propose a data dependent confidence interval for the ridge parameter k. The method of obtaining the confidence interval is illustrated with the help of a data set. A simulation study indicates that the empirical coverage probability of the suggested confidence intervals are quite high.     

Coverage‐adjusted Confidence Intervals for a Binomial Proportion

We consider the classic problem of interval estimation of a proportion p based on binomial sampling. The ‘exact’ Clopper–Pearson confidence interval for p is known to be unnecessarily conservative. We propose coverage adjustments of the Clopper–Pearson interval that incorporate prior or posterior beliefs into the interval. Using heatmap‐type plots for comparing confidence intervals, we show that the coverage‐adjusted intervals have satisfying coverage and shorter expected lengths than competing intervals found in the literature.     

On the robustness of empirical likelihood ratio confidence intervals for location

The authors examine the robustness of empirical likelihood ratio (ELR) confidence intervals for the mean and M‐estimate of location. They show that the ELR interval for the mean has an asymptotic breakdown point of zero. They also give a formula for computing the breakdown point of the ELR interval for M‐estimate. Through a numerical study, they further examine the relative advantages of the ELR interval to the commonly used confidence intervals based on the asymptotic distribution of the M‐estimate.     

Coverage accuracy for a mean without third moment

For constructing a confidence interval for the mean of a random variable with a known variance, one may prefer the sample mean standardized by the true standard deviation to the Student's t-statistic since the information of knowing the variance is used in the former way. In this paper, by comparing the leading error term in the expansion of the coverage probability, we show that the above statement is not true when the third moment is infinite. Our theory prefers the Student's t-statistic either when one-sided confidence intervals are considered for a heavier tail distribution or when two-sided confidence intervals are considered. Unlike other existing expansions for the Student's t-statistic, the derived explicit expansion for the case of infinite third moment can be used to estimate the coverage error so that bias correction becomes possible.     

Exact confidence limits for the probability of response in two-stage designs

In addition to point estimate for the probability of response in a two-stage design (e.g. Simon's two-stage design for binary endpoints), confidence limits should be computed and reported. The current method of inverting the p-value function to compute the confidence interval does not guarantee coverage probability in a two-stage setting. The existing exact approach to calculate one-sided limits is based on the overall number of responses to order the sample space. This approach could be conservative because many sample points have the same limits. We propose a new exact one-sided interval based on p-value for the sample space ordering. Exact intervals are computed by using binomial distributions directly, instead of a normal approximation. Both exact intervals preserve the nominal confidence level. The proposed exact interval based on the p-value generally performs better than the other exact interval with regard to expected length and simple average length of confidence intervals.