Constructing quantile confidence intervals using extended simple random sample in finite populations

This paper constructs quantile confidence intervals based on extended simple random sample (SRS) from a finite population, where ranks of population units are all known. Extended simple random sample borrows additional information from unmeasured observations in the population by conditioning on the population ranks of the measured units in SRS. The confidence intervals are improved using Rao-Blackwell theorem over the conditional distribution of sample ranks given the measured sample units. Empirical evidence shows that the proposed confidence intervals have shorter lengths than confidence intervals constructed from an SRS sample.     

Best exact nonparametric confidence intervals for quantiles

Well-known nonparametric confidence intervals for quantiles are of the form (X _{i : n} , X _{j : n} ) with suitably chosen order statistics X _{i : n} and X _{j : n} , but typically their coverage levels differ from those prescribed. It appears that the coverage level of the confidence interval of the form (X _{i : n} , X _{j : n} ) with random indices I and J can be rendered equal, exactly to any predetermined level γ?∈?(0, 1). Best in the sense of minimum E(J???I), i.e., ‘the shortest’, two-sided confidence intervals are constructed. If no two-sided confidence interval exists for a given γ, the most accurate one-sided confidence intervals are constructed.     

Better nonparametric bootstrap confidence intervals for the correlation coefficient

We respond to criticism leveled at bootstrap confidence intervals for the correlation coefficient by recent authors by arguing that in the correlation coefficient case, non–standard methods should be employed. We propose two such methods. The first is a bootstrap coverage coorection algorithm using iterated bootstrap techniques (Hall, 1986; Beran, 1987a; Hall and Martin, 1988) applied to ordinary percentile–method intervals (Efron, 1979), giving intervals with high coverage accuracy and stable lengths and endpoints. The simulation study carried out for this method gives results for sample sizes 8, 10, and 12 in three parent populations. The second technique involves the construction of percentile–t bootstrap confidence intervals for a transformed correlation coefficient, followed by an inversion of the transformation, to obtain “transformed percentile–t” intervals for the correlation coefficient. In particular, Fisher's z–transformation is used, and nonparametric delta method and jackknife variance estimates are used to Studentize the transformed correlation coefficient, with the jackknife–Studentized transformed percentile–t interval yielding the better coverage accuracy, in general. Percentile–t intervals constructed without first using the transformation perform very poorly, having large expected lengths and erratically fluctuating endpoints. The simulation study illustrating this technique gives results for sample sizes 10, 15 and 20 in four parent populations. Our techniques provide confidence intervals for the correlation coefficient which have good coverage accuracy (unlike ordinary percentile intervals), and stable lengths and endpoints (unlike ordinary percentile–t intervals).     

Empirical likelihood confidence intervals for nonparametric functional data analysis

We consider the problem of constructing confidence intervals for nonparametric functional data analysis using empirical likelihood. In this doubly infinite-dimensional context, we demonstrate the Wilk's phenomenon and propose a bias-corrected construction that requires neither undersmoothing nor direct bias estimation. We also extend our results to partially linear regression models involving functional data. Our numerical results demonstrate improved performance of the empirical likelihood methods over normal approximation-based methods.     

Robust regression and small sample confidence intervals

In assessing the behavior of robust regression estimates, the techniques of small sample asymptotics can be very useful. The results reported in this paper demonstrate the use of the small sample techniques in the construction of confidence intervals for robust regression. Several contrasting approaches are discussed and some numerical results are presented.     

Calculating nonparametric confidence intervals for quantiles using fractional order statistics

In this paper, we provide an easy-to-program algorithm for constructing the preselected 100(1 - alpha)% nonparametric confidence interval for an arbitrary quantile, such as the median or quartile, by approximating the distribution of the linear interpolation estimator of the quantile function Q L ( u ) = (1 - epsilon) X [ n u ] + epsilon X [ n u ] + 1 with the distribution of the fractional order statistic Q I ( u ) = Xn u , as defined by Stigler, where n = n + 1 and [ . ] denotes the floor function. A simulation study verifies the accuracy of the coverage probabilities. An application to the extreme-value problem in flood data analysis in hydrology is illustrated.     

Smoothed empirical likelihood confidence intervals for quantile regression parameters with auxiliary information

This paper develops a smoothed empirical likelihood (SEL)-based method to construct confidence intervals for quantile regression parameters with auxiliary information. First, we define the SEL ratio and show that it follows a Chi-square distribution. We then construct confidence intervals according to this ratio. Finally, Monte Carlo experiments are employed to evaluate the proposed method.     

Confidence intervals for nonparametric quantile regression: an emphasis on smoothing splines approach

In this paper we consider the problem of constructing confidence intervals for nonparametric quantile regression with an emphasis on smoothing splines. The mean‐based approaches for smoothing splines of Wahba (1983) and Nychka (1988) may not be efficient for constructing confidence intervals for the underlying function when the observed data are non‐Gaussian distributed, for instance if they are skewed or heavy‐tailed. This paper proposes a method of constructing confidence intervals for the unknown τth quantile function (0<τ<1) based on smoothing splines. In this paper we investigate the extent to which the proposed estimator provides the desired coverage probability. In addition, an improvement based on a local smoothing parameter that provides more uniform pointwise coverage is developed. The results from numerical studies including a simulation study and real data analysis demonstrate the promising empirical properties of the proposed approach.     

Pmc-optimal nonparametric quantile estimator

According to Pitman's Measure of Closeness, if T₁and T₂are two estimators of a real parameter $[d], then T₁is better than T₂if Po[d]{T₁-o[d] < T₂-0[d]} > 1/2 for all 0[d]. It may however happen that while T₁is better than T₂and T₂is better than T₃, T₃is better than T₁. Given q ? (0,1) and a sample X₁, X₂, ..., X_nfrom an unknown F ? F, an estimator T* = T*(X₁,X₂...X_n)of the q-th quantile of the distribution F is constructed such that P_F{F(T*)-q <[d] F(T)-q} >[d] 1/2 for all F?F and for all T€T, where F is a nonparametric family of distributions and T is a class of estimators. It is shown that T* =X_j:n'for a suitably chosen jth order statistic.     

Nonparametric confidence intervals for quantiles and quantile intervals from inversely sampled record breaking data in a multi-sample plan

There are a number of situations in which an observation is retained only if it is a record value, which include studies in industrial quality control experiments, destructive stress testing, meteorology, hydrology, seismology, athletic events and mining. When the number of records is fixed in advance, the data are referred to as inversely sampled record-breaking data. In this paper, we study the problems of constructing the nonparametric confidence intervals for quantiles and quantile intervals of the parent distribution based on record data. For a single record-breaking sample, the confidence coefficients of the confidence intervals for the pth quantile cannot exceed p and 1?p, on the basis of upper and lower records, respectively; hence, replication is required. So, we develop the procedure based on k independent record-breaking samples. Various cases have been studied and in each case, the optimal k and the exact nonparametric confidence intervals are obtained, and exact expressions for the confidence coefficients of these confidence intervals are derived. Finally, the results are illustrated by numerical computations.     

A comment on sample size calculations for binomial confidence intervals

In this article we examine sample size calculations for a binomial proportion based on the confidence interval width of the Agresti–Coull, Wald and Wilson Score intervals. We pointed out that the commonly used methods based on known and fixed standard errors cannot guarantee the desired confidence interval width given a hypothesized proportion. Therefore, a new adjusted sample size calculation method was introduced, which is based on the conditional expectation of the width of the confidence interval given the hypothesized proportion. With the reduced sample size, the coverage probability can still maintain at the nominal level and is very competitive to the converge probability for the original sample size.     

Small sample performance of jackknife confidence intervals for the james-stein estimator

The primary goal of this paper is to examine the small sample coverage probability and size of jackknife confidence intervals centered at a Stein-rule estimator. A Monte Carlo experiment is used to explore the coverage probabilities and lengths of nominal 90% and 95% delete-one and infinitesimal jackknife confidence intervals centered at the Stein-rule estimator; these are compared to those obtained using a bootstrap procedure.     

Expanded confidence intervals

Confidence intervals for location parameters are expanded (in either direction) to some “crucial” points and the resulting increase in the confidence coefficient investigated.Particaular crucial points are chosen to illuminate some hypothesis testing problems.Special results are dervied for the normal distribution with estimated variance and, in particular, for the problem of classifiying treatments as better or worse than a control.For this problem the usual two-sided Dunnett procedure is seen to be inefficient.Suggestions are made for the use of already published tables for this problem.Mention is made of the use of expanded confidence intervals for all pairwise comparisons of treatments using an “honest ordering difference” rather than Tukey's “honest siginificant difference”.     

Exact nonparametric confidence,prediction and tolerance intervals based on multi-sample Type-II right censored data

Exact nonparametric inference based on ordinary Type-II right censored samples has been extended here to the situation when there are multiple samples with Type-II censoring from a common continuous distribution. It is shown that marginally, the order statistics from the pooled sample are mixtures of the usual order statistics with multivariate hypergeometric weights. Relevant formulas are then derived for the construction of nonparametric confidence intervals for population quantiles, prediction intervals, and tolerance intervals in terms of these pooled order statistics. It is also shown that this pooled-sample approach assists in achieving higher confidence levels when estimating large quantiles as compared to a single Type-II censored sample with same number of observations from a sample of comparable size. We also present some examples to illustrate all the methods of inference developed here.     

Parametric and nonparametric confidence intervals for estimating the difference of means of two skewed populations

In this paper, we have reviewed and proposed several interval estimators for estimating the difference of means of two skewed populations. Estimators include the ordinary-t, two versions proposed by Welch [17] and Satterthwaite [15], three versions proposed by Zhou and Dinh [18], Johnson [9], Hall [8], empirical likelihood (EL), bootstrap version of EL, median t proposed by Baklizi and Kibria [2] and bootstrap version of median t. A Monte Carlo simulation study has been conducted to compare the performance of the proposed interval estimators. Some real life health related data have been considered to illustrate the application of the paper. Based on our findings, some possible good interval estimators for estimating the mean difference of two populations have been recommended for the researchers.     

Charting small sample characteristics of asymptotic confidence intervals for the binomial parameterp

Small sample properties of seven confidence intervals for the binomial parameterp (based on various normal approximations) and of the Clopper-Pearson interval are compared. Coverage probabilities and expected lower and upper limits of the intervals are graphically displayed as functions of the binomial parameterp for various sample sizes.     

Small sample confidence intervals for survival functions under the proportional hazards model

We develop a saddlepoint-based method for generating small sample confidence bands for the population surviival function from the Kaplan-Meier (KM), the product limit (PL), and Abdushukurov-Cheng-Lin (ACL) survival function estimators, under the proportional hazards model. In the process we derive the exact distribution of these estimators and developed mid-ppopulation tolerance bands for said estimators. Our saddlepoint method depends upon the Mellin transform of the zero-truncated survival estimator which we derive for the KM, PL, and ACL estimators. These transforms are inverted via saddlepoint approximations to yield highly accurate approximations to the cumulative distribution functions of the respective cumulative hazard function estimators and these distribution functions are then inverted to produce our saddlepoint confidence bands. For the KM, PL and ACL estimators we compare our saddlepoint confidence bands with those obtained from competing large sample methods as well as those obtained from the exact distribution. In our simulation studies we found that the saddlepoint confidence bands are very close to the confidence bands derived from the exact distribution, while being much easier to compute, and outperform the competing large sample methods in terms of coverage probability.     

A nonparametric coherent confidence procedure

ABSTRACT

Holm's step-down testing procedure starts with the smallest p-value and sequentially screens larger p-values without any information on confidence intervals. This article changes the conventional step-down testing framework by presenting a nonparametric procedure that starts with the largest p-value and sequentially screens smaller p-values in a step-by-step manner to construct a set of simultaneous confidence sets. We use a partitioning approach to prove that the new procedure controls the simultaneous confidence level (thus strongly controlling the familywise error rate). Discernible features of the new stepwise procedure include consistency with individual inference, coherence, and confidence estimations for follow-up investigations. In a simple simulation study, the proposed procedure (treated as a testing procedure), is more powerful than Holm's procedure when the correlation coefficient is large, and vice versa when it is small. In the data analysis of a medical study, the new procedure is able to detect the efficacy of Aspirin as a cardiovascular prophylaxis in a nonparametric setting.