Bootstrap confidence intervals

Nonparametric confidence bounds are obtained for a wide class of statistics using bootstrap. These results improve the errors in the probability estimates of the confidence intervals over the ones obtained by the normal approximation theory unconditionally.     

Coverage of generalized confidence intervals

Generalized confidence intervals provide confidence intervals for complicated parametric functions in many common practical problems. They do not have exact frequentist coverage in general, but often provide coverage close to the nominal value and have the correct asymptotic coverage. However, in many applications generalized confidence intervals do not have satisfactory finite sample performance. We derive expansions of coverage probabilities of one-sided generalized confidence intervals and use the expansions to explain the nonuniform performance of the generalized intervals. We then show how to use these expansions to obtain improved coverage by suitable calibration. The benefits of the proposed modification are illustrated via several examples.     

Improved confidence intervals for quantiles

We derive the Edgeworth expansion for the studentized version of the kernel quantile estimator. Inverting the expansion allows us to get very accurate confidence intervals for the pth quantile under general conditions. The results are applicable in practice to improve inference for quantiles when sample sizes are moderate.     

A note on studentized confidence intervals for the change-point

We study an AMOC time series model with an abrupt change in the mean and dependent errors that fulfill certain mixing conditions. It is known how to construct resampling confidence intervals using blocking techniques, but so far no studentizing has been considered. A simulation study shows that we obtain better intervals by studentizing. When studentizing dependent data, we need to use flat-top kernels for the estimation of the asymptotic variance. It turns out that this estimator taking possible changes into account behaves much better than the corresponding Bartlett estimator. Since the asymptotic distribution of change-point statistics for time-series depends on this value, having a good estimator under the null as well as alternatives is also essential for testing problems.     

Exact confidence coefficients of simultaneous confidence intervals for multinomial proportions

Simultaneous confidence intervals for multinomial proportions are useful in many areas of science. Since 1964, approximate simultaneous 1-α confidence intervals have been proposed for multinomial proportions. Although at each point in the parameter space, these confidence sets have asymptotic 1-α coverage probability, the exact confidence coefficients of these simultaneous confidence intervals for a fixed sample size are unknown before.In this paper, we propose a procedure for calculating exact confidence coefficients for simultaneous confidence intervals of multinomial proportions for any fixed sample size. With this methodology, exact confidence coefficients can be clearly derived, and the point at which the infimum of the coverage probability occurs can be clearly identified.     

Sequential confidence intervals for time series

Sequential estimation of parameters for time series observations is considered. The Chow-Robbins procedure is extended and Wald’s identity is proven for such data. Various confidence bands are defined. These give simultaneous confidence intervals for a sequence of sample sizes.     

Generalized fiducial confidence intervals for extremes

The generalized Pareto distribution is relevant to many situations when modeling extremes of random variables. In particular, peaks over threshold data approximately follow the generalized Pareto distribution. We use a fiducial framework to perform inference on the parameters and the extreme quantiles of the generalized Pareto. This inference technique is demonstrated both when the threshold is a known and unknown parameter. Assuming the threshold is a known parameter resulted in fiducial intervals with good empirical properties and asymptotically correct coverage. Likewise, our simulation results suggest that the fiducial intervals and point estimates compare favorably to the competing methods seen in the literature. The proposed intervals for the extreme quantiles when the threshold is unknown also have good empirical properties regardless of the underlying distribution of the data. Comparisons to a similar Bayesian method suggest that the fiducial intervals have better coverage and are similar in length with fewer assumptions. In addition to simulation results, the proposed method is applied to a data set from the NASDAQ 100. The data set is analyzed using the fiducial approach and its competitors for both cases when the threshold is known and unknown. R code for our procedure can be downloaded at .     

Asymptotic confidence intervals for Poisson regression

Let (X,Y) be a R^d×N₀-valued random vector where the conditional distribution of Y given X=x is a Poisson distribution with mean m(x). We estimate m by a local polynomial kernel estimate defined by maximizing a localized log-likelihood function. We use this estimate of m(x) to estimate the conditional distribution of Y given X=x by a corresponding Poisson distribution and to construct confidence intervals of level α of Y given X=x. Under mild regularity conditions on m(x) and on the distribution of X we show strong convergence of the integrated L₁ distance between Poisson distribution and its estimate. We also demonstrate that the corresponding confidence interval has asymptotically (i.e., for sample size tending to infinity) level α, and that the probability that the length of this confidence interval deviates from the optimal length by more than one converges to zero with the number of samples tending to infinity.     

Empirical likelihood based confidence intervals for copulas

Copula as an effective way of modeling dependence has become more or less a standard tool in risk management, and a wide range of applications of copula models appear in the literature of economics, econometrics, insurance, finance, etc. How to estimate and test a copula plays an important role in practice, and both parametric and nonparametric methods have been studied in the literature. In this paper, we focus on interval estimation and propose an empirical likelihood based confidence interval for a copula. A simulation study and a real data analysis are conducted to compare the finite sample behavior of the proposed empirical likelihood method with the bootstrap method based on either the empirical copula estimator or the kernel smoothing copula estimator.     

Better saddlepoint confidence intervals via bootstrap calibration

Confidence interval construction for parameters of lattice distributions is considered. By using saddlepoint formulas and bootstrap calibration, we obtain relatively short intervals and bounds with coverage errors, in contrast with and coverage errors for normal theory intervals and bounds when the population distribution is absolutely continuous. Closed form solutions are also provided for the cases of binomial and Poisson distributions. The method is illustrated by some simulation results.

     

On some sequential simultaneous confidence intervals procedures

Summary The purpose of this paper is two-fold: (i) to extend the simultaneous confidence intervals procedures (SCIP) of Healy [7] along the lines of Chow and Robbins [3] and (ii) to develop certain robust non-parametric SCIP based on the results of Sen [10] and Sen and Ghosh [11]; the allied efficiency results are also presented. Work supported by the National Institutes of Health, Grant GM-12868.     

Simultaneous confidence intervals for several inverse Gaussian populations

In this research, we propose simultaneous confidence intervals for all pairwise comparisons of means from inverse Gaussian distribution. Our method is based on fiducial generalized pivotal quantities for vector parameters. We prove that the constructed confidence intervals have asymptotically correct coverage probabilities. Simulation results show that the simulated Type-I errors are close to the nominal level even for small samples. The proposed approach is illustrated by an example.     

Nonparametric confidence intervals for functions of several distributions

Let F=(F₁...F_k) denote k unknown distribution functions and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja% Gaeyypa0ZaaeWaaeaaceWGgbGbaKaadaWgaaWcbaGaaGymaaqabaGc% caGGUaGaaiOlaiaac6caceWGgbGbaKaadaWgaaWcbaGaam4Aaaqaba% aakiaawIcacaGLPaaaaaa!3E24!\[\hat F = \left( {\hat F_1 ...\hat F_k } \right)\] their sample (empirical) functions based on random samples from them of sizes n ₁, ..., n _k. Let T(F) be a real functional of F. The cumulants of T(% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja% aaaa!35B2!\[\hat F\]) are expanded in powers of the inverse of n, the minimum sample size. The Edgeworth and Cornish-Fisher expansions for both the standardized and Studentized forms of T(% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja% aaaa!35B2!\[\hat F\]) are then given together with confidence intervals for T(F) of level 1–+O(n^-j/2) for any given in (0, 1) and any given j. In particular, confidence intervals are given for linear combinations and ratios of the means and variances of different populations without assuming any parametric form for their distributions.     

Fixed-width confidence intervals for contrasts in the means

The sequential procedure developed by Bhargava and Srivastava (1973, J. Roy. Statist. Soc. Ser. B, 35, 147–152) to construct fixed-width confidence intervals for contrasts in the means is further analyzed. Second-order approximations for the first two moments of the stopping time and the coverage probability associated with the sequential procedure, are obtained. A lower bound for the number of additional observations after stopping is derived, which ensures the mxact probability of coverage. Moreover, two-stage, three-stage and modified sequential procedures are proposed for the same estimation problem. Relative advantages and disadvantages of these sampling schemes are discussed and their properties are studied.     

Inner and outer fuzzy approximations of confidence intervals

We extend the notion of confidence region to fuzzy data, by defining a pair of fuzzy inner and outer confidence regions. We show the connection with previous proposals, as well as with recent studies on hypothesis testing with low quality data.     

Model-based likelihood ratio confidence intervals for survival functions

We introduce an adjusted likelihood ratio procedure for computing pointwise confidence intervals for survival functions from censored data. The test statistic, scaled by a ratio of two variance quantities, is shown to converge to a chi-squared distribution with one degree of freedom. The confidence intervals are seen to be a neighborhood of a semiparametric survival function estimator and are shown to have correct empirical coverage. Numerical studies also indicate that the proposed intervals have smaller estimated mean lengths in comparison to the ones that are produced as a neighborhood of the Kaplan-Meier estimator. We illustrate our method using a lung cancer data set.