Robust simultaneous confidence interval estimation of principal component loadings

We propose a method that integrates bootstrap into the forward search algorithm in the construction of robust confidence intervals for elements of the eigenvectors of the correlation matrix in the presence of outliers. Coverage probability of the bootstrap simultaneous confidence intervals was compared to the coverage probabilities of regular asymptotic confidence region and asymptotic confidence region based on the minimum covariance determinant (MCD) approach through a simulation study. The method produced more stable coverage probabilities for datasets with or without outliers and across several sample sizes compared to approaches based on asymptotic confidence regions.     

Comparison of Ordinary and Penalized Power-Divergence Test Statistics for Small and Moderate Samples in Three-Way Contingency Tables via Simulation

In this article, size and power properties of the ordinary and penalized power-divergence test statistics have been compared to test a nested sequence of log-linear models for various penalization and λ values. 2 × 2 × 2 contingency tables distributed according to a multinomial distribution are considered. The simulation study results reveal that the penalized power-divergence test statistics with penalization value of one and λ = ?1.5, ?2 are the best choices for small and moderate samples, respectively.     

Sieve bootstrapping the memory parameter in long-range dependent stationary functional time series

We consider a sieve bootstrap procedure to quantify the estimation uncertainty of long-memory parameters in stationary functional time series. We use a semiparametric local Whittle estimator to estimate the long-memory parameter. In the local Whittle estimator, discrete Fourier transform and periodogram are constructed from the first set of principal component scores via a functional principal component analysis. The sieve bootstrap procedure uses a general vector autoregressive representation of the estimated principal component scores. It generates bootstrap replicates that adequately mimic the dependence structure of the underlying stationary process. We first compute the estimated first set of principal component scores for each bootstrap replicate and then apply the semiparametric local Whittle estimator to estimate the memory parameter. By taking quantiles of the estimated memory parameters from these bootstrap replicates, we can nonparametrically construct confidence intervals of the long-memory parameter. As measured by coverage probability differences between the empirical and nominal coverage probabilities at three levels of significance, we demonstrate the advantage of using the sieve bootstrap compared to the asymptotic confidence intervals based on normality.

     

Hybrid regions for post-change mean in a sequence of normal variables

The hybrid bootstrap uses resampling ideas to extend the duality approach to the interval estimation for a parameter of interest when there are nuisance parameters. The confidence region constructed by the hybrid bootstrap may perform much better than the ordinary bootstrap region in a situation where the data provide substantial information about the nuisance parameter, but limited information about the parameter of interest. We apply this method to estimate the post-change mean after a change is detected by a stopping procedure in a sequence of independent normal variables. Since distribution theory in change point problems is generally a challenge, we use bootstrap simulation to find empirical distributions of test statistics and calculate critical thresholds. Both likelihood ratio and Bayesian test statistics are considered to set confidence regions for post-change means in the normal model. In the simulation studies, the performance of hybrid regions are compared with that of ordinary bootstrap regions in terms of the widths and coverage probabilities of confidence intervals.     

Simultaneous confidence intervals by iteratively adjusted alpha for relative effects in the one-way layout

A bootstrap based method to construct 1−α simultaneous confidence intervals for relative effects in the one-way layout is presented. This procedure takes the stochastic correlation between the test statistics into account and results in narrower simultaneous confidence intervals than the application of the Bonferroni correction. Instead of using the bootstrap distribution of a maximum statistic, the coverage of the confidence intervals for the individual comparisons are adjusted iteratively until the overall confidence level is reached. Empirical coverage and power estimates of the introduced procedure for many-to-one comparisons are presented and compared with asymptotic procedures based on the multivariate normal distribution.     

Bootstrap confidence intervals in mixtures of discrete distributions

The problem of building bootstrap confidence intervals for small probabilities with count data is addressed. The law of the independent observations is assumed to be a mixture of a given family of power series distributions. The mixing distribution is estimated by nonparametric maximum likelihood and the corresponding mixture is used for resampling. We build percentile-t and Efron percentile bootstrap confidence intervals for the probabilities and we prove their consistency in probability. The new theoretical results are supported by simulation experiments for Poisson and geometric mixtures. We compare percentile-t and Efron percentile bootstrap intervals with eight other bootstrap or asymptotic theory based intervals. It appears that Efron percentile bootstrap intervals outperform the competitors in terms of coverage probability and length.     

Bootstrap confidence intervals for smoothing splines and their comparison to bayesian confidence intervals

We construct bootstrap confidence intervals for smoothing spline estimates based on Gaussian data, and penalized likelihood smoothing spline estimates based on data from .exponential families. Several vari- ations of bootstrap confidence intervals are considered and compared. We find that the commonly used ootstrap percentile intervals are inferior to the T intervals and to intervals based on bootstrap estimation of mean squared errors. The best variations of the bootstrap confidence intervals behave similar to the well known Bayesian confidence intervals. These bootstrap confidence intervals have an average coverage probability across the function being estimated, as opposed to a pointwise property.     

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.     

Small-sample comparisons for powerdivergence goodness-of-fit statistics for symmetric and skewed simple null hypotheses

Power-divergence goodness-of-fit statistics have asymptotically a chi-squared distribution. Asymptotic results may not apply in small-sample situations, and the exact significance of a goodness-of-fit statistic may potentially be over- or under-stated by the asymptotic distribution. Several correction terms have been proposed to improve the accuracy of the asymptotic distribution, but their performance has only been studied for the equiprobable case. We extend that research to skewed hypotheses. Results are presented for one-way multinomials involving k = 2 to 6 cells with sample sizes N = 20, 40, 60, 80 and 100 and nominal test sizes f = 0.1, 0.05, 0.01 and 0.001. Six power-divergence goodness-of-fit statistics were investigated, and five correction terms were included in the study. Our results show that skewness itself does not affect the accuracy of the asymptotic approximation, which depends only on the magnitude of the smallest expected frequency (whether this comes from a small sample with the equiprobable hypothesis or a large sample with a skewed hypothesis). Throughout the conditions of the study, the accuracy of the asymptotic distribution seems to be optimal for Pearson's X² statistic (the power-divergence statistic of index u = 1) when k > 3 and the smallest expected frequency is as low as between 0.1 and 1.5 (depending on the particular k, N and nominal test size), but a computationally inexpensive improvement can be obtained in these cases by using a moment-corrected h² distribution. If the smallest expected frequency is even smaller, a normal correction yields accurate tests through the log-likelihood-ratio statistic G² (the power-divergence statistic of index u = 0).     

On the construction of Bayes–confidence regions

We obtain approximate Bayes–confidence intervals for a scalar parameter based on directed likelihood. The posterior probabilities of these intervals agree with their unconditional coverage probabilities to fourth order, and with their conditional coverage probabilities to third order. These intervals are constructed for arbitrary smooth prior distributions. A key feature of the construction is that log-likelihood derivatives beyond second order are not required, unlike the asymptotic expansions of Severini.     

Stationary bootstrapping for realized covariations of high frequency financial data

This paper studies the stationary bootstrap applicability for realized covariations of high frequency asynchronous financial data. The stationary bootstrap method, which is characterized by a block-bootstrap with random block length, is applied to estimate the integrated covariations. The bootstrap realized covariance, bootstrap realized regression coefficient and bootstrap realized correlation coefficient are proposed, and the validity of the stationary bootstrapping for them is established both for large sample and for finite sample. Consistencies of bootstrap distributions are established, which provide us valid stationary bootstrap confidence intervals. The bootstrap confidence intervals do not require a consistent estimator of a nuisance parameter arising from nonsynchronous unequally spaced sampling while those based on a normal asymptotic theory require a consistent estimator. A Monte-Carlo comparison reveals that the proposed stationary bootstrap confidence intervals have better coverage probabilities than those based on normal approximation.     

Bootstrap adaptive estimation: The trimmed-mean example

We consider the problem of choosing among a class of possible estimators by selecting the estimator with the smallest bootstrap estimate of finite sample variance. This is an alternative to using cross-validation to choose an estimator adaptively. The problem of a confidence interval based on such an adaptive estimator is considered. We illustrate the ideas by applying the method to the problem of choosing the trimming proportion of an adaptive trimmed mean. It is shown that a bootstrap adaptive trimmed mean is asymptotically normal with an asymptotic variance equal to the smallest among trimmed means. The asymptotic coverage probability of a bootstrap confidence interval based on such adaptive estimators is shown to have the nominal level. The intervals based on the asymptotic normality of the estimator share the same asymptotic result, but have poor small-sample properties compared to the bootstrap intervals. A small-sample simulation demonstrates that bootstrap adaptive trimmed means adapt themselves rather well even for samples of size 10.     

Nonparametric asymptotic confidence intervals for extreme quantiles

In this paper, we propose new asymptotic confidence intervals for extreme quantiles, that is, for quantiles located outside the range of the available data. We restrict ourselves to the situation where the underlying distribution is heavy-tailed. While asymptotic confidence intervals are mostly constructed around a pivotal quantity, we consider here an alternative approach based on the distribution of order statistics sampled from a uniform distribution. The convergence of the coverage probability to the nominal one is established under a classical second-order condition. The finite sample behavior is also examined and our methodology is applied to a real dataset.     

Corrected asymptotic distribution of statistics based on the multinomial law

Investigators and epidemiologists often use statistics based on the parameters of a multinomial distribution. Two main approaches have been developed to assess the inferences of these statistics. The first one uses asymptotic formulae which are valid for large sample sizes. The second one computes the exact distribution, which performs quite well for small samples. They present some limitations for sample sizes N neither large enough to satisfy the assumption of asymptotic normality nor small enough to allow us to generate the exact distribution. We analytically computed the 1/N corrections of the asymptotic distribution for any statistics based on a multinomial law. We applied these results to the kappa statistic in 2×2 and 3×3 tables. We also compared the coverage probability obtained with the asymptotic and the corrected distributions under various hypothetical configurations of sample size and theoretical proportions. With this method, the estimate of the mean and the variance were highly improved as well as the 2.5 and the 97.5 percentiles of the distribution, allowing us to go down to sample sizes around 20, for data sets not too asymmetrical. The order of the difference between the exact and the corrected values was 1/N² for the mean and 1/N³ for the variance.     

Minimum power-divergence estimator in three-way contingency tables

Cressie et al. (2000; 2003) introduced and studied a new family of statistics, based on the φ-divergence measure, for solving the problem of testing a nested sequence of loglinear models. In that family of test statistics the parameters are estimated using the minimum φ-divergence estimator which is a generalization of the maximum likelihood estimator. In this paper we study the minimum power-divergence estimator (the most important family of minimum φ-divergence estimator) for a nested sequence of loglinear models in three-way contingency tables under assumptions of multinomial sampling. A simulation study illustrates that the minimum chi-squared estimator is simultaneously the most robust and efficient estimator among the family of the minimum power-divergence estimator.     

Goodness‐of‐fit Test for Directional Data

In this paper, we study the problem of testing the hypothesis on whether the density f of a random variable on a sphere belongs to a given parametric class of densities. We propose two test statistics based on the L² and L¹ distances between a non‐parametric density estimator adapted to circular data and a smoothed version of the specified density. The asymptotic distribution of the L² test statistic is provided under the null hypothesis and contiguous alternatives. We also consider a bootstrap method to approximate the distribution of both test statistics. Through a simulation study, we explore the moderate sample performance of the proposed tests under the null hypothesis and under different alternatives. Finally, the procedure is illustrated by analysing a real data set based on wind direction measurements.     

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.     

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.     

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.