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Tolerance intervals in regression allow the user to quantify, with a specified degree of confidence, bounds for a specified proportion of the sampled population when conditioned on a set of covariate values. While methods are available for tolerance intervals in fully-parametric regression settings, the construction of tolerance intervals for nonparametric regression models has been treated in a limited capacity. This paper fills this gap and develops likelihood-based approaches for the construction of pointwise one-sided and two-sided tolerance intervals for nonparametric regression models. A numerical approach is also presented for constructing simultaneous tolerance intervals. An appealing facet of this work is that the resulting methodology is consistent with what is done for fully-parametric regression tolerance intervals. Extensive coverage studies are presented, which demonstrate very good performance of the proposed methods. The proposed tolerance intervals are calculated and interpreted for analyses involving a fertility dataset and a triceps measurement dataset. [ABSTRACT FROM AUTHOR]

This paper focuses on comparing two nonparametric methods of constructing confidence intervals: the maximum entropy (ME) and the empirical likelihood (EL). The objective is to estimate probability distributions given some moment conditions in the presence of length-biased sampling. Some simulation studies are conducted to indicate ME confidence intervals have better coverage probabilities in comparison with their EL counterparts, when the biased data are contaminated. At the end, the ME and EL methods are applied to analyze a set of real data on elderly residents of a retirement center. [ABSTRACT FROM AUTHOR]

In survival analysis and reliability theory, the mean past lifetime (MPL), arises in situations where the mean time elapsed since the failure of a component T, given that it has failed before time t, is of interest. For inference on the MPL function, some procedures have been proposed in the literature for the MPL function's estimator. In this paper, an empirical likelihood (EL) inference procedure of the MPL function is developed. In addition to that, we obtain the adjusted EL and mean EL confidence interval for the MPL function. The EL confidence intervals are compared through extensive simulation studies in terms of coverage probability and the average length of the confidence interval. The simulation studies showed that the proposed EL methods have better coverage probability and shorter average lengths than the normal approximation results. Finally, the proposed methods are illustrated by two real data analyses. [ABSTRACT FROM AUTHOR]

We, in this paper, investigate two-sample quantile difference by empirical likelihood method when the responses with high-dimensional covariates of the two populations are missing at random. In particular, based on sufficient dimension reduction technique, we construct three empirical log-likelihood ratios for the quantile difference between two samples by using inverse probability weighting imputation, regression imputation as well as augmented inverse probability weighting imputation, respectively, and prove their asymptotic distributions. At the same time, we give a test to check whether two populations have the same distribution. A simulation study is carried out to investigate finite sample behavior of the proposed methods too. [ABSTRACT FROM AUTHOR]

It is common practice for auditors to verify only a sample of recorded values to estimate the total error amount. Monetary-unit sampling is often used to over-sample large valued items which may be overstated. The aim is to compute an upper confidence bound for the total errors amount. Naïve bounds based on the central limit theorem are not suitable, because the distribution of errors are often very skewed. Auditors frequently use the Stringer bound which known to be too conservative. We propose to use weighted empirical likelihood bounds for Monetary-unit sampling. The approach proposed is different from mainstream empirical likelihood. A Monte–Carlo simulation study highlights the advantage of the proposed approach over the Stringer bound. [ABSTRACT FROM AUTHOR]

Generalized additive partially linear models enjoy the simplicity of GLMs and the flexibility of GAMs because they combine both parametric and nonparametric components. Based on spline-backfitted kernel estimator, we propose empirical likelihood (EL)-based pointwise confidence intervals and simultaneous confidence bands (SCBs) for the nonparametric component functions to make statistical inference. Simulation study strongly supports the asymptotic theory and shows that EL-based SCBs are much easier for implementation and have better performance than Wald-type SCBs. We apply the proposed method to a university retention study and provide SCBs for the effect of the students information. [ABSTRACT FROM AUTHOR]

The consistency of the jackknife estimator of the asymptotic covariance matrix of a differentiable function of the symmetrized Tyler matrix is proved for sampling from continuous distribution without any constraint on this sampled continuous distribution. It is shown that under mild regularity conditions the rank of the asymptotic covariance matrix attains maximum possible value. The results are applied to the construction of the asymptotic confidence interval for correlation coefficient. It is illustrated by simulations that for some distributions the proposed interval yields better results than the classical Z-transformation interval. [ABSTRACT FROM AUTHOR]

For the high-dimensional partially linear errors-in-function model with martingale difference errors, we, in this paper, propose an empirical log-likelihood ratio function for the regression parameter. The empirical log-likelihood ratio is proven to be asymptotically chi-squared. In addition, we propose penalized empirical likelihood (PEL) for the parameter. By using an appropriate penalty function, we show that PEL has the oracle property. Also, we obtain the PEL ratio is defined and its limiting distribution is asymptotically chi-squared under the null hypothesis. Moreover, simulation studies and a real data example are undertaken to assess the finite sample performance of our proposed method. [ABSTRACT FROM AUTHOR]

A credible band is the set of all functions between a lower and an upper bound that are constructed so that the set has prescribed mass under the posterior distribution. In a Bayesian analysis such a band is used to quantify the remaining uncertainty on the unknown function in a similar manner as a confidence band. We investigate the validity of a credible band in the nonparametric regression model with the prior distribution on the function given by a Gaussian process. We show that there are many true regression functions for which the credible band has the correct order of magnitude to be used as a confidence set. We also exhibit functions for which the credible band is misleading. [ABSTRACT FROM AUTHOR]

We consider nonparametric estimation of the ridge of a probability density function for multivariate linear processes with long-range dependence. We derive functional limit theorems for estimated eigenvectors and eigenvalues of the Hessian matrix. We use these results to obtain the weak convergence for the estimated ridge and asymptotic simultaneous confidence regions. [ABSTRACT FROM AUTHOR]

This study investigates simultaneous confidence bands for extremal quantile regressions using the spline method. We construct the spline estimator for intermediate order quantiles using a conventional quantile regression framework, and we obtain the extreme order quantile estimator by extrapolating the spline estimator for intermediate order quantiles. We establish the asymptotic normality of the spline and extrapolated estimators for intermediate and extreme order quantiles. By applying the volume of tube formula to the above two estimators, we construct simultaneous conditional quantile confidence bands for intermediate and extreme order quantiles. To confirm the performance of the proposed confidence bands, we use a Monte Carlo simulation and an example with real data. [ABSTRACT FROM AUTHOR]

522 C.N.P.G.Arachchige [3] References [1] C. N. P. G. Arachchige, M. Cairns and L. A. Prendergast, 'Interval estimators for ratios of independent quantiles and interquantile ranges', Comm. The objective of this thesis was to provide new distribution-free point and interval estimators for measures of spread, relative spread and skewness involving quantiles. In [1], the distribution-free point and interval estimators were introduced for ratios of independent quantiles to compare the locations of two independent samples. [Extracted from the article]

We survey some of the recent advances in mean estimation and regression function estimation. In particular, we describe sub-Gaussian mean estimators for possibly heavy-tailed data in both the univariate and multivariate settings. We focus on estimators based on median-of-means techniques, but other methods such as the trimmed-mean and Catoni's estimators are also reviewed. We give detailed proofs for the cornerstone results. We dedicate a section to statistical learning problems—in particular, regression function estimation—in the presence of possibly heavy-tailed data. [ABSTRACT FROM AUTHOR]

Inferential methods based on ranks present robust and powerful alternative methodology for testing and estimation. In this article, two objectives are followed. First, develop a general method of simultaneous confidence intervals based on the rank estimates of the parameters of a general linear model and derive the asymptotic distribution of the pivotal quantity. Second, extend the method to high dimensional data such as gene expression data for which the usual large sample approximation does not apply. It is common in practice to use the asymptotic distribution to make inference for small samples. The empirical investigation in this article shows that for methods based on the rank-estimates, this approach does not produce a viable inference and should be avoided. A method based on the bootstrap is outlined and it is shown to provide a reliable and accurate method of constructing simultaneous confidence intervals based on rank estimates. In particular it is shown that commonly applied methods of normal or t-approximation are not satisfactory, particularly for large-scale inferences. Methods based on ranks are uniquely suitable for analysis of microarray gene expression data since they often involve large scale inferences based on small samples containing a large number of outliers and violate the assumption of normality. A real microarray data is analyzed using the rank-estimate simultaneous confidence intervals. Viability of the proposed method is assessed through a Monte Carlo simulation study under varied assumptions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study the class of bivariate penalised splines that use tensor product splines and a smoothness penalty. Similar to Claeskens, G., Krivobokova, T., and Opsomer, J.D. [(2009), 'Asymptotic Properties of Penalised Spline Estimators', Biometrika, 96(3), 529–544] for the univariate penalised splines, we show that, depending on the number of knots and penalty, the global asymptotic convergence rate of bivariate penalised splines is either similar to that of tensor product regression splines or to that of thin plate splines. In each scenario, the bivariate penalised splines are found rate optimal in the sense of Stone, C.J. [(12, 1982), 'Optimal Global Rates of Convergence for Nonparametric Regression', The Annals of Statistics, 10(4), 1040–1053] for a corresponding class of functions with appropriate smoothness. For the scenario where a small number of knots is used, we obtain expressions for the local asymptotic bias and variance and derive the point-wise and uniform asymptotic normality. The theoretical results are applicable to tensor product regression splines. [ABSTRACT FROM AUTHOR]

We introduce a recursive kernel estimator of the hazard function in the framework of independent rightly censored data. We compute its bias and variance, and compare its mean squared error to those of non-recursive kernel estimators. We also establish its weak convergence rate and point out that, for estimation by confidence intervals, our recursive estimator performs better than the non-recursive ones. This is confirmed by the simulations study we give. [ABSTRACT FROM AUTHOR]

A plug-in estimator is proposed for a local measure of variance explained by regression, termed correlation curve in Doksum et al. (J Am Stat Assoc 89:571-582, 1994), consisting of a two-step spline-kernel estimator of the conditional variance function and local quadratic estimator of first derivative of the mean function. The estimator is oracally efficient in the sense that it is as efficient as an infeasible correlation estimator with the variance function known. As a consequence of the oracle efficiency, a smooth simultaneous confidence band (SCB) is constructed around the proposed correlation curve estimator and shown to be asymptotically correct. Simulated examples illustrate the versatility of the proposed oracle SCB which confirms the asymptotic theory. Application to a 1995 British Family Expenditure Survey data has found marginally significant evidence for a local version of Engel’s law, i.e., food budget share and household real income are inversely related (Hamilton in Am Econ Rev 91:619-630, 2001). [ABSTRACT FROM AUTHOR]

A compound Poisson process whose parameters are all unknown is observed at finitely many equispaced times. Nonparametric estimators of the jump and Lévy distributions are proposed and functional central limit theorems using the uniform norm are proved for both under mild conditions. The limiting Gaussian processes are identified and efficiency of the estimators is established. Kernel estimators for the mass function, the intensity and the drift are also proposed, their asymptotic properties including efficiency are analysed, and joint asymptotic normality is shown. Inference tools such as confidence regions and tests are briefly discussed. [ABSTRACT FROM AUTHOR]

We construct new pivotals to obtain confidence bounds and confidence intervals for the mean of a stationary process. These follow the approach based on estimating functions. The new pivotals are compared with the standard pivotal based on studentization. We study the first four cumulants of each of these pivotals and explain why the pivotals based on the estimating function approach result in better coverage probabilities. Some simulation results comparing these pivotals have been reported. [ABSTRACT FROM AUTHOR]

We study optimal conformity measures for various criteria of efficiency of set-valued classification in an idealised setting. This leads to an important class of criteria of efficiency that we call probabilistic and argue for; it turns out that the most standard criteria of efficiency used in literature on conformal prediction are not probabilistic unless the problem of classification is binary. We consider both unconditional and label-conditional conformal prediction. [ABSTRACT FROM AUTHOR]

We construct universal prediction systems in the spirit of Popper's falsifiability and Kolmogorov complexity and randomness. These prediction systems do not depend on any statistical assumptions (but under the IID assumption they dominate, to within the usual accuracy, conformal prediction). Our constructions give rise to a theory of algorithmic complexity and randomness of time containing analogues of several notions and results of the classical theory of Kolmogorov complexity and randomness. [ABSTRACT FROM AUTHOR]

The conformal prediction framework allows for specifying the probability of making incorrect predictions by a user-provided confidence level. In addition to a learning algorithm, the framework requires a real-valued function, called nonconformity measure, to be specified. The nonconformity measure does not affect the error rate, but the resulting efficiency, i.e., the size of output prediction regions, may vary substantially. A recent large-scale empirical evaluation of conformal regression approaches showed that using random forests as the learning algorithm together with a nonconformity measure based on out-of-bag errors normalized using a nearest-neighbor-based difficulty estimate, resulted in state-of-the-art performance with respect to efficiency. However, the nearest-neighbor procedure incurs a significant computational cost. In this study, a more straightforward nonconformity measure is investigated, where the difficulty estimate employed for normalization is based on the variance of the predictions made by the trees in a forest. A large-scale empirical evaluation is presented, showing that both the nearest-neighbor-based and the variance-based measures significantly outperform a standard (non-normalized) nonconformity measure, while no significant difference in efficiency between the two normalized approaches is observed. The evaluation moreover shows that the computational cost of the variance-based measure is several orders of magnitude lower than when employing the nearest-neighbor-based nonconformity measure. The use of out-of-bag instances for calibration does, however, result in nonconformity scores that are distributed differently from those obtained from test instances, questioning the validity of the approach. An adjustment of the variance-based measure is presented, which is shown to be valid and also to have a significant positive effect on the efficiency. For conformal regression forests, the variance-based nonconformity measure is hence a computationally efficient and theoretically well-founded alternative to the nearest-neighbor procedure. [ABSTRACT FROM AUTHOR]

In this article, we employ the jackknife empirical likelihood (JEL) method to construct the confidence regions for the difference of the means of twod-dimensional samples. Compared with traditional EL for the two-sample mean problem, JEL is extremely simpler to use in practice and is more effective in computing. Based on the JEL ratio test, a version of Wilks’ theorem is developed. Furthermore, to improve the coverage accuracy of confidence regions, a Bartlett correction is applied. The effectiveness of the proposed method is demonstrated by a simulation study and a real data analysis. [ABSTRACT FROM AUTHOR]

The ordinary Wilcoxon signed rank test table provides confidence intervals for the median of one population. Adjusted Wilcoxon signed rank test tables which can provide confidence intervals for the median and the 10th percentile of one population are created in this paper. Base-(n + 1) number system and theorems about property of symmetry of the adjusted Wilcoxon signed rank test statistic are derived for programming. Theorem 1 states that the adjusted Wilcoxon signed rank test statistic are symmetric aroundn(n+ 1)/4. Theorem 2 states that the adjusted Wilcoxon signed rank test statistic with the same number of negative ranks m are symmetric around m(n+1)/2. 87.5% and 85% confidence intervals of the median are given in the table for n = 12, 13,…, 29 to create approximated 95% confidence intervals of the ratio of medians for two independent populations. 95% and 92.5% confidence intervals of the 10th percentile are given in the table for n = 26, 27, 28, 29 to create approximated 95% confidence regions of the ratio of the 10th percentiles for two independent populations. Finally two large datasets from wood industry will be partitioned to verify the correctness of adjusted Wilcoxon signed rank test tables for small samples. [ABSTRACT FROM AUTHOR]

Often decision makers have to cope with a programming problem with unknown quantitities. Then they will estimate these quantities and solve the problem as it then appears-the 'approximate problem'. Thus there is a need to establish conditions which will ensure that the solutions to the approximate problem will come close to the solutions to the true problem in a suitable manner. Confidence sets, i.e. sets that cover the true sets with a given prescribed probability, provide useful quantitative information. In this paper we consider multiobjective problems and derive confidence sets for the sets of efficient points, weakly efficient points, and the corresponding solution sets. Besides the crucial convergence conditions for the objective and/or constraint functions, one approach for the derivation of confidence sets requires some knowledge about the true problem, which may be not available. Therefore also another method, called relaxation, is suggested. This approach works without any knowledge about the true problem. The results are applied to the Markowitz model of portfolio optimization. [ABSTRACT FROM AUTHOR]

In this paper, we develop exact nonparametric inference for quantiles of the population based on the sequence of k -record values, conditionally on the corresponding observed inter- k -record times, to increase the precision of inference, as compared to nonparametric unconditional procedures. The effect of including the additional information contained in the inter- k -record times is evaluated by means of an empirical study. The conditional tolerance and prediction intervals for a future k -record from another independent sequence are obtained as well. A real data set is also used to illustrate the inferential results developed here. [ABSTRACT FROM AUTHOR]

Nonparametric statistics for distribution functions F or densities f = F ′ under qualitative shape constraints constitutes an interesting alternative to classical parametric or entirely nonparametric approaches. We contribute to this area by considering a new shape constraint: F is said to be bi-log-concave, if both log F and log ( 1 − F ) are concave. Many commonly considered distributions are compatible with this constraint. For instance, any c.d.f. F with log-concave density f = F ′ is bi-log-concave. But in contrast to log-concavity of f , bi-log-concavity of F allows for multimodal densities. We provide various characterisations. It is shown that combining any nonparametric confidence band for F with the new shape constraint leads to substantial improvements, particularly in the tails. To pinpoint this, we show that these confidence bands imply non-trivial confidence bounds for arbitrary moments and the moment generating function of F . [ABSTRACT FROM AUTHOR]

Statistical depth functions are well-known nonparametric tools for analysing multivariate data. Halfspace depth is most frequently used, and while it has many desirable properties, it is dependent on global characteristics of the underlying distribution. In some circumstances, however, it may be desirable to take into account more local and intrinsic characteristics of the data. To this end, we introduce weighted halfspace depths in which the indicator function of closed halfspace is replaced by a more general weight function. Our approach, which calls in part on functions associated with conic sections, encompasses as special cases the notions of sample halfspace depth and kernel density estimation. We give several illustrations and prove the strong uniform consistency of weighted halfspace depth incorporating mild conditions on the weight function. [ABSTRACT FROM AUTHOR]