inference on impulse response functions in structural var models skepticism toward traditional identifying assumptions based on exclusion restrictions has led to a surge in the use of structural var models in which structural shocks are identified by restricting the sign of the responses of selected macroeconomic aggregates to these shocks researchers commonly report the vector of pointwise posterior medians of the impulse responses as a measure of central tendency of the estimated response functions along with pointwise 68 percent posterior error bands it can be shown that this approach cannot be used to characterize the central tendency of the structural impulse response functions we propose an alternative method of summarizing the evidence from sign identified var models designed to enhance their practical usefulness our objective is to characterize the most likely admissible model s within the set of structural var models that satisfy the sign restrictions we show how the set of most likely structural response functions can be computed from the posterior mode of the joint distribution of admissible models both in the fully identified and in the partially identified case and we propose a highest posterior density credible set that characterizes the joint uncertainty about this set our approach can also be used to resolve the long standing problem of how to conduct joint inference on sets of structural impulse response functions in exactly identified var models we illustrate the differences between our approach and the traditional approach for the analysis of the effects of monetary policy shocks and of the effects of oil demand and oil supply shocks

a large sample study of cox s regression model strong consistency and asymptotic normality are established for the maximum partial likelihood estimate of the regression parameter in cox s regression model estimates are also derived for the underlying cumulative hazard function and survival distribution we establish the asymptotic normality of these estimates and calculate the limiting variances

nonparametric likelihood ratio confidence bands for quantile functions from incomplete survival data the purpose of this paper is to derive confidence bands for quantile functions using a nonparametric likelihood ratio approach the method is easy to implement and has several appealing properties it applies to right censored and left truncated data and it does not involve density estimation or even require the existence of a density previous approaches e g bootstrap have imposed smoothness conditions on the density the performance of the proposed method is investigated in a monte carlo study and an application to real data is given

searching and pebbling abstract we relate the search number of an undirected graph g with the minimum and maximum of the progressive pebble demands of the directed acyclic graphs obtained by orienting g towards this end we introduce node searching a slight variant of searching in which an edge is cleared by placing searchers on both of its endpoints we also show that the minimum number of searchers necessary to node search a graph equals its vertex separator plus one

local empirical likelihood inference for varying coefficient density ratio models based on case control data in this article we develop a varying coefficient density ratio model for case control studies the case and control samples come from two different distributions under the model assumption the ratio of the two densities is related to the linear combination of covariates with varying coefficients through a known function a special case is the exponential tilt model where the log ratio of the two densities is a linear function of covariates we propose a local empirical likelihood el approach to estimate the nonparametric coefficient functions under some regularity assumptions the proposed estimators are shown to be consistent and asymptotically normally distributed the sieve empirical likelihood ratio selr test statistic for detecting whether the varying coefficients are really constant and other related hypotheses is constructed and it follows approximately a chi squared distribution we introduce a modified bootstrap procedure to estimate the null distribution of the selr when sample size is s

nonparametric estimates of regression quantiles and their local bahadur representation let x y be a random vector such that x is d dimensional y is real valued and y theta x varepsilon where x and varepsilon are independent and the alpha th quantile of varepsilon is 0 alpha is fixed such that 0 0 and set r p m 2p d where m is a nonnegative integer smaller than p let t theta denote a derivative of theta of order m it is proved that there exists a pointwise estimate hat t n of t theta based on a set of i i d observations x 1 y 1 cdots s n y n that achieves the optimal nonparametric rate of convergence n r under appropriate regularity conditions further a local bahadur type representation is shown to hold for the estimate hat t n and this is used to obtain some useful asymptotic results

empirical likelihood ratio confidence regions than are those based on student s t statistic an effective method is presented for computing empirical profile likelihoods for the mean of a vector random variable the method is a reduction by convex duality to an unconstrained minimization of a convex function on a low dimensional domain algorithms exist for finding the unique global minimum at a superlinear rate of convergence a byproduct is a noncombinatorial algorithm for determining whether a given point lies within the convex hull of a finite set of points the multivariate empirical likelihood regions are justified for functions of several means such as variances correlations and regression parameters and for statistics with linear estimating equations an algorithm is given for computing profile empirical likelihoods for these statistics

data driven confidence intervals for diffusion process using double smoothing empirical likelihood abstract in this paper we propose an innovative estimation method for the volatility function in diffusion process and construct the empirical likelihood confidence interval for it compared with double smoothing local constant estimator double smoothing local linear estimator proposed in this paper is an inventive estimation method moreover we find that the empirical likelihood confidence interval constructed with the approximate estimating equation is much better than the one based on the estimating equation since the latter undermines the predictability of the estimator accordingly new algorithm for simulation is proposed through monte carlo simulation and empirical analysis both these approaches are superior to traditional asymptotic normality confidence interval in terms of coverage rate etc

multicommodity max flow min cut theorems and their use in designing approximation algorithms in this paper we establish max flow min cut theorems for several important classes of multicommodity flow problems in particular we show that for any n node multicommodity flow problem with uniform demands the max flow for the problem is within an o log n factor of the upper bound implied by the min cut the result which is existentially optimal establishes an important analogue of the famous 1 commodity max flow min cut theorem for problems with multiple commodities the result also has substantial applications to the field of approximation algorithms for example we use the flow result to design the first polynomial time polylog n times optimal approximation algorithms for well known np hard optimization problems such as graph partitioning min cut linear arrangement crossing number vlsi layout and minimum feedback arc set applications of the flow results to path routing problems network reconfiguration communication in distributed networks scientific computing and rapidly mixing markov chains are also described in the paper categories and subject descriptors f 2 2 analysis of algorithms and problem complexity

empirical likelihood ratio confidence intervals for a single functional summary the empirical distribution function based on a sample is well known to be the maximum likelihood estimate of the distribution from which the sample was taken in this paper the likelihood function for distributions is used to define a likelihood ratio function for distributions it is shown that this empirical likelihood ratio function can be used to construct confidence intervals for the sample mean for a class of m estimates that includes quantiles and for differentiable statistical functionals the results are nonparametric extensions of wilks s 1938 theorem for parametric likelihood ratios the intervals are illustrated on some real data and compared in a simulation to some bootstrap confidence intervals and to intervals based on student s t statistic a hybrid method that uses the bootstrap to determine critical values of the likelihood ratio is introduced

empirical likelihood based inference in generalized random coefficient autoregressive model with conditional moment restrictions abstract this paper concentrates on the parameter estimating strategy for the generalized random coefficient autoregressive grca model in the presence of the auxiliary information we propose a weighted least squares estimate for the model parameters and empirical likelihood el based weights are obtained through using these auxiliary information the asymptotic distribution of our proposed estimator is normal distribution and the asymptotic variance is reduced compared to the least square ls estimator therefore our method yields more efficient estimates we also carry out some simulation experiments to assess the performance of the suggested estimator and illustrate the usefulness of this method through the analysis of a real time series data sets

large sample properties of weighted monte carlo estimators a general approach to improving simulation accuracy uses information about auxiliary control variables with known expected values to improve the estimation of unknown quantities we analyze weighted monte carlo estimators that implement this idea by applying weights to independent replications the weights are chosen to constrain the weighted averages of the control variables we distinguish two cases unbiased and biased depending on whether the weighted averages of the controls are constrained to equal their expected values or some other values in both cases the number of constraints is usually smaller than the number of replications so there may be many feasible weights we select maximally uniform weights by minimizing a separable convex function of the weights subject to the control variable constraints estimators of this form arise sometimes implicitly in several settings including at least two in finance calibrating a model to market data as in the work of avellaneda et al 2001 and calculating conditional expectations to price american options we analyze properties of these estimators as the number of replications increases we show that in the unbiased case weighted monte carlo reduces asymptotic variance and that all convex objective functions within a large class produce estimators that are very close to each other in a strong sense in contrast in the biased case the choice of objective function does matter we show explicitly how the choice of objective determines the limit to which the estimator converges

asymptotic optimality of empirical likelihood for testing moment restrictions we show by example that empirical likelihood and other commonly used tests for moment restrictions are unable to control the exponential rate at which the probability of a type i error tends to zero unless the possible distributions for the observed data are restricted appropriately from this it follows that for the optimality claim for empirical likelihood in kitamura 2001 to hold additional assumptions and qualifications are required under stronger assumptions than those in kitamura 2001 we establish the following optimality result i empirical likelihood controls the rate at which the probability of a type i error tends to zero and ii among all procedures for which the probability of a type i error tends to zero at least as fast empirical likelihood maximizes the rate at which the probability of a type ii error tends to zero for most alternatives this result further implies that empirical likelihood maximizes the rate at which the probability of a type ii error tends to zero for all alternatives among a class of tests that satisfy a weaker criterion for their type i error probabilities

monge kantorovich depth quantiles ranks and signs we propose new concepts of statistical depth multivariate quantiles ranks and signs based on canonical transportation maps between a distributionof interest on ird and a reference distribution on the d dimensionalunit ball the new depth concept called monge kantorovich depth specializesto halfspace depth in the case of elliptical distributions but for more generaldistributions differs from the latter in the ability for its contours to account fornon convex features of the distribution of interest we propose empirical counterpartsto the population versions of those monge kantorovich depth contours quantiles ranks and signs and show their consistency by establishing a uniformconvergence property for transport maps which is of independent interest

empirical likelihood based inference in missing response problems and its application in observational studies summary the problem of missing response data is ubiquitous in medical and social science studies in the case of responses that are missing at random depending on some covariate information analyses focused only on the complete data may lead to biased results various debias methods have been extensively studied in the literature particularly the weighting method that was motivated by horvitz and thompson s estimators to improve efficiency robins rotnitzky and zhao proposed augmented estimating equations based on corrected complete case analyses a nice feature of the augmented method is its double robustness i e the estimator that is derived from the augmented method is asymptotically unbiased if either the underlying missing data mechanism or the underlying regression function is correctly specified furthermore the augmented estimator can achieve full efficiency if both the missing data mechanism and the regression function are correctly specified in general however it is very difficult to specify the regression function correctly especially when the dimension of covariates is high this is the so called curse of dimensionality problem the augmented estimator has much lower efficiency if the working regression model is not close to the true regression model in this paper the empirical likelihood method is employed to seek a constrained empirical likelihood estimation of mean response with the assumption that responses are missing at random the empirical likelihood based estimators enjoy the double robustness property moreover it is possible that the empirical likelihood based inference can produce asymptotically unbiased and efficient estimators even if the true regression function is not completely known simulation results indicate that the empirical likelihood based estimators are very robust to a misspecification of the propensity score and dominate other competitors in the sense of having smaller mean square errors methods that are developed in this paper have a nice application in observational causal inferences the propensity score is used to adjust for differences in pretreatment variables in the estimation of average treatment effects

bayesian estimation and comparison of moment condition models in this article we develop a bayesian semiparametric analysis of moment condition models by casting the problem within the exponentially tilted empirical likelihood etel framework we use this f

a new nuisance parameter elimination method with application to the unordered homologous chromosome pairs problem motivated by applications of the case control model or exponential tilting model in the unordered homologous chromosome pairs problem in genetic studies and in the interim analysis in double blinded clinical trials we develop a new nuisance parameter elimination method based on the empirical shannon s mutual information the asymptotic behaviors of the maximum empirical shannon s mutual information estimation and the empirical shannon s mutual information test are similar to those of the maximum likelihood estimation and the likelihood ratio test respectively interestingly we have found a connection between the empirical shannon s mutual information and the profile empirical likelihood owen 1988 under some constraints in the test of the null hypothesis that the unordered pairs come from the same distribution the maximum shannon s mutual information estimation has a degenerate information matrix as a result we have to expand the empirical shannon s mutual information test statistic up to the four

improved approximation algorithms for minimum weight vertex separators we develop the algorithmic theory of vertex separators and its relation to the embeddings of certain metric spaces unlike in the edge case we show that embeddings into l 1 and even euclidean embeddings are insufficient but that the additional structure provided by many embedding theorems does suffice for our purposes we obtain an o sqrt log n approximation for minimum ratio vertex cuts in general graphs based on a new semidefinite relaxation of the problem and a tight analysis of the integrality gap which is shown to be theta sqrt log n we also prove an optimal o log k approximate max flow min vertex cut theorem for arbitrary vertex capacitated multicommodity flow instances on k terminals for uniform instances on any excluded minor family of graphs we improve this to o 1 and this yields a constant factor approximation for minimum ratio vertex cuts in such graphs previously this was known only for planar graphs and for general excluded minor families the best known ratio was o log n these results have a number of applications we exhibit an o sqrt log n pseudoapproximation for finding balanced vertex separators in general graphs in fact we achieve an approximation ratio of o sqrt log opt where opt is the size of an optimal separator improving over the previous best bound of o log opt likewise we obtain improved approximation ratios for treewidth in any graph of treewidth k we show how to find a tree decomposition of width at most o k sqrt log k whereas previous algorithms yielded o k log k for graphs excluding a fixed graph as a minor which includes e g bounded genus graphs we give a constant factor approximation for the treewidth this in turn can be used to obtain polynomial time approximation schemes for several problems in such graphs

empirical likelihood semiparametric regression analysis for longitudinal data a semiparametric regression model for longitudinal data is considered the empirical likelihood method is used to estimate the regression coefficients and the baseline function and to construct confidence regions and intervals it is proved that the maximum empirical likelihood estimator of the regression coefficients achieves asymptotic efficiency and the estimator of the baseline function attains asymptotic normality when a bias correction is made two calibrated empirical likelihood approaches to inference for the baseline function are developed we propose a groupwise empirical likelihood procedure to handle the inter series dependence for the longitudinal semiparametric regression model and employ bias correction to construct the empirical likelihood ratio functions for the parameters of interest this leads us to prove a nonparametric version of wilks theorem compared with methods based on normal approximations the empirical likelihood does not require consistent estimators for the asymptotic variance and bias a simulation compares the empirical likelihood and normal based methods in terms of coverage accuracies and average areas lengths of confidence regions intervals

making and evaluating point forecasts typically point forecasting methods are compared and assessed by means of an error measure or scoring function with the absolute error and the squared error being key examples the individual scores are averaged over forecast cases to result in a summary measure of the predictive performance such as the mean absolute error or the mean squared error i demonstrate that this common practice can lead to grossly misguided inferences unless the scoring function and the forecasting task are carefully matched effective point forecasting requires that the scoring function be specified ex ante or that the forecaster receives a directive in the form of a statistical functional such as the mean or a quantile of the predictive distribution if the scoring function is specified ex ante the forecaster can issue the optimal point forecast namely the bayes rule if the forecaster receives a directive in the form of a functional it is critical that the scoring function be consistent for it in the sense that t

vector quantile regression an optimal transport approach we propose a notion of conditional vector quantile function and a vector quantile regression a conditional vector quantile function cvqf of a random vector y taking values in d given covariates z z taking values in k is a map u qy z u z which is monotone in the sense of being a gradient of a convex function and such that given that vector u follows a reference non atomic distribution fu for instance uniform distribution on a unit cube in d the random vector qy z u z has the distribution of y conditional on z z moreover we have a strong representation y qy z u z almost surely for some version of u the vector quantile regression vqr is a linear model for cvqf of y given z under correct specification the notion produces strong representation y u f z for f z denoting a known set of transformations of z where u u f z is a monotone map the gradient of a convex function and the quantile regression coefficients u u have the interpretations analogous to that of the standard scalar quantile regression as f z becomes a richer class of transformations of z the model becomes nonparametric as in series modelling a key property of vqr is the embedding of the classical monge kantorovich s optimal transportation problem at its core as a special case in the classical case where y is scalar vqr reduces to a version of the classical qr and cvqf reduces to the scalar conditional quantile function several applications to diverse problems such as multiple engel curve estimation and measurement of financial risk are considered

on a geometric notion of quantiles for multivariate data abstract an extension of the concept of quantiles in multidimensions that uses the geometry of multivariate data clouds has been considered the approach is based on blending as well as generalization of the key ideas used in the construction of spatial median and regression quantiles both of which have been extensively studied in the literature these geometric quantiles are potentially useful in constructing trimmed multivariate means as well as many other l estimates of multivariate location and they lead to a directional notion of central and extreme points in a multidimensional setup such quantiles can be defined as meaningful and natural objects even in infinite dimensional hilbert and banach spaces and they yield an effective generalization of quantile regression in multiresponse linear model problems desirable equivariance properties are shown to hold for these multivariate quantiles and issues related to their computation for data in finite dimensional spaces are discussed n 1 2 consistenc

spatial quantile estimation of multivariate threshold time series models abstract in this paper we study spatial quantile regression estimation of multivariate threshold time series models bahadur s representations for our estimators are established which naturally lead to asymptotic normality of the estimators simulations and a real example are used to evaluate the performance of the proposed estimators

multivariate quantiles and multiple output regression quantiles from l1 optimization to halfspace depth a new multivariate concept of quantile based on a directional version of koenker and bassett s traditional regression quantiles is introduced for multivariate location and multiple output regression problems in their empirical version those quantiles can be computed efficiently via linear programming techniques consistency bahadur representation and asymptotic normality results are established most importantly the contours generated by those quantiles are shown to coincide with the classical halfspace depth contours associated with the name of tukey this relation does not only allow for efficient depth contour computations by means of parametric linear programming but also for transferring from the quantile to the depth universe such asymptotic results as bahadur representations finally linear programming duality opens the way to promising developments in depth related multivariate rank based inference

fixed smoothing asymptotics in the generalized empirical likelihood estimation framework this paper concerns the fixed smoothing asymptotics for two commonly used estimators in the generalized empirical likelihood estimation framework for time series data namely the continuous updating estimator and the maximum blockwise empirical likelihood estimator for continuously updating generalized method of moments gmm estimator we show that the results for the two step gmm estimator in sun 2014a continue to hold under suitable assumptions for continuous updating estimator obtained through solving a saddle point problem newey and smith 2004 and the maximum blockwise empirical likelihood estimator kitamura 1997 we show that their fixed smoothing asymptotic distributions up to an unknown linear transformation are mixed normal based on these results we derive the asymptotic distributions of the specification tests including the over identification testing and testing on parameters under the fixed smoothing asymptotics where the corresponding limiting distributions are nonstandard yet pivotal simulation studies show that i the fixed smoothing asymptotics provides better approximation to the sampling distributions of the continuous updating estimator and the maximum blockwise empirical likelihood estimator as compared to the standard normal approximation the testing procedures based on the fixed smoothing critical values are more accurate in size than the conventional chi square based tests ii the continuously updating gmm estimator is asymptotically more efficient and the corresponding specification tests are generally more powerful than the other two competitors finite sample results from an empirical data analysis are also reported

moment conditions and bayesian non parametrics models phrased through moment conditions are central to much of modern inference here these moment conditions are embedded within a non parametric bayesian set up handling such a model is not probabilistically straightforward as the posterior has support on a manifold we solve the relevant issues building new probability and computational tools by using hausdorff measures to analyse them on real and simulated data these new methods which involve simulating on a manifold can be applied widely including providing bayesian analysis of quasi likelihoods linear and non linear regression missing data and hierarchical models

on bartlett correction of empirical likelihood in the presence of nuisance parameters summary lazar mykland 1999 showed that an empirical likelihood defined by two estimating equations with a nuisance parameter need not be bartlett correctable this paper shows that bartlett correction of empirical likelihood in the presence of a nuisance parameter depends critically on the way the nuisance parameter is removed when formulating the likelihood for the parameter of interest we establish in the broad framework of estimating functions that the empirical likelihood is still bartlett correctable if the nuisance parameter is profiled out given the value of the parameter of interest

testing conditional independence via empirical likelihood we construct two classes of smoothed empirical likelihood ratio tests for the conditional independence hypothesis by writing the null hypothesis as an infinite collection of conditional moment restrictions indexed by a nuisance parameter one class is based on the cdf another is based on smoother functions we show that the test statistics are asymptotically normal under the null hypothesis and a sequence of pitman local alternatives we also show that the tests possess an asymptotic optimality property in terms of average power simulations suggest that the tests are well behaved in finite samples applications to some economic and financial time series indicate that our tests reveal some interesting nonlinear causal relations which the traditional linear granger causality test fails to detect

nonparametric confidence bands for a quantile comparison function this article develops statistical methods useful for comparing the marginals f and g of a bivariate distribution such comparisons are appropriate in the analysis of matched pair data involving a treatment and a control or two treatments the quantile comparison functionq g 1 f proposed by lehmann as a generalized measure of treatment effect in fully randomized experiments is shown to be applicable and also useful in the analysis of matched pair data nonparametric simultaneous confidence bands for q are constructed using asymptotic and permutation methods the copula of the bivariate distribution plays a significant role in the methodology application of the results is illustrated on a dataset from an industrial experiment

bootstrapping general empirical measures it is proved that the bootstrapped central limit theorem for empirical processes indexed by a class of functions mathscr f and based on a probability measure p holds a s if and only if mathscr f in mathrm clt p and int f 2 dp infty where f sup f in mathscr f f and it holds in probability if and only if mathscr f in mathrm clt p thus for a large class of statistics no local uniformity of the clt about p is needed for the bootstrap to work consistency of the bootstrap the bootstrapped law of large numbers is also characterized these results are proved under certain weak measurability assumptions on mathscr f

improving semiparametric estimation by using surrogate data summary the paper considers estimating a parameter that defines an estimating function u y x for an outcome variable y and its covariate x when the outcome is missing in some of the observations we assume that in addition to the outcome and the covariate a surrogate outcome is available in every observation the efficiency of existing estimators for depends critically on correctly specifying the conditional expectation of u given the surrogate and the covariate when the conditional expectation is not correctly specified which is the most likely scenario in practice the efficiency of estimation can be severely compromised even if the propensity function of missingness is correctly specified we propose an estimator that is robust against the choice of the conditional expectation via an empirical likelihood we demonstrate that the estimator proposed achieves a gain in efficiency whether the conditional score is correctly specified or not when the conditional score is correctly specified the estimator reaches the semiparametric variance bound within the class of estimating functions that are generated by u the practical performance of the estimator is evaluated by using simulation and a data set that is based on the 1996 us presidential election

efficient information theoretic inference for conditional moment restrictions abstract the generalised method of moments estimator may be substantially biased in finite samples especially so when there are large numbers of unconditional moment conditions this paper develops a class of first order equivalent semi parametric efficient estimators and tests for conditional moment restrictions models based on a local or kernel weighted version of the cressie read power divergence family of discrepancies this approach is similar in spirit to the empirical likelihood methods of kitamura et al 2004 empirical likelihood based inference in conditional moment restrictions models econometrica 72 1667 1714 and tripathi and kitamura 2003 testing conditional moment restrictions annals of statistics 31 2059 2095 these efficient local methods avoid the necessity of explicit estimation of the conditional jacobian and variance matrices of the conditional moment restrictions and provide empirical conditional probabilities for the observations

unified empirical likelihood ratio tests for functional concurrent linear models and the phase transition from sparse to dense functional data summary we consider the problem of testing functional constraints in a class of functional concurrent linear models where both the predictors and the response are functional data measured at discrete time points we propose test procedures based on the empirical likelihood with bias corrected estimating equations to conduct both pointwise and simultaneous inferences the asymptotic distributions of the test statistics are derived under the null and local alternative hypotheses where sparse and dense functional data are considered in a unified framework we find a phase transition in the asymptotic null distributions and the orders of detectable alternatives from sparse to dense functional data specifically the tests proposed can detect alternatives of n order when the number of repeated measurements per curve is of an order larger than n 0 with n being the number of curves the transition points 0 for pointwise and simultaneous tests are different and both are smaller than the transition point in the estimation problem simulation studies and real data analyses are conducted to demonstrate the methods proposed

intractability of clique width parameterizations we show that edge dominating set hamiltonian cycle and graph coloring are w 1 hard parameterized by clique width it was an open problem explicitly mentioned in several papers whether any of these problems is fixed parameter tractable when parameterized by the clique width that is solvable in time g k cdot n o 1 on n vertex graphs of clique width k where g is some function of k only our results imply that the running time o n f k of many clique width based algorithms is essentially the best we can hope for up to a widely believed assumption from parameterized complexity namely fpt neq w 1

the bayesian bootstrap the bayesian bootstrap is the bayesian analogue of the bootstrap instead of simulating the sampling distribution of a statistic estimating a parameter the bayesian bootstrap simulates the posterior distribution of the parameter operationally and inferentially the methods are quite similar because both methods of drawing inferences are based on somewhat peculiar model assumptions and the resulting inferences are generally sensitive to these assumptions neither method should be applied without some consideration of the reasonableness of these model assumptions in this sense neither method is a true bootstrap procedure yielding inferences unaided by external assumptions

confidence tubes for multiple quantile plots via empirical likelihood the nonparametric empirical likelihood approach is used to obtain simultaneous confidence tubes for multiple quantile plots based on k independent possibly right censored samples these tubes are asymptotically distribution free except when both k 3 and censoring is present pointwise versions of the confidence tubes however are asymptotically distribution free in all cases the various confidence tubes are valid under minimal conditions the proposed methods are applied in three real data examples 1 introduction the quantile quantile q q plot is a well known and attractive graphical method for comparing two distributions especially when confidence limits are added in this paper we develop q q plot methods for the comparison of two or more distributions from randomly censored data more specifically we consider the problem of finding simultaneous confidence tubes for multiple quantile plots for brevity multi q plots from k independent samples of possibly right censored survival times the multi q plot is defined to be the k dimensional curve q1 p qk p parameterized by 0 p 1 where qj is the quantile function of the jth distribution it specializes to the ordinary q q plot in the two sample case the comparison of quantile functions is particularly useful for the analysis of survival data in biomedical settings frail and strong individuals corresponding to low and high values of p often respond to different treatments in different ways so treatment effects can be hard to determine from comparison of mean or median survival times alone see for example doksum 1974 the approach developed here allows comparison of treatments simultaneously across all frailty levels our approach is based on the nonparametric empirical likelihood method this method was originally developed by thomas and grunkemeier 1975 and owen 1988 1990 as a way of improving upon wald type confidence regions there now exists a substantial literature on empirical likelihood indicating that it is widely viewed as a desirable and natural approach to

interval estimation of value at risk based on garch models with heavy tailed innovations arch and garch models are widely used to model financial market volatilities in risk management applications considering a garch model with heavy tailed innovations we characterize the limiting distribution of an estimator of the conditional value at risk var which corresponds to the extremal quantile of the conditional distribution of the garch process we propose two methods the normal approximation method and the data tilting method for constructing confidence intervals for the conditional var estimator and assess their accuracies by simulation studies finally we apply the proposed approach to an energy market data set

intentionally biased bootstrap methods summary a class of weighted bootstrap techniques called biased bootstrap or b bootstrap methods is introduced it is motivated by the need to adjust empirical methods such as the uniform bootstrap in a surgical way to alter some of their features while leaving others unchanged depending on the nature of the adjustment the b bootstrap can be used to reduce bias or to reduce variance or to render some characteristic equal to a predetermined quantity examples of the last application include a b bootstrap approach to hypothesis testing in nonparametric contexts where the b bootstrap enables simulation under the null hypothesis even when the hypothesis is false and a b bootstrap competitor to tibshirani s variance stabilization method an example of the bias reduction application is adjustment of nadaraya watson kernel estimators to make them competitive with local linear smoothing other applications include density estimation under constraints outlier trimming sensitivity analysis skewness or kurtosis reduction and shrinkage

asymptotic refinements of a misspecification robust bootstrap for gel estimators abstract i propose a nonparametric iid bootstrap procedure for the empirical likelihood the exponential tilting and the exponentially tilted empirical likelihood estimators that achieves asymptotic refinements for t tests and confidence intervals and wald tests and confidence regions based on such estimators furthermore the proposed bootstrap is robust to model misspecification i e it achieves asymptotic refinements regardless of whether the assumed moment condition model is correctly specified or not this result is new because asymptotic refinements of the bootstrap based on these estimators have not been established in the literature even under correct model specification monte carlo experiments are conducted in dynamic panel data setting to support the theoretical finding as an application bootstrap confidence intervals for the returns to schooling of hellerstein and imbens 1999 are calculated the result suggests that the returns to schooling may be higher

empirical likelihood inference for the area under the roc curve for a continuous scale diagnostic test the most commonly used summary index of the receiver operating characteristic curve roc is the area under the curve auc that measures the accuracy of the diagnostic test in this article we propose an empirical likelihood el approach for the inference on the auc first we define an el ratio for the auc and show that its limiting distribution is a scaled chi square distribution we then obtain an el based confidence interval for the auc using the scaled chi square distribution this el inference for the auc can be extended to stratified samples and the resulting limiting distribution is a weighted sum of independent chi square distributions additionally we conduct simulation studies to compare the relative performance of the proposed el based interval with the existing normal approximation based intervals and bootstrap intervals for the auc

quantile calculus and censored regression quantile regression has been advocated in survival analysis to assess evolving covariate effects however challenges arise when the censoring time is not always observed and may be covariate dependent particularly in the presence of continuously distributed covariates in spite of several recent advances existing methods either involve algorithmic complications or impose a probability grid the former leads to difficulties in the implementation and asymptotics whereas the latter introduces undesirable grid dependence to resolve these issues we develop fundamental and general quantile calculus on cumulative probability scale in this article upon recognizing that probability and time scales do not always have a one to one mapping given a survival distribution these results give rise to a novel estimation procedure for censored quantile regression based on estimating integral equations a numerically reliable and efficient progressive localized minimization plmin algorithm is proposed for the computation this procedure reduces exactly to the kaplan meier method in the k sample problem and to standard uncensored quantile regression in the absence of censoring under regularity conditions the proposed quantile coefficient estimator is uniformly consistent and converges weakly to a gaussian process simulations show good statistical and algorithmic performance the proposal is illustrated in the application to a clinical study

fugitive search games on graphs and related parameters abstract the goal of a fugitive search game on a graph is to trap a fugitive that hides on the vertices of the graph by systematically placing searchers on the vertices the fugitive is assumed to have complete knowledge of the graph and of the searchers moves but is restricted to move only along paths whose vertices are not guarded by searchers the search number of the graph is the least number of searchers necessary to trap the fugitive variants of the fugitive search game have been related to important graph parameters like treewidth and pathwidth in this paper we introduce a class of fugitive search games where the searchers do not see the fugitive and the fugitive can only move just before a searcher is placed on the vertex it occupies letting the fugitive s speed i e the maximum number of edges the fugitive can traverse at a single move vary we get different games we show that if the speed of the fugitive is unbounded then the search number minus 1 is equal to the treewidth of the graph while if the speed is 1 then the search number minus 1 is equal to a polynomially computable graph parameter which is called width or alternatively linkage and is studied in the context of the constraint satisfaction and boolean satisfiability problems we also show that in the above two cases the search number remains the same even if we consider only search strategies that at every step further restrict the fugitive s possible resorts this monotonicity property is usually expressed as recontamination does not help moreover we give an equivalent characterization of the search number for any given fugitive speed in terms of an elimination ordering of the vertices of the graph using this characterization we show that for any graph if the length of any chordless cycle is bounded by a constant s s 3 then the treewidth of the graph plus 1 is equal to the search number for fugitive speed s 2

of quantiles and expectiles consistent scoring functions choquet representations and forecast rankings summary in the practice of point prediction it is desirable that forecasters receive a directive in the form of a statistical functional for example forecasters might be asked to report the mean or a quantile of their predictive distributions when evaluating and comparing competing forecasts it is then critical that the scoring function used for these purposes be consistent for the functional at hand in the sense that the expected score is minimized when following the directive we show that any scoring function that is consistent for a quantile or an expectile functional can be represented as a mixture of elementary or extremal scoring functions that form a linearly parameterized family scoring functions for the mean value and probability forecasts of binary events constitute important examples the extremal scoring functions admit appealing economic interpretations of quantiles and expectiles in the context of betting and investment problems the choquet type mixture representations give rise to simple checks of whether a forecast dominates another in the sense that it is preferable under any consistent scoring function in empirical settings it suffices to compare the average scores for only a finite number of extremal elements plots of the average scores with respect to the extremal scoring functions which we call murphy diagrams permit detailed comparisons of the relative merits of competing forecasts

empirical likelihood as a goodness of fit measure summary the method of empirical likelihood can be viewed as one of allocating probabilities to an n cell contingency table so as to minimise a goodness of fit criterion it is shown that when the cressie read power divergence statistic is used as the criterion confidence regions enjoying the same convergence rates as those found for empirical likelihood can be obtained for the entire range of values of the cressie read parameter including 1 maximum entropy 0 empirical likelihood and 1 pearson s z2 it is noted that in the power divergence family empirical likelihood is the only member which is bartlettcorrectable however simulation results suggest that for the mean using a scaled f distribution yields more accurate coverage levels for moderate sample sizes

a note on the empirical likelihood confidence band for hazards ratio with covariate adjustment in medical studies comparing two treatments in the presence of censored data the stratified cox model is an important tool that has the ability to flexibly handle non proportional hazards while allowing parsimonious covariate adjustment in order to capture the cumulative treatment effect the ratio of the treatment specific cumulative baseline hazards is often used as a measure of the treatment effect pointwise and simultaneous confidence bands associated with the estimated ratio provide a global picture of how the treatment effect evolves over time recently dong and matthews 2012 biometrics 68 408 418 proposed to construct a pointwise confidence interval for the ratio using a plug in type empirical likelihood approach however their result on the limiting distribution of the empirical likelihood ratio is generally incorrect and the resulting confidence interval is asymptotically undercovering in this article we derive the correct limiting distribution for the likelihood ratio statistic we also present simulation studies to demonstrate the effectiveness of our approach

approximation algorithms for digraph width parameters several problems that are np hard on general graphs are efficiently solvable on graphs with bounded treewidth efforts have been made to generalize treewidth and the related notion of pathwidth to digraphs directed treewidth dag width and kelly width are some such notions which generalize treewidth whereas directed pathwidth generalizes pathwidth each of these digraph width measures have an associated decomposition structure in this paper we present approximation algorithms for all these digraph width parameters in particular we give an o log n approximation algorithm for directed treewidth and an o log 3 2 n approximation algorithm for directed pathwidth dag width and kelly width our algorithms construct the corresponding decompositions whose widths agree with the above mentioned approximation factors

data tilting for time series summary we develop a general methodology for tilting time series data attention is focused on a large class of regression problems where errors are expressed through autoregressive processes the class has a range of important applications and in the context of our work may be used to illustrate the application of tilting methods to interval estimation in regression robust statistical inference and estimation subject to constraints the method can be viewed as empirical likelihood with nuisance parameters

posterior consistency of nonparametric conditional moment restricted models this paper addresses the estimation of the nonparametric conditional moment restricted model that involves an infinite dimensional parameter g0 we estimate it in a quasi bayesian way based on the limited information likelihood and investigate the impact of three types of priors on the posterior consistency i truncated prior priors supported on a bounded set ii thin tail prior a prior that has very thin tail outside a growing bounded set and iii normal prior with nonshrinking variance in addition g0 is allowed to be only partially identified in the frequentist sense and the parameter space does not need to be compact the posterior is regularized using a slowly growing sieve dimension and it is shown that the posterior converges to any small neighborhood of the identified region we then apply our results to the nonparametric instrumental regression model finally the posterior consistency using a random sieve dimension parameter is studied

uniqueness of the spatial median on montre que si une mesure de probabilite sur un espace euclidien n est pas concentree sur une droite alors sa mediane spatiale est unique

an approach to multivariate covariate dependent quantile contours with application to bivariate conditional growth charts multivariate quantile contours are useful in numerous applications and have been studied in different contexts however no easy solutions exist when dynamic and conditional quantile contours are needed without strong distributional assumptions in this article we propose a new form of bivariate quantile contours and a two stage estimation procedure to take time effect into account the proposed procedure relies on quantile regression for longitudinal data and is flexible to include potentially important covariates as necessary in addition we propose a visual model assessment tool and discuss a practical guideline for model selection the performance of the proposed methodology is demonstrated by a simulation study as well as an application to joint height weight screening of young children in the united states we construct bivariate growth charts by a nested sequence of age dependent and covariate varying quantile contours of height and weight and use it to locate an individual subject s percentile

the spatial distribution in infinite dimensional spaces and related quantiles and depths the spatial distribution has been widely used to develop various nonparametric procedures for finite dimensional multivariate data in this paper we investigate the concept of spatial distribution for data in infinite dimensional banach spaces many technical difficulties are encountered in such spaces that are primarily due to the non compactness of the closed unit ball in this work we prove some glivenko cantelli and donsker type results for the empirical spatial distribution process in infinite dimensional spaces the spatial quantiles in such spaces can be obtained by inverting the spatial distribution function a bahadur type asymptotic linear representation and the associated weak convergence results for the sample spatial quantiles in infinite dimensional spaces are derived a study of the asymptotic efficiency of the sample spatial median relative to the sample mean is carried out for some standard probability distributions in function spaces the spatial distribution can be used to define the spatial depth in infinite dimensional banach spaces and we study the asymptotic properties of the empirical spatial depth in such spaces we also demonstrate the spatial quantiles and the spatial depth using some real and simulated functional data

on empirical likelihood statistical functions abstract we consider the empirical likelihood method for estimation of distribution and quantile functions where side information is incorporated through moment conditions we systematically study the asymptotic properties of the estimators such as the uniform strong laws of large numbers and weak convergence over classes of functions two monte carlo examples are also given to illustrate the practical utility of the method

nonparametric estimation of convex models via mixtures we present a general approach to estimating probability measures constrained to lie in a convex set we represent constrained measures as mixtures of simple known extreme measures and so the problem of estimating a constrained measure becomes one of estimating an unconstrained mixing measure convex constraints arise in many modeling situations such as estimation of the mean and estimation under stochastic ordering constraints we describe mixture representation techniques for these and other situations and discuss applications to maximum likelihood and bayesian estimation

asymptotic equivalence between the cox estimator and the general ml estimators of regression and survival parameters in the cox model the usual approach to estimating regression parameters in the cox regression model uses the partial likelihood if the covariates are not time dependent the model can be stated in terms of the survival function which allows one to derive a generalized likelihood containing both regression and survival curve parameters it is shown that in the absence of ties an estimator results which is asymptotically equivalent to the partial likelihood estimator a joint information matrix leads simply to standard errors for both regression and survival curve parameters which are asymptotically correct

constrained empirical likelihood confidence regions in nonignorable covariate missing data problems missing covariates in regression analysis are a pervasive problem in medical social and economic researches we study empirical likelihood confidence regions for unconstrained and constrained regression parameters in a nonignorable covariate missing data problem for an assumed conditional mean regression model we assume that some covariates are fully observed but other covariates are missing for some subjects by exploitation of a probability model of missingness and a working conditional score model from a semiparametric perspective we build a system of unbiased estimating equations where the number of equations exceeds the number of unknown parameters based on the proposed estimating equations we introduce unconstrained and constrained empirical likelihood ratio statistics to construct empirical likelihood confidence regions for the underlying regression parameters without and with constraints we establish the asymptotic distributions of the proposed empirical likelihood ratio statistics simulation results show that the proposed empirical likelihood methods have a better finite sample performance than other competitors in terms of coverage probability and interval length finally we apply the proposed empirical likelihood methods to the analysis of a data set from the us national health and nutrition examination survey

inference of the youden index and associated threshold using empirical likelihood for quantiles the youden index is a widely used measure in the framework of medical diagnostics where the effectiveness of a biomarker screening marker or predictor for classifying a disease status is studied when the biomarker is continuous it is important to determine the threshold or cut off point to be used in practice for the discrimination between diseased and healthy populations we introduce two methods aimed at estimating the youden index and its associated threshold the first one is a modified version of a recent approach based on the delta method and the second one is based on the adjusted empirical likelihood for quantiles in the setting of a two sample problem we also include cis for both of them in the simulation study we compare both methods under different scenarios finally a real example of prostatic cancer well known in the literature is analysed to provide the reader with a better understanding of the new methodology copyright 2011 john wiley sons ltd

a partial k arboretum of graphs with bounded treewidth the notion of treewidth has seen to be a powerful vehicle for many graph algorithmic studies this survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class also some mutual relations between such classes are discussed

a test for model specification of diffusion processes we propose a test for model specification of a parametric diffusion process based on a kernel estimation of the transitional density of the process the empirical likelihood is used to formulate a statistic for each kernel smoothing bandwidth which is effectively a studentized l2 distance between the kernel transitional density estimator and the parametric transitional density implied by the parametric process to reduce the sensitivity of the test on smoothing bandwidth choice the final test statistic is constructed by combining the empirical likelihood statistics over a set of smoothing bandwidths to better capture the finite sample distribution of the test statistic and data dependence the critical value of the test is obtained by a parametric bootstrap procedure properties of the test are evaluated asymptotically and numerically by simulation and by a real data example

digraph measures kelly decompositions games and orderings we consider various well known equivalent complexity measures for graphs such as elimination orderings k trees and cops and robber games and study their natural translations to digraphs we show that on digraphs the translations of these measures are also equivalent and induce a natural connectivity measure we introduce a decomposition for digraphs and an associated width kelly width which is equivalent to the aforementioned measure we demonstrate its usefulness by exhibiting potential applications including polynomial time algorithms for np complete problems on graphs of bounded kelly width and complexity analysis of asymmetric matrix factorization finally we compare the new width to other known decompositions of digraphs

jackknife empirical likelihood empirical likelihood has been found very useful in many different occasions however when applied directly to some more complicated statistics such as u statistics it runs into serious computational difficulties in this paper we introduce a so called jackknife empirical likelihood jel method the new method is extremely simple to use in practice in particular the jel is shown to be very effective in handling one and two sample u statistics the jel can be potentially useful for other nonlinear statistics

on fractile transformation of covariates in regression the need for comparing two regression functions arises frequently in statistical applications comparison of the usual regression functions is not very meaningful in situations where the distributions and the ranges of the covariates are different for the populations for instance in econometric studies the prices of commodities and people s incomes observed at different time points may not be on comparable scales due to inflation and other economic factors in this article we describe a method of standardizing the covariates and estimating the transformed regression function which then become comparable we develop smooth estimates of the fractile regression function and study its statistical properties analytically as well as numerically we also provide a few real examples that illustrate the difficulty in comparing the usual regression functions and motivate the need for the fractile transformation our analysis of the real examples leads to new and useful statistical conclusions that are missed b

semiparametric roc surfaces for continuous diagnostic tests based on two test measurements we propose a semiparametric method for estimating roc surfaces for continuous diagnostic tests based on two test measurements such a three class diagnostic problem based on two test measurements arises naturally from some dna amplification related diagnostic scenarios simulation results show that our proposed semiparametric roc surface estimator is more efficient than the nonparametric counterpart and is quite comparable with the parametric counterpart when model assumption of the data is correctly specified for the parametric method moreover when the parametric model assumption is not true our proposed semiparametric method is superior to both nonparametric and parametric methods some discussions and a simulated example along with r programs for implementation of the proposed method are also provided

on m estimators and normal quantiles this paper explores a class of robust estimators of normal quantiles filling the gap between maximum likelihood estimators and empirical quantiles our estimators are linear combinations of m estimators their asymptotic variances can be arbitrarily close to variances of the maximum likelihood estimators compared with empirical quantiles the new estimators offer considerable reduction of variance at near normal probability distributions

testing for spherical symmetry of a multivariate distribution rotationally invariant tests based on test statistics of the von mises type are proposed under the hypothesis of spherical symmetry of a multivariate distribution the tests are distribution free when the hypothesis of spherical symmetry is true the asymptotic distribution of the test statistics are derived under the null hypothesis and under any fixed alternative a simple criterion for consistency is given the results are illustrated by numerous examples of test statistics which give rise to tests being consistent against all alternatives

m estimation convexity and quantiles the paper develops a class of extensions of the univariate quantile function to the multivariate case m quantiles related in a certain way to m parameters of a probability distribution and their m estimators the spatial geometric quantiles recently introduced by koltchinskii and dudley and by chaudhuri as well as the regression quantiles of koenker and basset are the examples of the m quantile function discussed in the paper we study the main properties of m quantiles and develop the asymptotic theory of empirical m quantiles we use m quantiles to extend l parameters and l estimators to the multivariate case to introduce a bootstrap test for spherical symmetry of a multivariate distribution and to extend the notion of regression quantiles to multiresponse linear regression models

empirical likelihood methods with weakly dependent processes this paper studies the method of empirical likelihood in models with weakly dependent processes in such cases if the likelihood function is formulated as if the data process were independent obviously empirical likelihood fails we propose to use empirical likelihood of blocks of observations to solve this problem in a nonparametric manner this method of blockwise empirical likelihood preserves the dependence of data and the resulting likelihood ratios can be used to construct asymptotically valid confidence intervals we consider general estimating equations for which an efficient estimator is derived by maximizing blockwise empirical likelihood we also introduce blocks of blocks empirical likelihood to conduct inference for parameters of the infinite dimensional joint distribution of data the smooth function model is used for such cases we show that blockwise empirical likelihood of the smooth function model with weakly dependent processes is bartlett correctable a wide variety of problems such as time series regressions and spectral densities can be treated using our methodology

nonparametric likelihood ratio confidence intervals we consider construction of two sided nonparametric confidence intervals in a smooth function model setting a nonparametric likelihood approach based on stein s least favourable family is proposed as an alternative to empirical likelihood the approach enjoys the same asymptotic properties as empirical likelihood but is analytically and computationally less cumbersome the simplicity of the method allows us to propose and analyse asymptotic and bootstrapping techniques as a means of reducing coverage error to levels comparable with those obtained by more computationally intensive techniques such as the iterated bootstrap a simulation study confirms that coverage error may be substantially reduced by simple analytic adjustment of the nonparametric likelihood interval and that bootstrapping the distribution of the nonparametric likelihood ratio results in very desirable coverage accuracy

likelihood ratio based confidence bands for survival functions abstract thomas and grunkemeier introduced a nonparametric likelihood ratio approach to confidence interval estimation of survival probabilities based on right censored data we construct simultaneous confidence bands using this approach the boundaries of the bands are contained within 0 1 a procedure essentially equivalent to a bias correction is developed the resulting increase in coverage accuracy is illustrated by an example and a simulation study we look at various versions of log likelihood ratio based confidence bands and compare them to the hall wellner band and nair s equal precision band we also construct likelihood ratio based bands for cumulative hazard functions

random sieve likelihood and general regression models abstract consider a semiparametric regression model y f x where f is a known function is an unknown vector consists of a random error and possibly of some unobserved variables and the distribution f of x is unspecified this article introduces in a general setting new methodology for estimating and f the proposed method constructs a profile likelihood defined on random level sets a random sieve the proposed method is related to empirical likelihood but is more generally applicable four examples are discussed including a quadratic model high dimensional semiparametric regression a nonparametric random effects model and linear regression with right censored data simulation results and asymptotic analysis support the utility and effectiveness of the proposed method

convex models mle and misspecification we analyze the asymptotic behavior of maximum likelihood estimators mle in convex dominated models when the true distribution generating the independent data does not necessarily belong to the model inspired by the hellinger distance and its properties we introduce a family of divergences contrast functions which allow a unified treatment of well and misspecified convex models convergence and rates of convergence of the mle with respect to our divergences are obtained from inequalities satisfied by these divergences and results from empirical process theory uniform laws of large numbers and maximal inequalities as a particular case we recover existing results for hellinger convergence of mle in well specified convex models four examples are considered mixtures of discrete distributions monotone densities decreasing failure rate distributions and a finite dimensional parametric model

estimation with survey data under nonignorable nonresponse or informative sampling nonresponse is a very common phenomenon in survey sampling nonignorable nonresponse that is a response mechanism that depends on the values of the variable having nonresponse is the most difficult type of nonresponse to handle this article studies a likelihood based estimation method for data with nonignorable nonresponse the likelihood is semiparametric in the sense that it consists of a parametric component for the response mechanism such as a logistic model and a nonparametric component for the distribution of the variable of interest and the covariates we show how auxiliary information can be augmented to improve the efficiency of estimators asymptotic distributions for the resulting parameter estimates are derived a simulation study shows that the proposed method gives promising results the method can also be applied to problems with two phase sampling in which the second phase sampling is informative

testing conditional moment restrictions let x z be a pair of observable random vectors we construct a new smoothed empirical likelihood based test for the hypothesis e g z x 0 w p 1 where g is a vector of known functions and an unknown finite dimensional parameter we show that the test statistic is asymptotically normal under the null hypothesis and derive its asymptotic distribution under a sequence of local alternatives furthermore the test is shown to possess an optimality property in large samples simulation evidence suggests that it also behaves well in small samples

concavity and estimation on donne des conditions simplifiees pour la consistance et la normalite asymptotique d estimations m deduites par maximisation de moyennes de fonctions concaves aleatoires identiquement distribuees et independantes des applications sont donnees pour l estimation de maximum de vraisemblance

on the bootstrap and likelihood based confidence regions summary we describe a method for constructing likelihood based confidence regions for a vector parameter using the bootstrap and nonparametric density estimation the technique is illustrated by application to a numerical example and its theoretical properties are elucidated it is argued that likelihood based regions should not be approximated by ellipses if we are to have any hope of capturing first order departures from normality bootstrap algorithms for constructing simultaneous confidence intervals for the components of a vector parameter are also presented advantages of the percentile t method over the ordinary percentile method are demonstrated in a multivariate setting

merging information for semiparametric density estimation summary the density ratio model specifies that the likelihood ratio of m 1 probability density functions with respect to the mth is of known parametric form without reference to any parametric model we study the semiparametric inference problem that is related to the density ratio model by appealing to the methodology of empirical likelihood the combined data from all the samples leads to more efficient kernel density estimators for the unknown distributions we adopt variants of well established techniques to choose the smoothing parameter for the density estimators proposed