scalar curvature of a metric with unit volume the problem of finding riemannian metrics on a closed manifold with prescribed scalar curvature function is now fairly well understood from the works of kazdan and warner in 1970 s 10 and references cited in it in this paper we shall consider the same problem under a constraint on the volume for this purpose it is useful to introduce an invariant p m of a smooth closed manifold m which will be called the yamabe number of m defined as the supremum of m c of all conformal classes c of riemannian metrics on m

current status data with competing risks consistency and rates of convergence of the mle we study nonparametric estimation of the sub distribution functions for current status data with competing risks our main interest is in the nonparametric maximum likelihood estimator mle and for comparison we also consider a simpler naive estimator both types of estimators were studied by jewell van der laan and henneman biometrika 2003 90 183 197 but little was known about their large sample properties we have started to fill this gap by proving that the estimators are consistent and converge globally and locally at rate n1 3 we also show that this local rate of convergence is optimal in a minimax sense the proof of the local rate of convergence of the mle uses new methods and relies on a rate result for the sum of the mles of the sub distribution functions which holds uniformly on a fixed neighborhood of a point our results are used in groeneboom maathuis and wellner ann statist 2008 36 1064 1089 to obtain the local limiting distributions of the estimators

learning a decision boundary from stochastic examples incremental algorithms with and without queries even if it is not possible to reproduce a target input output relation a learning machine should be able to minimize the probability of making errors a practical learning algorithm should also be simple enough to go without memorizing example data if possible incremental algorithms such as error backpropagation satisfy this requirement we propose incremental algorithms that provide fast convergence of the machine parameter to its optimal choice o with respect to the number of examples t we will consider the binary choice model whose target relation has a blurred boundary and the machine whose parameter specifies a decision boundary to make the output prediction the question we wish to address here is how fast can approach o depending upon whether in the learning stage the machine can specify inputs as queries to the target relation or the inputs are drawn from a certain distribution if queries are permitted the machine can achieve the fastest convergence o 2 o t 1 if not o t 1 convergence is generally not attainable for learning without queries we showed in a previous paper that the error minimum algorithm exhibits a slow convergence o 2 o t 2 3 we propose here a practical algorithm that provides a rather fast convergence o t 4 5 it is possible to further accelerate the convergence by using more elaborate algorithms the fastest convergence turned out to be o lnt 2t 1 this scaling is considered optimal among possible algorithms and is not due to the incremental nature of our algorithm

asymptotic efficiency in semi parametric models with censoring abstract we consider a regression model subject to bivariate censoring in which the errors are independent of the explanatory variables our objective is to drop the assumption of bivaraite normality and obtain an information bound on asymptotic efficiency when the error distribution is unrestricted except for smoothness and regularity conditions this bound has simple form if there are no exclusion restrictions on the selection equation then positive information requires a restriction on the regression slope parameters and a continuous distribution for a component of the explanatory variables we also consider a binary choice model under the weak assumption that the error distribution has zero median conditional on the explanatory variables here the semi parametric information bound is zero hence although a consistent estimator exists it is not possible to attain covergence at rate 1 n

maximum likelihood estimation of a multi dimensional log concave density fn is shown to have smaller mean integrated squared error compared with kernel based methods even when we allow the use of a theoret ical optimal fixed bandwidth for the kernel estimator that would not be available in practice we also present a real data clustering example which shows that our methodology can be used in conjunction with the expectation maximization algorithm to fit finite mixtures of log concave densities

inference in semiparametric binary response models with interval data this paper studies the semiparametric binary response model with interval data investigated by manski and tamer 2002 in this partially identified model we propose a new estimator based on mt s modified maximum score mms method by introducing density weights to the objective function which allows us to develop asymptotic properties of the proposed set estimator for inference we show that the density weighted mms estimator converges at a nearly cube root n rate we propose an asymptotically valid inference procedure for the identified region based on subsampling monte carlo experiments provide supports to our inference procedure

on h on the complement of c1 α curves abstract let be a quasiconformal mapping on the plane with complex dilatation we show that if satisfies a certain carleson measure condition then one can transfer h on the upper half plane onto the corresponding space in the complement of the quasicircle r and that this condition on characterizes c 1 curves

new donsker classes several classes of functions are shown to be donsker by an argument based on partitioning the sample space one example is the class of all nondecreasing functions f r r such that 0 f f for a given function f with f 2 dp p

the behavior of the npmle of a decreasing density near the boundaries of the support we investigate the behavior of the nonparametric maximum likelihood estimator f n for a decreasing density f near the boundaries of the support of f we establish the limiting distribution of f n n where we need to distinguish between different values of 0 a 1 similar results are obtained for the upper endpoint of the support in the case it is finite this yields consistent estimators for the values of f at the boundaries of the support the limit distribution of these estimators is established and their performance is compared with the penalized maximum likelihood estimator

the sorted effects method discovering heterogeneous effects beyond their averages the partial ceteris paribus effects of interest in nonlinear and interactive linear models are heterogeneous as they can vary dramatically with the underlying observed or unobserved covariates despite the apparent importance of heterogeneity a common practice in modern empirical work is to largely ignore it by reporting average partial effects or at best average effects for some groups while average effects provide very convenient scalar summaries of typical effects by definition they fail to reflect the entire variety of the heterogeneous effects in order to discover these effects much more fully we propose to estimate and report sorted effects a collection of estimated partial effects sorted in increasing order and indexed by percentiles by construction the sorted effect curves completely represent and help visualize the range of the heterogeneous effects in one plot they are as convenient and easy to report in practice as the conventional average partial effects they also serve as a basis for classification analysis where we divide the observational units into most or least affected groups and summarize their characteristics we provide a quantification of uncertainty standard errors and confidence bands for the estimated sorted effects and related classification analysis and provide confidence sets for the most and least affected groups the derived statistical results rely on establishing key new mathematical results on hadamard differentiability of a multivariate sorting operator and a related classification operator which are of independent interest we apply the sorted effects method and classification analysis to demonstrate several striking patterns in the gender wage gap we find that this gap is particularly strong for married women ranging from 60 to 0 between the 2 and 98 percentiles as a function of observed and unobserved characteristics while the gap for never married women ranges from 40 to 20 the most adversely affected women tend to be married do not have college degrees work in sales and have high levels of potential experience

confidence sets for the date of a single break in linear time series regressions this paper considers the problem of constructing confidence sets for the date of a single break in a linear time series regression we establish analytically and by small sample simulation that the current standard method in econometrics for constructing such confidence intervals has a coverage rate far below nominal levels when breaks are of moderate magnitude given that breaks of moderate magnitude are a theoretically and empirically relevant phenomenon we proceed to develop an appropriate alternative we suggest constructing confidence sets by inverting a sequence of tests each of the tests maintains a specific break date under the null hypothesis and rejects when a break occurs elsewhere by inverting a certain variant of a locally best invariant test we ensure that the asymptotic critical value does not depend on the maintained break date a valid confidence set can hence be obtained by assessing which of the sequence of test statistics exceeds a single number

testing for threshold effects in regression models in this article we develop a general method for testing threshold effects in regression models using sup likelihood ratio lr type statistics although the sup lr type test statistic has been considered in the literature our method for establishing the asymptotic null distribution is new and nonstandard the standard approach in the literature for obtaining the asymptotic null distribution requires that there exist a certain quadratic approximation to the objective function the article provides an alternative novel method that can be used to establish the asymptotic null distribution even when the usual quadratic approximation is intractable we illustrate the usefulness of our approach in the examples of the maximum score estimation maximum likelihood estimation quantile regression and maximum rank correlation estimation we establish consistency and local power properties of the test we provide some simulation results and also an empirical application to tipping in racial segregation this ar

semiparametric estimation of the random utility model with rank ordered choice data abstract we propose semiparametric methods for estimating random utility models using rank ordered choice data our primary method is the generalized maximum score gms estimator with partially rank ordered data the gms estimator allows for arbitrary forms of interpersonal heteroskedasticity with fully rank ordered data the gms estimator becomes considerably more flexible allowing for random coefficients and alternative specific heteroskedasticity and correlations the gms estimator has a non standard asymptotic distribution and a convergence rate of n 1 3 we proceed to construct its smoothed version which is asymptotically normal with a faster convergence rate of n d 2 d 1 where d 2 increases in the strength of smoothness assumptions

inference of signs of interaction effects in simultaneous games with incomplete information this paper studies the inference of interaction effects in discrete simultaneous games with incomplete information we propose a test for the signs of state dependent interaction effects that does not require parametric specifications of players payoffs the distributions of their private signals or the equilibrium selection mechanism the test relies on the commonly invoked assumption that players private signals are independent conditional on observed states the procedure is valid in but does not rely on the presence of multiple equilibria in the data generating process dgp as a by product we propose a formal test for multiple equilibria in the dgp we also implement the test using data on radio programming of commercial breaks in the united states and infer stations incentives to synchronize their commercial breaks our results support the earlier finding by sweeting 2009 that stations have stronger incentives to coordinate and air commercials at the same time during rush hours and in smaller markets

robust out of sample inference this paper presents analytical empirical and simulation results concerning inference about the moments of nondifferentiable functions of out of sample forecasts and forecast errors special attention is given to the measurement of a model s predictive ability using the test of equal mean absolute error tests for equal mean absolute error and mean square error are used to evaluate predictions of excess returns to the s p 500 composite simulations indicate that appropriately constructed tests for equal mean absolute error can provide more accurately sized and more powerful tests than inappropriately constructed tests for equal mean absolute error and mean square error

boundedly pivotal structural change tests in continuous updating gmm with strong weak identification and completely unidentified cases abstract this paper develops structural change tests in the continuous updating gmm framework that are robust to weak identification we propose likelihood ratio like anderson rubin 1949 estimation of the parameters of a single equation in a complete system of stochastic equations annals of mathematical statistics 20 46 63 and kleinbergen 2005 testing parameters in gmm without assuming that they are identified manuscript brown university types of tests since the limits of the test statistics are not nuisance parameter free bounds for the limit of the test statistics are derived the bounds are nuisance parameter free and robust to identification problems simulations show that the anderson rubin 1949 and kleinbergen 2005 testing parameters in gmm without assuming that they are identified manuscript brown university type of tests have very good small sample properties in the case of weak instruments the sup lm test of andrews 1993a tests for parameter instability and structural change with unknown change point econometrica 61 821 856 rejects the true null of parameter stability more than the nominal level and also has low power in the weak instrument setup

nonparametric tests for constant tail dependence with an application to energy and finance the present paper proposes new tests for detecting structural breaks in the tail dependence of multivariate time series using the concept of tail copulas to obtain asymptotic properties we derive a new limit result for the sequential empirical tail copula process moreover consistency of both the tests and a change point estimator are proven we analyze the finite sample behavior of the tests by monte carlo simulations finally and crucial from a risk management perspective we apply the new findings to datasets from energy and financial markets

two stage plans for estimating the inverse of a monotone function this study investigates two stage plans based on nonparametric procedures for estimating an inverse regression function at a given point specifically isotonic regression is used at stage one to obtain an initial estimate followed by another round of isotonic regression in the vicinity of this estimate at stage two it is shown that such two stage plans accelerate the convergence rate of one stage procedures and are superior to existing two stage procedures that use local parametric approximations at stage two when the available budget is moderate and or the regression function is ill behaved both wald and likelihood ratio type confidence intervals for the threshold value of interest are investigated and the latter are recommended in applications due to their simplicity and robustness the developed plans are illustrated through a comprehensive simulation study and an application to car fuel efficiency data

maximum score estimation of disequilibrium models and the role of anticipatory price setting abstract in this paper a general model of disequilibrium quantity determination under median regression assumptions is combined with weak assumptions on the price adjustment to yield a consistent maximum score estimator in contrast to sapra 1986 the supply and demand parameters are separately identifiable and the estimator employs all of the observed data the price adjustment assumptions can be motivated by the anticipatory pricing model of green and laffont 1981 in contrast to the green laffont model however our assumptions only require that long run walrasian prices have positive probability as an illustrative application maximum score estimates of a model of the us commercial loan market are computed and compared to maximum likelihood estimates monte carlo results are also reported

on the bootstrap of the maximum score estimator this paper shows that the bootstrap does not consistently estimate the asymptotic distribution of the maximum score estimator the theory developed also applies to other estimators within a cube root convergence class for some single parameter estimators in this class the results suggest a simple method for inference based upon the bootstrap

persistence of fractional brownian motion with moving boundaries and applications we consider various problems related to the persistence probability of fractional brownian motion fbm which is the probability that the fbm x stays below a certain level until time t recently oshanin et al 2012 arxiv 1209 3313v2 have studied a physical model where persistence properties of fbm are shown to be related to scaling properties of a quantity jn called the steady state current it turns out that for this analysis it is important to determine persistence probabilities of fbm with a moving boundary we show that one can add a boundary of logarithmic order to an fbm without changing the polynomial rate of decay of the corresponding persistence probability which proves a result needed in oshanin et al 2013 phys rev lett at press arxiv 1209 3313v2 moreover we complement their findings by considering the continuous time version of jt finally we use the results for moving boundaries in order to improve estimates by molchan 1999 commun math phys 205 97 111 concerning the persistence properties of other quantities of interest such as the time when an fbm reaches its maximum on the time interval 0 1 or the last zero in the interval 0 1

convex clustering via l1 fusion penalization summary we study the large sample behaviour of a convex clustering framework which minimizes the sample within cluster sum of squares under an l1 fusion constraint on the cluster centroids this recently proposed approach has been gaining in popularity however its asymptotic properties have remained mostly unknown our analysis is based on a novel representation of the sample clustering procedure as a sequence of cluster splits determined by a sequence of maximization problems we use this representation to provide a simple and intuitive formulation for the population clustering procedure we then demonstrate that the sample procedure consistently estimates its population analogue and we derive the corresponding rates of convergence the proof conducts a careful simultaneous analysis of a collection of m estimation problems whose cardinality grows together with the sample size on the basis of the new perspectives gained from the asymptotic investigation we propose a key post processing modification of the original clustering framework we show both theoretically and empirically that the resulting approach can be successfully used to estimate the number of clusters in the population using simulated data we compare the proposed method with existing number of clusters and modality assessment approaches and obtain encouraging results we also demonstrate the applicability of our clustering method to the detection of cellular subpopulations in a single cell virology study

binary quantile regression with local polynomial smoothing abstract manski 1975 1985 proposed the maximum score estimator for the binary choice model under a weak conditional median restriction that converges at the rate of n 1 3 and the standardized version has a nonstandard distribution by imposing additional smoothness conditions horowitz 1992 proposed a smoothed maximum score estimator that often has large finite sample biases and is quite sensitive to the choice of smoothing parameter in this paper we propose a novel framework that leads to a local polynomial smoothing based estimator our estimator possesses finite sample and asymptotic properties typically associated with the local polynomial regression in addition our local polynomial regression based estimator can be extended to the panel data setting simulation results suggest that our estimators may offer significant improvement over the smoothed maximum score estimators

functional limit theorems for u processes un processus u est une collection de statistiques u indicees par une famille de noyaux symetriques on utilise la theorie limite classique pour demontrer pour le processus u des analogues du theoreme central limite pour les processus empiriques

asymptotic stochastic programs consider a stochastic program with unique solution by the notion of epi convergence in distribution in local coordinates we define the asymptotic stochastic program associated to it stochastic programs may be classified according to the type of the pertaining asymptotic program in particular three such types are studied in detail the normal shift program the wiener type program and the poisson hyperplane program conditions for the convergence to each of the three types are given

asymptotic theory for nonparametric estimation of survival curves under order restrictions f t g f t all t the nonparametric maximum likelihood estimators in these problems were derived by brunk franck hansen and hogg and dykstra in the present paper we derive their large sample distributions we present two sets of proofs depending on whether or not the data are right censored when centered and scaled by n 1r2 the estimators con verge in distribution to limiting processes related to the concave majorant of brownian motion the limiting distributions are not known in closed form but can be simulated for the purpose of forming asymptotic point wise confidence limits

semiparametric estimation of a binary response model with a change point due to a covariate threshold this paper is concerned with semiparametric estimation of a threshold binary response model the estimation method considered in the paper is semiparametric since the parameters for a regression function are finite dimensional while allowing for heteroskedasticity of unknown form in particular the paper considers manski 1975 1985 s maximum score estimator the model in this paper is irregular because of a change point due to an unknown threshold in a covariate this irregularity coupled with the discontinuity of the objective function of the maximum score estimator complicates the analysis of the asymptotic behavior of the estimator sufficient conditions for the identification of parameters are given and the consistency of the estimator is obtained it is shown that the estimator of the threshold parameter is n consistent and the estimator of the remaining regression parameters is cube root n consistent furthermore we obtain the asymptotic distribution of the estimators it turns out that a suitably normalized estimator of the regression parameters converges weakly to the distribution to which it would converge weakly if the true threshold value were known and likewise for the threshold estimator

asymptotics of reweighted estimators of multivariate location and scatter we investigate the asymptotic behavior ofa weighted sample mean and covariance where the weights are determined by the mahalanobis distances with respect to initial robust estimators we derive an explicit expansion for the weighted estimators from this expansion it can be seen that reweighting does not improve the rate ofconvergence ofthe initial estimators we also show that ifone uses smooth s estimators to determine the weights the weighted estimators are asymptotically normal finally we will compare the efficiency and local robustness of the reweighted s estimators with two other improvements of s estimators estimators and constrained m estimators

semiparametric identification of binary decision games of incomplete information with correlated private signals abstract this paper studies the identification and estimation of a static binary decision game of incomplete information we make no parametric assumptions on the joint distribution of private signals and allow them to be correlated we show that the parameters of interest can be point identified subject to a scale normalization under mild support requirements for the regressors publicly observed state variables and errors private signals following manski and tamer 2002 we propose a maximum score type estimator for the structural parameters and establish the asymptotic properties of the estimator

semiparametric estimation of a simultaneous game with incomplete information abstract this paper studies the problem of estimating the normal form payoff parameters of a simultaneous discrete game where the realization of such payoffs is not common knowledge the paper contributes to the existing literature in two ways first by making a comparison with the complete information case it formally describes a set of conditions under which allowing for private information in payoffs facilitates the identification of various features of the game second focusing on the incomplete information case it presents an estimation procedure based on the equilibrium properties of the game that relies on weak semiparametric assumptions and relatively flexible information structures which allow players to condition their beliefs on signals whose exact distribution function is unknown to the researcher the proposed estimators recover unobserved beliefs by solving a semiparametric sample analog of the population bayesian nash equilibrium conditions the asymptotic features of such estimators are studied for the case in which the distribution of unobserved shocks is known and the case in which it is unknown in both instances equilibrium uniqueness is assumed to hold only in a neighborhood of the true parameter value and for a subset z of realizations of the signals multiple equilibria are allowed elsewhere in the parameter space and no equilibrium selection theory is involved extensions to games where beliefs are conditioned on unobservables as well as general games with many players and actions are also discussed an empirical application of a simple capital investment game is included

robust inference for threshold regression models abstract this paper considers robust inference in threshold regression models when the practitioners do not know whether at the threshold point the true specification has a kink or a jump nesting previous works that assume either continuity or discontinuity at the threshold we find that the parameter values under the kink restriction are irregular points of the hessian matrix destroying the asymptotic normality and inducing the cube root convergence rate for the threshold estimate however we are able to obtain the same asymptotic distribution as hansen 2000 for the quasi likelihood ratio statistic for the unknown threshold we propose to construct confidence intervals for the threshold by bootstrap test inversion finite sample performances of the proposed procedures are examined through monte carlo simulations and an economic empirical application is given

universal donsker classes and metric entropy when mathfrak f is a universal donsker class then for independent indetically distributed i i d observation mathbf x 1 ldots mathbf x n with an unknown law p for any mathfrak f i in mathfrak f i 1 ldots m quad n 1 2 left mathfrak f 1 left mathbf x 1 right ldots mathfrak f i left mathbf x n right right 1 leq i leq m is asymptotically normal with mean vector n 1 2 left int mathfrak f i left mathbf x n right d mathbf p left x right right 1 leq i leq m and covariance matrix int mathfrak f i mathfrak f j d mathbf p int mathfrak f id mathbf p int mathfrak f jd mathbf p uniformly for mathfrak f i in mathfrak f then for certain statistics formed frome the mathfrak f i left mathbf x k right even where mathfrak f i may be chosen depending on the mathbf x k there will be asymptotic distribution as n rightarrow infty for example for mathbf x 2 statistics where f i are indicators of disjoint intervals depending suitably on mathbf x 1 ldots mathbf x n whose union is the real line mathbf x 2 quadratic forms have limiting distributions roy 1956 and watson 1958 which may however not be mathbf x 2 distributions and may depend on p chernoff and lehmann 1954 universal donsker classes of sets are up to mild measurability conditions just classes satisfying the vapnik cervonenkis comdinatorial conditions defined later in this section donsker the vapnik cervonenkis combinatorial conditions defined later in this section durst and dudley 1981 and dudley 1984 chapter 11 the use of such classes allows a variety of extensions of the roy watson results to general multidimensional sample spaces pollard 1979 and moore and subblebine 1981 vapnik and cervonenkis 1974 indicated application of their families of sets to classification pattern recognition problems more recently the classes have been applied to tree structured classifiacation breiman friedman olshen and stone 1984 chapter 12

quasiconformal mappings and chord arc curves given a quasiconformal mapping p on the plane what conditions on its dilatation guarantee that p r is rectifiable and pir is locally abso lutely continuous we show in this paper that if satisfies certain quadratic carleson measure conditions with small norm then p r is a chord arc curve with small constant and p x p 0 g0 ea t dt for x e r with a e bmo having small norm conversely given any such map from r c we show that it has an extension to c with the right kind of dilatation similar results hold with r replaced by a chord arc curve examples are given that show that it would be hard to improve these results applications are given to conformal welding and the theorem of coifman and meyer on the real analyticity of the riemann mapping on the manifold of chord arc curves let p be a quasiconformal map of the plane onto itself thus p is a homeomorphism with locally integrable distributional derivatives and ps upz where u e l c il ul100 1 here we use the notations fz af azf 2 z id f fz 2 aa ida f this function u is called the complex dilatation of p the mapping theorem for q c maps states that for each u e l c il ul100 1 there is a q c map p on c with dilatation u and p is unique up to normalization a basic problem is to understand how geometric properties of p are reflected in u for example one would like to have natural conditions on u which imply that p r is rectifiable and p ir is absolutely continuous this question arises naturally when considering problems in conformal mappings and conformal welding in this paper we obtain such estimates for the mapping theorem and we also give some applications our results involve bmo aoo weights chord arc curves and carleson measures and so we first review the appropriate definitions a locally integrable function f on r lies in bmo if llfil sup lil if x fil dx received by the editors june 1 1986 1980 mathematics subject classification 1985 revision primary 30g60 secondary 42b20 30e20

convergence of the cr yamabe flow we consider the cr yamabe flow on a compact strictly pseudoconvex cr manifold m of real dimension 2n 1 we prove convergence of the cr yamabe flow when n 1 or m is spherical

a likelihood ratio test against stochastic ordering in several populations abstract the likelihood ratio test is often used to test hypotheses involving a stochastic ordering distribution theory for the likelihood ratio test has been developed only for two stochastically ordered distributions for testing equality of distributions against a stochastic ordering in several populations this paper derives the null asymptotic distribution of the likelihood ratio test statistic which is characterized by minimization problems and has no closed form a monte carlo simulation is conducted to study the limiting distribution because the limiting distribution depends on the specific values of the unknown distributions under the null hypothesis asymptotic and bootstrap approaches are proposed to overcome practical difficulties and implement tests based on the likelihood principle asymptotic validities for these tests are established and simulations are carried out to check their performances for finite sample sizes the tests are applied to an example involving data for survival time f

rates of convergence for estimating regression coefficients in heteroskedastic discrete response models abstract in this paper we consider estimation of discrete response models exhibiting conditional heteroskedasticity of a multiplicative form where the latent error term is assumed to be the product of an unknown scale function and a homoskedastic error term it is first shown that for estimation of the slope coefficients in a binary choice model under this type of restriction the semiparametric information bound is zero even when the homoskedastic error term is parametrically specified hence it is impossible to attain the parametric convergence rate for the parameters of interest however for ordered response models where the response variable can take at least three different values the parameters of interest can be estimated at the parametric rate under the multiplicative heteroskedasticity assumption two estimation procedures are proposed the first estimator based on a parametric restriction on the homoskedastic component of the error term is a two step maximum likelihood estimators where the unknown scale function is estimated nonparametrically in the first stage the second procedure which does not require the parametric restriction estimates the parameters by a kernel weighted least squares procedure under regularity conditions which are standard in the literature both estimators are shown to be n consistent and asymptotically normal

proof of the positive mass theorem ii the positive mass theorem states that for a nontrivial isolated physical system the total energy which includes contributions from both matter and gravitation is positive this assertion was demonstrated in our previous paper in the important case when the space time admits a maximal slice here this assumption is removed and the general theorem is demonstrated abstracts of the results of this paper appeared in 11 and 13

optimal third root asymptotic bounds in the statistical estimation of thresholds this paper is concerned with estimating the intersection point of two densities given a sample of both of the densities this problem arises in classification theory the main results provide lower bounds for the probability of the estimation errors to be large on a scale determined by the inverse cube root of the sample size as corollaries we obtain probabilistic bounds for the prediction error in a classification problem the key to the proof is an entropy estimate the lower bounds are based on bounds for general estimators which are applicable in other contexts as well furthermore we introduce a class of optimal estimators whose errors asymptotically meet the border permitted by the lower bounds

quasiconformal mappings and ahlfors david curves we show that if p is a quasiconformal mapping of the plane whose dilatation satisfies a certain quadratic carleson measure condition relative to an ahlfors david curve f then p is differentiable almost everywhere on f and log ip l e bmo when f is chord arc we show that its image is a bishopjones curve if the carleson norm is small then we show that p is absolutely continuous on f the image of f is an ahlfors david curve and p ea where a e bmo with a small norm

the length of the shorth soit n 0 1 la longueur de l fraction la plus courte de l echantillon ordonne x 1 n x 2 n x n n on utilise le concept de derivees compactes des fonctionnelles statistiques pour etudier le comportement asymptotique de n lorsque n

sharp asymptotics for isotonic regression the asymptotic behavior of the isotonic estimator of a monotone regression function that is the least squares estimator under monotonicity restriction is investigated in particular it is proved that the 1 distance between the isotonic estimator and the true function is of magnitude n 1 3 moreover it is proved that a centered version of this 1 distance converges at the n1 2 rate to a gaussian variable with fixed variance

estimates for 8706 μ 1 and calderon s theorem on the cauchy integral etude du caractere borne de l operateur integral de cauchy a l aide d une transformation quasi conforme

games with incomplete information played by bayesian players i iii this article originally appeared in management science november 1967 volume 14 number 3 pp 159 182 published by the institute of management sciences the paper develops a new theory for the analysis of games with incomplete information where the players are uncertain about some important parameters of the game situation such as the payoff functions the strategies available to various players the information other players have about the game etc however each player has a subjective probability distribution over the alternative possibilities in most of the paper it is assumed that these probability distributions entertained by the different players are mutually consistent in the sense that they can be regarded as conditional probability distributions derived from a certain basic probability distribution over the parameters unknown to the various players but later the theory is extended also to cases where the different players subjective probability distributions fail to satisfy this consistency assumption in cases where the consistency assumption holds the original game can be replaced by a game where nature first conducts a lottery in accordance with the basic probability distribution and the outcome of this lottery will decide which particular subgame will be played i e what the actual values of the relevant parameters will be in the game yet each player will receive only partial information about the outcome of the lottery and about the values of these parameters however every player will know the basic probability distribution governing the lottery thus technically the resulting game will be a game with complete information it is called the bayes equivalent of the original game part i of the paper describes the basic model and discusses various intuitive interpretations for the latter part ii shows that the nash equilibrium points of the bayes equivalent game yield bayesian equilibrium points for the original game finally part iii considers the main properties of the basic probability distribution

u processes rates of convergence on introduit un nouveau processus stochastique une collection de statistiques u indicees par une famille de noyaux symetriques on obtient des conditions pour la convergence presque sure uniforme d une suite de tels processus

mixed rates asymptotics a general method is presented for deriving the limiting behavior of estimators that are defined as the values of parameters optimizing an empirical criterion function the asymptotic behavior of such estimators is typically deduced from uniform limit theorems for rescaled and reparametrized criterion functions the new method can handle cases where the standard approach does not yield the complete limiting behavior of the estimator the asymptotic analysis depends on a decomposition of criterion functions into sums of components with different rescalings the method is explained by examples from lasso type estimation k means clustering shorth estimation and partial linear models

semiparametric analysis of discrete response asymptotic properties of the maximum score estimator abstract this paper is concerned with the estimation of the model med y x x from a random sample of observations on sgn y x manski 1975 introduced the maximum score estimator of the normalized parameter vector in the present paper strong consistency is proved it is also proved that the maximum score estimate lies outside any fixed neighborhood of with probability that goes to zero at exponential rate

the asymptotics of s estimators in the linear regression model we consider the consistency and weak convergence of s estimators in the linear regression model sufficient conditions for consistency with varying dimension are given which are sufficiently weak to cover for example polynomial trends and i i d carriers a weak convergence theorem for the hampel rousseeuw least median of squares estimator is obtained and it is shown under rather general conditions that the correct norming factor is n 1 3 finally the asymptotic normality of s estimators with a smooth rho function is obtained again under weak conditions on the carriers

least median of squares regression abstract classical least squares regression consists of minimizing the sum of the squared residuals many authors have produced more robust versions of this estimator by replacing the square by something else such as the absolute value in this article a different approach is introduced in which the sum is replaced by the median of the squared residuals the resulting estimator can resist the effect of nearly 50 of contamination in the data in the special case of simple regression it corresponds to finding the narrowest strip covering half of the observations generalizations are possible to multivariate location orthogonal regression and hypothesis testing in linear models

a group bridge approach for variable selection in multiple regression problems when covariates can be naturally grouped it is important to carry out feature selection at the group and within group individual variable levels simultaneously the existing methods including the lasso and group lasso are designed for either variable selection or group selection but not for both we propose a group bridge approach that is capable of simultaneous selection at both the group and within group individual variable levels the proposed approach is a penalized regularization method that uses a specially designed group bridge penalty it has the oracle group selection property in that it can correctly select important groups with probability converging to one in contrast the group lasso and group least angle regression methods in general do not possess such an oracle property in group selection simulation studies indicate that the group bridge has superior performance in group and individual variable selection relative to several existing methods

estimation and confidence regions for parameter sets in econometric models this paper develops a framework for performing estimation and inference in econometric models with partial identification focusing particularly on models characterized by moment inequalities and equalities applications of this framework include the analysis of game theoretic models revealed preference restrictions regressions with missing and corrupted data auction models structural quantile regressions and asset pricing models specifically we provide estimators and confidence regions for the set of minimizers i of an econometric criterion function q in applications the criterion function embodies testable restrictions on economic models a parameter value that describes an economic model satisfies these restrictions if q attains its minimum at this value interest therefore focuses on the set of minimizers called the identified set we use the inversion of the sample analog q n of the population criterion q to construct estimators and confidence regions for the identified set and develop consistency rates of convergence and inference results for these estimators and regions to derive these results we develop methods for analyzing the asymptotic properties of sample criterion functions under set identification

two step estimation of semiparametric censored regression models root n consistent estimators of the regression coecients in the linear censored regression model under conditional quantile restrictions on the error terms were pro posed by powell journal of econometrics 25 1984 303 325 32 1986a 143 155 while those estimators have desirable asymptotic properties under weak regularity conditions simulation studies have shown these estimators to exhibit a small sam ple bias in the opposite direction of the least squares bias for censored data this paper introduces two step estimators for these models which minimize convex objec tive functions and are designed to overcome thisnite sample bias the paper gives regularity conditions under which the proposed two step estimators are consistent and asymptotically normal a monte carlo study compares thenite sample behavior of the proposed methods with their one step counterparts 2001 elsevier science s a all rights reserved

estimation of a panel data sample selection model we consider the problem of estimation in a panel data sample selection model where both the selection and the regression equation of interest contain unobservable individual specific effects we propose a two step estimation procedure which differences out both the sample selection effect and the unobservable individual effect from the equation of interest in the first step the unknown coefficients of the selection equation are consistently estimated the estimates are then used to estimate the regression equation of interest the estimator proposed in this paper is consistent and asymptotically normal with a rate of convergence that can be made arbitrarily close to n 1 2 depending on the strength of certain smoothness assumptions the finite sample properties of the estimator are investigated in a small monte carlo simulation

brownian motion with a parabolic drift and airy functions let w t t be two sided brownian motion originating from zero and let v a be defined by v a sup t w t t a 2 is maximal then v a a is a markovian jump process running through the locations of maxima of two sided brownian motion with respect to the parabolas fa t t a 2 we give an analytic expression for the infinitesimal generators of the processes a in terms of airy functions in theorem 4 1 this makes it possible to develop asymptotics for the global behavior of a large class of isotonic estimators i e estimators derived under order restrictions an example of this is given in groeneboom 1985 where the asymptotic distribution of the standardized l1 distance between a decreasing density and the grenander maximum likelihood estimator of this density is determined on our way to theorem 4 1 we derive some other results for example we give an analytic expression for the joint density of the maximum and the location of the maximum of the process w t ct2 t where c is an aribrary positive constant we also determine the laplace transform of the integral over a brownian excursion these last results also have recently been derived by several other authors using a variety of methods

asymptotic distributions of the maximal depth estimators for regression and multivariate location we derive the asymptotic distribution of the maximal depth regression estimator recently proposed in rousseeuw and hubert the estimator is obtained by maximizing a projection based depth and the limiting distribution is characterized through a max min operation of a continuous process the same techniques can be used to obtain the limiting distribution of some other depth estimators including tukey s deepest point based on half space depth results for the special case of two dimensional problems have been available but the earlier arguments have relied on some special geometric properties in the low dimensional space this paper completes the extension to higher dimensions for both regression and multivariate location models

pairwise difference estimation of incomplete information games abstract this paper contributes to the literature on econometric estimation of incomplete information games with nash equilibrium behavior by introducing a two step estimation procedure that makes no parametric assumptions about the distribution of unobservable payoffs shocks instead its asymptotic properties rely on assuming only that these distributions satisfy an invertibility condition and that the underlying equilibrium selection mechanism is degenerate our methodology relies on a pairwise differencing procedure which unlike aradillas lopez 2010 does not require computing the equilibria of the game furthermore if normal form payoffs are linear in the parameters of interest our procedure results in an estimator with a closed form expression we contribute to the pairwise differencing econometric literature by introducing the first model where both the control variables being matched and the regressors in the index function parameterized by contain nonparametric functions in particular the asymptotic theory developed in aradillas lopez et al 2007 does not cover this setting we describe conditions under which nonparametrically estimated plug ins yield a n consistent and asymptotically normal estimator for the parameter of interest a consistent specification test based on semiparametric residuals is also developed it appears to be the first test of this type for a model involving nonparametric or generated regressors several extensions of our method are also discussed a series of monte carlo experiments are used to investigate the properties of our estimator and our specification test

inference on regressions with interval data on a regressor or outcome this paper examines inference on regressions when interval data are available on one variable the other variables being measured precisely let a population be characterized by a distribution p y x v v 0 v 1 where y r 1 x r k and the real variables v v 0 v 1 satisfy v 0 v v 1 let a random sample be drawn from p and the realizations of y x v 0 v 1 be observed but not those of v the problem of interest may be to infer e y x v or e v x this analysis maintains interval i monotonicity m and mean independence mi assumptions i p v 0 v v 1 1 m e y x v is monotone in v mi e y x v v 0 v 1 e y x v no restrictions are imposed on the distribution of the unobserved values of v within the observed intervals v 0 v 1 it is found that the immi assumptions alone imply simple nonparametric bounds on e y x v and e v x these assumptions invoked when y is binary and combined with a semiparametric binary regression model yield an identification region for the parameters that may be estimated consistently by a modified maximum score mms method the immi assumptions combined with a parametric model for e y x v or e v x yield an identification region that may be estimated consistently by a modified minimum distance mmd method monte carlo methods are used to characterize the finite sample performance of these estimators empirical case studies are performed using interval wealth data in the health and retirement study and interval income data in the current population survey

identification and estimation of a discrete game of complete information we discuss the identification and estimation of discrete games of complete information following bresnahan and reiss 1990 1991 a discrete game is a generalization of a standard discrete choice model where utility depends on the actions of other players using recent algorithms to compute all of the nash equilibria to a game we propose simulation based estimators for static discrete games we demonstrate that the model is identified under weak functional form assumptions using exclusion restrictions and an identification at infinity approach monte carlo evidence demonstrates that the estimator can perform well in moderately sized samples as an application we study entry decisions by construction contractors to bid on highway projects in california we find that an equilibrium is more likely to be observed if it maximizes joint profits has a higher nash product uses mixed strategies and is not pareto dominated by another equilibrium

identification and estimation of games with incomplete information using excluded regressors the existing literature on binary games with incomplete information assumes that either payo functions or the distribution of private information are nitely parameterized to obtain point identi cation in contrast we show that given excluded regressors payofunctions and the distribution of private information can both be nonparametrically point identi ed an excluded regressor for player i is a su ciently varying state variable that does not aect other players utility and is additively separable from other components in i s payo we show how excluded regressors satis fying these conditions arise in contexts such as entry games between rms as variation in observed components of xed costs our identi cation proofs are constructive so consistent nonparametric estimators can be readily based on them for a semiparametric model with linear payos we propose root n consistent and asymptotically normal estimators for parameters in players payos finally we extend our approach to accommodate the existence of multiple bayesian nash equilibria in the data generating process without assuming equilibrium selection rules

convergence of the yamabe flow for arbitrary initial energy we consider the yamabe flow frac partial g partial t r g r g g where g is a riemannian metric on a compact manifold m r g denotes its scalar curvature and r g denotes the mean value of the scalar curvature we prove convergence of the yamabe flow if the dimension n satisfies 3 leq n leq 5 or the initial metric is locally conformally flat

a consistent bootstrap procedure for the maximum score estimator abstract in this paper we propose a new model based smoothed bootstrap procedure for making inference on the maximum score estimator of manski 1975 1985 and prove its consistency we provide a set of sufficient conditions for the consistency of any bootstrap procedure in this problem we compare the finite sample performance of different bootstrap procedures through simulation studies the results indicate that our proposed smoothed bootstrap outperforms other bootstrap schemes including the m out of n bootstrap additionally we prove a convergence theorem for triangular arrays of random variables arising from binary choice models which may be of independent interest

likelihood ratio tests for monotone functions we study the problem of testing for equality at a fixed point in the setting of nonparametric estimation of a monotone function the likelihood ratio test for this hypothesis is derived in the particular case of interval censoring or current status data and its limiting distribution is obtained the limiting distribution is that of the integral of the difference of the squared slope processes corresponding to a canonical version of the problem involving brownian motion t 2 and greatest convex minorants thereof inversion of the family of tests yields pointwise confidence intervals for the unknown distribution function we also study the behavior of the statistic under local and fixed alternatives 1 introduction we shall consider likelihood ratio tests and the corresponding confidence intervals in a class of problems involving nonparametric estimation of a monotone function the problem in each case involves testing the null hypothesis h 0 that the monotone function has a particular value at afixed point t 0 in the domain of the function of course with each testing problem there is a related problem of finding confidence intervals here are some examples of the problems we have in mind

cube root asymptotics we establish a new functional central limit theorem for empirical processes indexed by classes of functions in a neighborhood of a fixed parameter point an n 1 3 rescaling of the parameter is compensated for by an n 2 3 rescaling of the empirical measure resulting in a limiting gaussian process by means of a modified continuous mapping theorem for the location of the maximizing value we deduce limit theorems for several statistics defined by maximization or constrained minimization of a process derived from the empirical measure these statistics include the short rousseeuw s least median of squares estimator manski s maximum score estimator and the maximum likelihood estimator for a monotone density the limit theory depends on a simple new sufficient condition for a gaussian process to achieve its maximum almost surely at a unique point

estimation of network parameters in semiparametric stochastic perceptron it was reported kabashima and shinomoto 1992 that estimators of a binary decision boundary show asymptotically strange behaviors when the probability model is ill posed or semiparametric we give a rigorous analysis of this phenomenon in a stochastic perceptron by using the estimating function method a stochastic perceptron consists of a neuron that is excited depending on the weighted sum of inputs but its probability distribution form is unknown here it is shown that there exists no n consistent estimator of the threshold value h that is no estimator h that converges to h in the order of 1 n as the number n of observations increases therefore the accuracy of estimation is much worse in this semiparametric case with an unspecified probability function than in the ordinary case on the other hand it is shown that there is a n consistent estimator of the synaptic weight vector these results elucidate strange behaviors of learning curves in a semiparametric statistical model

hitting times for gaussian processes this is the publisher s version also available electronically from http www jstor org stable 25450587 seq 1 page scan tab contents