a short and elementary proof of the main bahadur kiefer theorem a short proof of the lower bound in the strong version of the famous theorem 1a in kiefer 1970 on the bahadur kiefer process is presented the proof is elementary and in particular does not use strong approximations

relative invariants of 2 simple prehomogeneous vector spaces of type i abstract in the present paper by using equivariant maps we construct explicitly relative invariants of the 2 simple prehomogeneous vector spaces of type i which are classified in t kimura s kasai m inuzuka o yasukura a classification of 2 simple prehomogeneous vector spaces of type i j algebra 114 1988 369 400 this is a continuation of our previous work t kogiso g miyabe m kobayashi t kimura non regular 2 simple prehomogeneous vector spaces of type i and their relative invariants j algebra 251 2002 27 69 t kogiso g miyabe m kobayashi t kimura relative invariants of some 2 simple prehomogeneous vector spaces math comp 72 2003 865 889

nonregular 2 simple prehomogeneous vector spaces of type i and their relative invariants after the fundamental theorem for nonregular reductive prehomogeneous vector spaces was proved by a gyoja tsukuba j math 14 1990 437 457 the construction of the theory of nonregular prehomogeneous vector spaces became an interesting problem however only spn gl2 1 2 1 m 2n 3 is known as an interesting example of nonregular reductive prehomogeneous vector spaces in this paper we give some other interesting examples by constructing all relative invariants of nonregular 2 simple prehomogeneous vector spaces of type i we consider everything over the complex number field c

singular invariant tempered distributions on regular prehomogeneous vector spaces revue des espaces vectoriels prehomogenes distributions temperees invariantes distributions temperees avec un parametre meromorphe puissances complexes des invariants relatifs soit p x un polynome homogene irreductible a coefficients reels sur un espace vectoriel v soit s x v p x 0 se decompose en un nombre fini de g 1 orbites avec g 1 g gl v p g x p x on pose v 1 uv 2 u uv e v s la decomposition en composantes connexes on donne une condition necessaire et suffisante pour qu une distribution temperee g 1 invariante a support dans s puisse s obtenir comme une combinaison lineaire finie de coefficients de laurent negatifs de la distribution p x s vd i

strong laws for local quantile processes we show that increments of size h n from the uniform quantile and uniform empirical processes in the neighborhood of a fixed point t 0 0 1 may have different rates of almost sure convergence to 0 in the range where h n 0 and nh n log n in particular when h n n with 0 0 the best possible uniform almost sure rate of approximation of the uniform quantile process by a normed kiefer process is not better than o n 1 4 log n e

the multifractal nature of levy processes we show that the sample paths of most levy processes are multifractal functions and we determine their spectrum of singularities

the b function and the holonomy diagram of a regular simple prehomogeneous vector space gl 1 2 spin 10 half spin rep vector rep the purpose of this paper is to investigate the micro local structure and to calculate the b function of a regular simple prehomogeneous vector space gl 1 2 spin 10 half spin rep vector rep by constructing the holonomy diagram

sharp rates of convergence of maximum likelihood estimators in nonparametric models sharp rates of convergence of maximum likelihood estimators are established in models which are defined by probability densities having bounded derivatives this result is achieved by making use of local properties of the empirical distribution function

a law of the logarithm for kernel density estimators in this paper we derive a law of the logarithm for the maximal deviation between a kernel density estimator and the true underlying density function extensions to higher derivatives are included the results are applied to get optimal window widths with respect to almost sure uniform convergence

a direct approach to inference in nonparametric and semiparametric quantile models we construct a generic confidence interval for a conditional quantile via the direct approach it avoids estimating the conditional density function of the dependent variable given the covariate and is asymptotically valid for any conditional quantile any conditional quantile estimator and any data structure provided that certain weak convergence of the conditional quantile process holds for the original quantile estimator we also construct a generic confidence band for the conditional quantile function across a range of covariate values by using yang stute estimator and two semiparametric quantile functions we demonstrate the flexibility and simplicity of the direct approach the advantages of our new confidence intervals are borne out in a simulation study

limit theorems and inequalities for the uniform empirical process indexed by intervals the uniform empirical process u is considered as a process indexed by intervals powerful new exponential bounds are established for the process indexed by both points and intervals these bounds trivialize the proof of the chibisov o reilly theorem concerning the convergence of the process with respect to 11 q 11 metrics and are used to prove an interval analogue of the chibisov o reilly theorem a strong limit theorem related to the well known holder condition for brownian bridge u is also proved connections with related work of csdki eicker jaeschke and stute are mentioned as an application we introduce a new statistic for testing uniformity which is the natural interval analogue of the classical anderson darling statistic

functional limit laws for the increments of kaplan meier product limit processes and applications we prove functional limit laws for the increment functions of empirical processes based upon randomly right censored data the increment sizes we consider are classified into different classes covering the whole possible spectrum we apply these results to obtain a description of the strong limiting behavior of a series of estimators of local functionals of lifetime distributions in particular we treat the case of kernel density and hazard rate estimators

a classification of 2 simple prehomogeneous vector spaces of type i let p g gl v be a rational representation of a connected linear algebraic group g on a finite dimensional vector space v all defined over an algebraically closed field k of characteristic zero if v has a zariskidense g orbit we call a triplet g p v a prehomogeneous oector space abbrev p v when p is irreducible such p v s have been classified in 11 since then it has turned out gradually that the complete classification of reductive p v s i e p v s with reductive groups g is an extremely laborious task therefore it is natural to classify some restricted class of p v s e g 2 to get some insight into the general situation a p v g p v is called a 2 simple p v when 1 g gl 1 x g x g with simple algebraic groups g and gz 2 p is the composition of a rational representation p of g x g of the form p p1 0 p pkol a a l l z t with k s t l where pi tv resp pi t are nontrivial irreducible representations of g resp g and the scalar multiplications gz 1 on each irreducible component v for i 1 f where v i v we say that a 2 simple p v g p v is of type zif k l and at least one of gl l xg xg pi pi i 1 k is a nontrivial p v see definition 5 p 43 in 11 on the other hand if k l and all gl l xg xg p op i l k are trivial p v s it is called a 2 simple p v of type ii in 3 all 2 simple p v s of type ii has been already classified in this paper we shall classify all 2simple p v s of type i thus together with 3 we complete a classification of all 2 simple p v s for example the fact that all irreducible p v s are castling equivalent to 2 simple p v s or to x m x y m x g 2 1 1 i i n 1 with m 2 3 see 1 indicates the importance of 2 simple p v s for simplicity we write g p v or g p instead of g p v

on some representations of the bootstrap the bootstrap statistic hat theta n hat theta n is represented as the sum of two terms one of which has the same distribution as hat theta n theta and the other is a relatively negligible remainder the representation is then applied to the problems of estimating a distribution and a quantile the study includes the tail part of the distribution and quantiles for which the convergence rates could be faster than the usual n 1 2 rate

etale representations for reductive algebraic groups with one dimensional center abstract a complex vector space v is a prehomogeneous g module if g acts rationally on v with a zariski open orbit the module is called etale if dim v dim g we study etale modules for reductive algebraic groups g with one dimensional center for such g even though every etale module is a regular prehomogeneous module its irreducible submodules have to be non regular for these non regular prehomogeneous modules we determine some strong constraints on the ranks of their simple factors this allows us to show that there do not exist etale modules for g gl 1 s s with s simple

a classification of the representations of reductive algebraic groups which admit only a finite number of orbits introduction let p g gl v be a finite dimensional rational representation of a semi simple algebraic group g all defined over c then we have p pi epe and v v1i d ve where pi is an irreducible representation of g on vi for i 1 f let p be the action of g gl 1 i x g on v which is the composition of p and scalar multiplications gl 1 ion each irreducible component vi i 1 f in this paper we shall classify all the triplet g p v such that v decomposes into a finite union of g orbits prof v g kac 3 4 and the first author 7 14 but only the necessary conditions are written in 14 have already completed independently the case when p is irreducible i e f 1 prof v g kac has also completed the case when each irreducible components are gl n x gl m al al v n v m see 5 which is the development of the results of prof p gabriel 1 note that if v decomposes into the finite union of g orbits then one of the orbits must be zariski dense and hence g p v must be a prehomogeneous vector space abbrev p v we call such a p v a finite prehomogeneous vector space abbrev f p this paper consists of the following sections

a uniform functional law of the logarithm for the local empirical process we prove a uniform functional law of the logarithm for the local empirical process to accomplish this we combine techniques from classical and abstract empirical process theory gaussian distributional approximation and probability on banach spaces the body of techniques we develop should prove useful to the study of the strong consistency of d variate kernel type nonparametric function estimators

total positive curvature of hypersurfaces with convex boundary we prove that if is a compact hypersurface in euclidean space rn its boundary lies on the boundary of a convex body c and meets c orthogonally from the outside then the total positive curvature of is bigger than or equal to half the area of the sphere sn 1 also we obtain necessary and sufficient conditions for the equality to hold

a new conformal invariant and its applications to the willmore conjecture and the first eigenvalue of compact surfaces let m be a compact riemannian manifold with a fixed conformal structure then we introduce the concept of conformal volume of m in the following manner for each branched conformal immersion q9 of m into the unit sphere s n we consider the set of all branched conformal immersions obtained by composi t ion of qo with the conformal automorphisms of s we let vc n qg be the max imum volume of these branched immersions the conformal volume of m is defined to be the infimum of v n q0 where qo ranges over all branched conformal immersions of m into the unit sphere s in this paper we study the case when m is a compact surface and we call the conformal volume of m to be the conformal area of m we demonst ra te that this conformal invariant is non trivial in fact we prove that if there exists a minimal immersion of m into s where coordinate functions are first eigenfunctions then the conformal area of m is given by the area of m with respect to the induced metric this enables us to compute the conformal area for several surfaces for example the conformal area of r p z is 6n and the conformal area of the square torus is 27c 2 we believe that the computa t ion of the conformal area for general surfaces will be very impor tant in studying the geometry of compact surfaces we demonstrate this claim by applying the concept of conformal area to two different branches of surface theory the first application is to study the total curvature of a compact surface in r this problem has a long history fenchel and fary 9 proved that for a closed curve o in r slk 27c where k is its curvature then milnor 12

sequential fixed width confidence intervals for a nonparametric density function in this paper sequential fixed width confidence intervals for a nonparametric density function are derived these are formed from certain values of a kernel density estimator the efficiency of such procedures is measured in terms of the expected stopping time

relative invariants of some 2 simple prehomogeneous vector spaces in this paper we shall construct explicitly irreducible relative invariants of two 2 simple prehomogeneous vector spaces together with a preprint by the same authors this completes the list of all relative invariants of regular 2 simple prehomogeneous vector spaces of type i

on false discovery control under dependence a popular framework for false discovery control is the random effects model in which the null hypotheses are assumed to be independent this paper generalizes the random effects model to a conditional dependence model which allows dependence between null hypotheses the dependence can be useful to characterize the spatial structure of the null hypotheses asymptotic properties of false discovery proportions and numbers of rejected hypotheses are explored and a large sample distributional theory is obtained

nearest neighbor estimation of a bivariate distribution under random censoring we consider the problem of estimating the bivariate distribution of the random vector x y when y may be subject to random censoring the censoring variable c is allowed to depend on x but it is assumed that y and c are conditionally independent given x x the estimate of the bivariate distribution is obtained by averaging estimates of the conditional distribution of y given x x over a range of values of x the weak convergence of the centered estimator multiplied by n 1 2 is obtained and a closed form expression for the covariance function of the limiting process is given it is shown that the proposed estimator is optimal in the beran sense this is similar to an optimality property the product limit estimator enjoys using the proposed estimator of the bivariate distribution an extension of the least squares estimator to censored data polynomial regression is obtained and its asymptotic normality established

strong laws for the maximal k spacing when k c log n we consider the maximal k spacing m n mathop max limits 0 underset raise0 3em hbox smash scriptscriptstyle leqslant i underset raise0 3em hbox smash scriptscriptstyle leqslant n 1 k left u ni k u ni right where u n1 u nn are the order statistics of an i i d sample of size n from the uniform distribution on 0 1 and u n0 0 u nn 1 1 the integer k is allowed to vary with n at a rate not exceeding log n we obtain laws of the iterated logarithm for all the k s in the given range for small k the methods used in the proofs are borrowed from extreme value theory for larger k the techniques are reminiscent of those used in the proof of the erdos renyi theorem

nonparametric checks for single index models in this paper we study goodness of fit testing of single index models the large sample behavior of certain score type test statistics is investigated as a by product we obtain asymptotically distribution free maximin tests for a large class of local alternatives furthermore characteristic function based goodness of fit tests are proposed which are omnibus and able to detect peak alternatives simulation results indicate that the approximation through the limit distribution is acceptable already for moderate sample sizes applications to two real data sets are illustrated

local convergence of empirical measures in the random censorship situation with application to density and rate estimators on etudie les deviations locales de la mesure empirique definie par l estimateur de kaplan meier 1958 pour la fonction de survie

on the asymptotics of penalized splines summavry we study the asymptotic behaviour of penalized spline estimators in the univariate case we use b splines and a penalty is placed on mth order differences of the coefficients the number of knots is assumed to converge to infinity as the sample size increases we show that penalized splines behave similarly to nadaraya watson kernel estimators with equivalent kernels depending upon m the equivalent kernels we obtain for penalized splines are the same as those found by silverman for smoothing splines the asymptotic distribution of the penalized spline estimator is gaussian and we give simple expressions for the asymptotic mean and variance provided that it is fast enough the rate at which the number of knots converges to infinity does not affect the asymptotic distribution the optimal rate of convergence of the penalty parameter is given penalized splines are not design adaptive

goodness of fit tests based on kullback leibler discrimination information we present a general methodology for developing asymptotically distribution free goodness of fit tests based on the kullback leibler discrimination information the tests are shown to be omnibus within an extremely large class of nonparametric global alternatives and to have good local power the proposed test procedure is a nonparametric extension of the classical neyman pearson log likelihood ratio test and is based on mth order spacings between order statistics cross validated by the observed log likelihood the developed method also generalizes cox s procedure of testing separate families and covers virtually all parametric families of distributions encountered in statistics it can also be viewed as a procedure based on sum log functionals of nonparametric density quantile estimators cross validated by the log likelihood with its good power properties the method provides an extremely simple and potentially much better alternative to the classical empirical distribution function edf based test procedures the important problem of selecting the order of spacings m in practice is also considered and a method based on maximizing the sample entropy constrained by the observed log likelihood is proposed this data driven method of choosing m is demonstrated by monte carlo simulations to be more powerful than deterministic choices of m and thus provides a practically useful tool for implementing our test procedure

transfer of tail information in censored regression models consider a heteroscedastic regression model y m x sigma x varepsilon where the functions m and sigma are smooth and varepsilon is independent of x the response variable y is subject to random censoring but it is assumed that there exists a region of the covariate x where the censoring of y is light under this condition it is shown that the assumed nonparametric regression model can be used to transfer tail information from regions of light censoring to regions of heavy censoring crucial for this transfer is the estimator of the distribution of varepsilon based on nonparametric regression residuals whose weak convergence is obtained the idea of transferrring tail information is applied to the estimation of the conditional distribution of y given x x with information on the upper tail borrowed from the region of light censoring and to the estimation of the bivariate distribution p x leq x y leq y with no regions of undefined mass the weak convergence of the two estimators is obtained by products of this investigation include the uniform consistency of the conditional kaplan meier estimator and its derivative the location and scale estimators and the estimators of their derivatives

some p v equivalences and a classification of 2 simple prehomogeneous vector spaces of type a classification of 2 simple prehomogeneous vector spaces is completed by using some p v equivalences together with 3 some part is very different from the previous classification of the irreducible or simple cases 1 2 and some new method is necessary this result shows the difficult point of a classification problem of reductive prehomogeneous vector spaces introduction in m sato and t kimura 1 all irreducible prehomogeneous vector spaces abbreviated irreducible p v s are classified up to castling equivalence and it is proved that any irreducible p v is castling equivalent to a 2 simple p v or to sl m x sl m x gl 2 a1 x a1 x a1 with m 2 3 therefore as a step to a classification of the general case it is natural to classify all nonirreducible 2 simple p v s in this paper we shall classify all 2 simple p v s of type ii and give the complete list of them up to strong equivalence in the sense of definition 4 p 36 in 1 however we need various p v equivalences cf propositions 1 7 1 12 1 34 1 36 1 37 theorem 1 16 and its corollary corollary 1 26 definition 1 31 remark 1 32 to carry out the classification and hence first we shall prove them we also investigate the regularity in some cases since all 2 simple p v s of type i are already classified in 3 this completes the classification of all 2 simple p v s here we say that g p v or simply g p is a 2 simple p v if it is a prehomogeneous vector space of the following form g gl 1 k s t xg x g2 where g1 and g2 are simple algebraic groups p the composition of the scalar multiplications gl l k s on each irreducible component and the representation p p1 p 1 pk p k a1 s 1 1 t1 t ofthegroupg1 x g2 where pi aj resp p tj are nontrivial irreducible representations of g1 resp g2 we write g p v instead of g p v for simplicity to avoid confusion we write instead of clearly each gl 1 x g1 x g2 pi p i 1 k is an irreducible 2 simple p v if at least one of them is a nontrivial p v see definition 1 1 we call it a 2 simple p v of type i on the other hand if all of them are trivial p v s we call it a 2 simple p v of type ii in this case we have g2 sl n received by the editors march 3 1986 and in revised form july 30 1986 1980 mathematics subject classification 1985 revision primary 11r20 11r29 secondary 11n25 11r18 01988 american mathematical society 0002 9947 88 1 00 25 per page

isoperimetric inequalities in euclidean convex bodies in this paper we consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a convex body i e a compact convex set in euclidean space with interior points we shall not impose any regularity assumption on the boundary of the convex set amongst other results we shall prove the equivalence between hausdorff and lipschitz convergence the continuity of the isoperimetric profile with respect to the hausdorff distance and the convergence in hausdorff distance of sequences of isoperimetric regions and their free boundaries we shall also describe the behavior of the isoperimetric profile for small volume and the behavior of isoperimetric regions for small volume

uniform in bandwidth consistency of kernel type function estimators we introduce a general method to prove uniform in bandwidth consistency of kernel type function estimators examples include the kernel density estimator the nadaraya watson regression estimator and the conditional empirical process our results may be useful to establish uniform consistency of data driven bandwidth kernel type function estimators

strong limit theorems for oscillation moduli of the uniform empirical process let u n t n tfrac 1 2 gamma n t t 0 leqq t leqq 1 denote the uniform empirical process based on the first n of a sequence 1 2 of iid uniform 0 1 random variables where gamma n t n 1 sum limits i 1 n 1 0 t xi i is the empirical distribution function the oscillation modulus of u n is defined by omega n a sup u n t h u n t 0 leqq t leqq 1 h h leqq a and the lipschitz 1 2 modulus of u n is defined by tilde omega n a sup u n t h u n t h tfrac 1 2 0 leqq t leqq 1 h a leqq h leqq 1

rates of growth and sample moduli for weighted empirical processes indexed by sets probability inequalities are obtained for the supremum of a weighted empirical process indexed by a vapnik cervonenkis class c of sets these inequalities are particularly useful under the assumption p c c p c t 0 as t 0 they are used to obtain almost sure bounds on the rate of growth of the process as the sample size approaches infinity to find an asymptotic sample modulus for the unweighted empirical process and to study the ratio pn p of the empirical measure to the actual measure

on the fractal nature of empirical increments we prove that the set of points where exceptional oscillations of empirical and related processes occur infinitely often is a random fractal and evaluate its hausdorff dimension

testing for covariate effects in the fully nonparametric analysis of covariance model traditional inference questions in the analysis of covariance mainly focus on comparing different factor levels by adjusting for the continuous covariates which are believed to also exert a significant effect on the outcome variable testing hypotheses about the covariate effects although of substantial interest in many applications has received relatively limited study in the semiparametric nonparametric setting in the context of the fully nonparametric analysis of covariance model of akritas et al we propose methods to test for covariate main effects and covariate factor interaction effects the idea underlying the proposed procedures is that covariates can be thought of as factors with many levels the test statistics are closely related to some recent developments in the asymptotic theory for analysis of variance when the number of factor levels is large the limiting normal distributions are established under the null hypotheses and local alternatives by asymptotically approximating a new class

on almost sure convergence of conditional empirical distribution functions on etudie la convergence presque sure d une distribution empirique conditionnelle de type noyau a la fois en norme sup et en norme sup ponderee

kiefer s theorem via the hungarian construction a proof of kiefer s theorem estimating the difference between the empirical and quantile processes is given by appealing to the hungarian construction

a quality index based on data depth and multivariate rank tests let f and g be the distribution functions of two given populations on rp p 1 we introduce and study a parameter q q f g which measures the overall outlyingness of population g relative to population f the parameter q can be defined using any concept of data depth its value ranges from 0 to 1 and is 5 when f and g are identical we show that within the dass of elliptical distributions when g departs from f in location or g has a larger spread or both the value of q dwindles down from 5 hence q can be used to detect the loss of accuracy or precision of a manufacturing process and thus it should serve as an important measure in quality assurance this in fact is the reason why we refer to q as a quality index in this article in addition to studying the properties of q we provide an exact rank test for testing q 5 vs q 5 this can be viewed as a multivariate analog of wilcoxon s rank sum test the tests proposed here have power against location change and scale increase simultaneo

on residual empirical processes of stochastic regression models with applications to time series motivated by gaussian tests for a time series we are led to investigate the asymptotic behavior of the residual empirical processes of stochastic regression models these models cover the fixed design regression models as well as general ar q models since the number of the regression coefficients is allowed to grow as the sample size increases the obtained results are also applicable to nonlinear regression and stationary ar models in this paper we first derive an oscillation like result for the residual empirical process then we apply this result to autoregressive time series in particular for a stationary ar process we are able to determine the order of the number of coefficients of a fitted ar q n model and obtain the limiting gaussian processes for an unstable ar q process we show that if the characteristic polynomial has a unit root 1 then the limiting process is no longer gaussian for the explosive case one of our side results also provides a short proof for the brownian bridge results given by koul and levental

a law of the logarithm for kernel quantile density estimators received february 1992 revised february 1993 1this manuscript was prepared using computer facilities supported in part by nsf grants dms 89 05292 dms 87 03942 dms 86 01732 and dms 84 04941 and by the university of chicago block fund ams 1991 subject classifications primary 60f15 secondary 62g05 62g30

ancova methods for heteroscedastic nonparametric regression models we consider an ancova design in which the relationship between the response yi and the covariate xi in cell factor level combination i satisfies the model yi mi xi i xi i where the error term i is assumed to be independent of xi and mi and i are respectively a smooth but unknown regression and scale function this model can be viewed as a generalization of the nonparametric ancova model of young and bowman as such it is a useful alternative for parametric or semiparametric ancova models whenever modeling assumptions such as proportional odds normality of the error terms linearity or homoscedasticity appear suspect we develop test statistics for the hypotheses of no main effects no interaction effects and no simple effects which adjust for the covariate values as defined by akritas arnold and du the asymptotic distribution of the test statistics is obtained its small sample behavior is studied by means of simulations and a real dataset is analyzed

sequential confidence bands for densities this paper proposes a fully sequential procedure for constructing a fixed width confidence band for an unknown density on a finite interval and shows the procedure has the desired coverage probability asymptotically as the width of the band approaches zero the procedure is based on a result of bickel and rosenblatt its implementation in the sequential setting cannot be obtained using anscombe s theorem because the normalized maximal deviations between the kernel estimate and the true density are not uniformly continuous in probability u c i p instead we obtain a slightly weaker version of the u c i p property and a correspondingly stronger convergence property of the stopping rule these together yield the desired results