transformations to reduce boundary bias in kernel density estimation we consider kernel estimation of a univariate density whose support is a compact interval if the density is non zero at either boundary then the usual kernel estimator can be seriously biased reflection at a boundary removes some bias but unless the first derivative of the density is 0 at the boundary the estimator with reflection can still be much more severely biased at the boundary than in the interior we propose to transform the data to a density that has its first derivative equal to 0 at both boundaries the density of the transformed data is estimated and an estimate of the density of the original data is obtained by a change of variables the transformation is selected from a parametric family which is allowed to be quite general in our theoretical study we propose algorithms where the transformation is either a quartic polynomial a b cumulative density function cdf or a linear combination of a polynomial and a b cdf the last two types of transformation are designed to accommodate possible poles at the boundaries the first two algorithms are tested on simulated data and compared with an adjusted kernel method of rice we find that our proposal performs similarly to rice s for densities with one sided derivatives at the boundaries unlike rice s method our proposal is guaranteed to produce non negative estimates this can be a distinct advantage when the density is 0 at either boundary our algorithm for densities with poles outperforms the rice adjustment when the density has a pole at a boundary

on automatic boundary corrections many popular curve estimators based on smoothing have difficulties caused by boundary effects these effects are visually disturbing in practice and can play a dominant role in theoretical analysis local polynomial regression smoothers are known to correct boundary effects automatically some analogs are implemented for density estimation and the resulting estimators also achieve automatic boundary corrections in both settings of density and regression estimation we investigate best weight functions for local polynomial fitting at the endpoints and find a simple solution the solution is universal for general degree of local polynomial fitting and general order of estimated derivative furthermore such local polynomial estimators are best among all linear estimators in a weak minimax sense and they are highly efficient even in the usual linear minimax sense

multivariate locally weighted least squares regression nonparametric regression using locally weighted least squares was first discussed by stone and by cleveland recently it was shown by fan and by fan and gijbels that the local linear kernel weighted least squares regression estimator has asymptotic properties making it superior in certain senses to the nadaraya watson and gasser muller kernel estimators in this paper we extend their results on asymptotic bias and variance to the case of multivariate predictor variables we are able to derive the leading bias and variance terms for general multivariate kernel weights using weighted least squares matrix theory this approach is especially convenient when analysing the asymptotic conditional bias and variance of the estimator at points near the boundary of the support of the predictors we also investigate the asymptotic properties of the multivariate local quadratic least squares regression estimator discussed by cleveland and devlin and in the univariate case higher order polynomial fits and derivative estimation

miscellanea kernel type density estimation on the unit interval we consider kernel type methods for the estimation of a density on 0 1 which eschew explicit boundary correction we propose using kernels that are symmetric in their two arguments these kernels are conditional densities of bivariate copulas we give asymptotic theory for the version of the new estimator using gaussian copula kernels and report on simulation comparisons of it with the beta kernel density estimator of chen 1 we also provide automatic bandwidth selection in the form of rule of thumb bandwidths for both estimators as well as its competitive integrated squared error performance advantages of the new approach include its greater range of possible values at 0 and 1 the fact that it is a bona fide density and that the individual kernels and resulting estimator are comprehensible in terms of a single simple picture

amos type bounds for modified bessel function ratios we systematically investigate lower and upper bounds for the modified bessel function ratio formula see text by functions of the form formula see text in case formula see text is positive for all formula see text or equivalently where formula see text or formula see text is a negative integer for formula see text we give an explicit description of the set of lower bounds and show that it has a greatest element we also characterize the set of upper bounds and its minimal elements if formula see text the minimal elements are tangent to formula see text in exactly one point formula see text and have formula see text as their lower envelope we also provide a new family of explicitly computable upper bounds finally if formula see text is a negative integer we explicitly describe the sets of lower and upper bounds and give their greatest and least elements respectively

turan type inequalities for struve functions some turan type inequalities for struve functions of the first kind are deduced by using various methods developed in the case of bessel functions of the first and second kind new formulas like mittag leffler expansion infinite product representation for struve functions of the first kind are obtained which may be of independent interest moreover some complete monotonicity results and functional inequalities are deduced for struve functions of the second kind these results complement naturally the known results for a particular case of lommel functions of the first kind and for modified struve functions of the first and second kind

bounds for ratios of modified bessel functions and associated turan type inequalities abstract new sharp inequalities for the ratios of bessel functions of consecutive orders are obtained using as main tool the first order difference differential equations satisfied by these functions many already known inequalities are also obtainable and most of them can be either improved or the range of validity extended it is shown how to generate iteratively upper and lower bounds which are converging sequences in the case of the i functions few iterations provide simple and effective upper and lower bounds for approximating the ratios i x i 1 x and the condition numbers x i x i x for any x 0 for the ratios k x k 1 x the same is possible but with some restrictions on using these bounds turan type inequalities are established extending the range of validity of some known inequalities and obtaining new inequalities as well for instance it is shown that k 1 x k 1 x k x 2 1 x 0 1 1 and that the inequality is the best possible this proves and improves an existing conjecture

a new type of sharp bounds for ratios of modified bessel functions abstract the bounds for the ratios of first and second kind modified bessel functions of consecutive orders are important quantities appearing in a large number of scientific applications we obtain new bounds which are accurate in a large region of parameters and which are sharper than previous bounds the new bounds are obtained by a qualitative analysis of the riccati equation satisfied by these ratios a procedure is considered in which the bounds obtained from the analysis of the riccati equation are used to define a new function satisfying a new riccati equation which yields sharper bounds similar ideas can be applied to other functions

exact boundary controllability for quasilinear wave equations a unified constructive method is given to establish the local exact boundary controllability for one dimensional quasilinear wave equations with boundary conditions of different types

exact controllability for multidimensional semilinear hyperbolic equations in this paper we obtain a global exact controllability result for a class of multidimensional semilinear hyperbolic equations with a superlinear nonlinearity and variable coefficients for this purpose we establish an observability estimate for the linear hyperbolic equation with an unbounded potential in which the crucial observability constant is estimated explicitly by a function of the norm of the potential such an estimate is obtained by a combination of a pointwise estimate and a global carleman estimate for the hyperbolic differential operators and analysis on the regularity of the optimal solution to an auxiliary optimal control problem

inequalities for bessel functions we derive some inequalities for the cylinder function c x defined by cv x jv x cos yv x sin 0 where j x and y x denote the bessel functions of the first and second kind respectively our chief tools are three recurrence relations for c x and an integral formula for j x

smooth optimum kernel estimators near endpoints summary kernel estimators for smooth curves like density spectral density or regression functions require modifications when estimating near endpoints of the support both for practical and asymptotic reasons the construction of such boundary kernels as solutions of a variational problem is addressed and representations in orthogonal polynomials are given including explicit solutions for the most important cases based on explicit formulae for certain functionals of the kernels it is shown that local bandwidth variation might be indicated near boundaries various bandwidth variation schemes are discussed and

on the observability inequalities for exact controllability of wave equations with variable coefficients this paper deals with boundary exact controllability for the dynamics governed by the wave equation with nonconstant coefficients in the principal part subject to dirichlet or neumann boundary controls the observability inequalities are established by the riemannian geometry method under some geometric condition for the dirichlet problem and for the neumann problem respectively next a number of nontrivial examples are presented to verify the observability inequality in particular a counterexample is given without boundary exact controllability where the control is exerted on the whole boundary

inequalities for some special functions abstract inequalities for bessel functions modified bessel functions and of their ratios are obtained these results are either sharper or hold under weaker conditions than had been known earlier similar inequalities are proved for the function w v t ti v t i v 1 t which is useful in some problems of finite elasticity in the sequel inequalities for confluent hypergeometric functions and of their ratios are established

energy methods for quasilinear hyperbolic initial boundary value problems applications to elastodynamics the aim of this paper is to show how energy methods can be used to establish local in time existence of smooth solutions to certain initial boundary value problems for systems of quasilinear hyperbolic partial differential equations our particular objective is to treat the equations of nonlinear elastodynamics

bounds for modified bessel functions abstract in this paper we derive inequalities for the modified bessel functions i v t and k v t in some cases our results are completely new in other cases we improve or extend known results similar inequalities are proved for the function w v ti v t i v 1 t which is useful in some problems of finite elasticity

local exact boundary controllability for nonlinear wave equations this paper deals with the local exact boundary controllability for dynamics governed by nonlinear wave equations subject to dirichlet neumann or any other kind of boundary controls which result in well posedness of the corresponding initial boundary value problem a constructive method is developed the local exact boundary controllability for semilinear wave equations is constructed in the case of both three odd and two even space dimensions and the boundary control is time optimal when the space dimension is three odd especially the local exact boundary controllability is established for quasi linear wave equations in several space dimensions by using the constructive method

asymmetric kernel regression kernel regression is one model that has been applied to explain or design radial basis neural networks practical application of the kernel regression method has shown that bias errors caused by the boundaries of the data can seriously effect the accuracy of this type of regression this paper investigates the correction of boundary error by substituting an asymmetric kernel function for the symmetric kernel function at data points close to the boundary the asymmetric kernel function allows a much closer approach to the boundary to be achieved without adversely effecting the noise filtering properties of the kernel regression

a bandwidth selector for local linear density estimators local linear density estimators achieve automatic boundary corrections and enjoy some typical optimal properties proper choice of the smoothing parameters is crucial for their performance a data based bandwidth selector is developed in the spirit of plug in rules consistency and asymptotic normality of the selected bandwidth are demonstrated the bandwidth is very efficient regardless of whether there are non smooth boundaries in the support of the density or not

an improved transformation based kernel estimator of densities on the unit interval the kernel density estimator kde suffers boundary biases when applied to densities on bounded supports which are assumed to be the unit interval transformations mapping the unit interval to the real line can be used to remove boundary biases however this approach may induce erratic tail behaviors when the estimated density of transformed data is transformed back to its original scale we propose a modified transformation based kde that employs a tapered and tilted back transformation we derive the theoretical properties of the new estimator and show that it asymptotically dominates the naive transformation based estimator while maintains its simplicity we then propose three automatic methods of smoothing parameter selection our monte carlo simulations demonstrate the good finite sample performance of the proposed estimator especially for densities with poles near the boundaries an example with real data is provided

transformations in density estimation abstract for the density estimation problem the global window width kernel density estimator does not perform well when the underlying density has features that require different amounts of smoothing at different locations in this article we propose to transform the data with the intention that a global window width is more appropriate for the density of the transformed data the density estimate of the original data is the back transform by change of variables of the global window width estimate of the transformed data s density we explore choosing the transformation from suitable parametric families data based selection rules for the choice of transformations and the window width are discussed application to real and simulated data demonstrates the usefulness of our proposals

kernels for nonparametric curve estimation summary the choice of kernels for the nonparametric estimation of regression functions and of their derivatives is investigated explicit expressions are obtained for kernels minimizing the asymptotic variance or the asymptotic integrated mean square error imse the present proof of the optimality of the latter kernels is restricted up to order k 5 these kernels are also of interest for the nonparametric estimation of probability densities and spectral densities a finite sample study indicates that higher order kernels asymptotically improving the rate of convergence may become attractive for realistic finite sample size suitably modified kernels are considered for estimating at the extremities of the data in a way which allows to retain the order of the bias found for interior points

inequalities for modified bessel functions and their integrals abstract simple inequalities for some integrals involving the modified bessel functions i x and k x are established we also obtain a monotonicity result for k x and a new lower bound that involves gamma functions for k 0 x

relative efficiencies of kernel and local likelihood density estimators summary local likelihood methods enjoy advantageous properties such as good performance in the presence of edge effects that are rarely found in other approaches to nonparametric density estimation however as we argue in this paper standard kernel methods can have distinct advantages when edge effects are not present we show that whereas the integrated variances of the two methods are virtually identical the integrated squared bias of a conventional kernel estimator is less than that of a local log linear estimator by as much as a factor of 4 moreover the greatest bias improvements offered by kernel methods occur when they are needed most i e when the effect of bias is particularly high similar comparisons can also be made when high degree local log polynomial fits are assessed against high order kernel methods for example although as is well known high degree local polynomial fits offer potentially infinite efficiency gains relative to their kernel competitors the converse is also true indeed the asymptotic value of the integrated squared bias of a local log quadratic estimator can exceed any given constant multiple of that for the competing kernel method in all cases the densities that suffer problems in the context of local log likelihood methods can be chosen to be symmetric either unimodal or bimodal either infinitely or compactly supported and to have arbitrarily many derivatives as functions on the real line they are not pathological however our results reveal quantitative differences between global performances of local log polynomial estimators applied to unimodal or multimodal distributions

sharp sufficient conditions for the observation control and stabilization of waves from the boundary for the observation or control of solutions of second order hyperbolic equation in mathbb r t times omega ralston s construction of localized states comm pure appl math 22 1969 pp

bounds for modified struve functions of the first kind and their ratios abstract we obtain a simple two sided inequality for the ratio l x l 1 x in terms of the ratio i x i 1 x where l x is the modified struve function of the first kind and i x is the modified bessel function of the first kind this result allows one to use the extensive literature on bounds for i x i 1 x to immediately deduce bounds for l x l 1 x we note some consequences and obtain further bounds for l x l 1 x by adapting techniques used to bound the ratio i x i 1 x we apply these results to obtain new bounds for the condition numbers x l x l x the ratio l x l y and the modified struve function l x itself amongst other results we obtain two sided inequalities for x l x l x and l x l y that are given in terms of x i x i x and i x i y respectively which again allows one to exploit the substantial literature on bounds for these quantities the results obtained in this paper complement and improve existing bounds in the literature

bound states in one and two spatial dimensions in this article we study the number of bound states for potentials in one and two spatial dimensions we first show that in addition to the well known fact that an arbitrarily weak attractive potential has a bound state it is easy to construct examples where weak potentials have an infinite number of bound states these examples have potentials which decrease at infinity faster than expected using somewhat stronger conditions we derive explicit bounds on the number of bound states in one dimension using known results for the three dimensional zero angular momentum a change of variables which allows us to go from the one dimensional case to that of two dimensions results in a bound for the zero angular momentum case finally we obtain a bound on the total number of bound states in two dimensions first for the radial case and then under stronger conditions for the noncentral case

new bounds for the generalized marcum q function in this paper we study the generalized marcum q function q nu a b where a nu 0 and b ges 0 our aim is to extend the results of corazza and ferrari ieee trans inf theory vol 48 pp 3003 3008 2002 to the generalized marcum q function in order to deduce some new tight lower and upper bounds the key tools in our proofs are some monotonicity properties of certain functions involving the modified bessel function of the first kind and some classical inequalities i e the cauchy buniakowski schwarz and chebyshev integral inequalities these bounds are shown to be very tight for large b i e the relative errors of our bounds converge to zero as b increases both theoretical analysis and numerical results are provided to show the tightness of our bounds

exact boundary controllability for quasi linear hyperbolic systems using a result on the existence and uniqueness of the semiglobal c1 solution to the mixed initial boundary value problem for first order quasi linear hyperbolic systems with general nonlinear boundary conditions we establish the exact boundary controllability for quasi linear hyperbolic systems if the c1 norm of initial and final states is small enough

on pseudodata methods for removing boundary effects in kernel density estimation we suggest a method for boundary correcting kernel density estimators based on generating pseudodata beyond the extremities of the density s support the estimator produced in this way enjoys optimal orders of bias and variance right up to the ends of the support and it may be used with kernels of arbitrary order our method is considerably more adaptive than the common data reflection approach which is not really appropriate for kernels of order 2 or more since it does not adequately correct boundary bias in a simulation study for densities on 0 1 with positive slope at x 0 our method was found to have lower mean squared error at x 0 than the boundary kernel method especially for small sample sizes our technique may be used in conjunction with plug in or least squares cross validation methods of bandwidth selection and has an analogue in the context of estimating a point process intensity

an improved estimator of the density function at the boundary abstract we propose a new method of boundary correction for kernel density estimation the technique is a kind of generalized reflection method involving reflecting a transformation of the data the transformation depends on a pilot estimate of the logarithmic derivative of the density at the boundary in simulations the new method is seen to clearly outperform an earlier generalized reflection idea it also has overall advantages over boundary kernel methods and a nonnegative adaptation thereof although the latter are competitive in some situations we also present the theory underlying the new methodology

tight bounds for the generalized marcum q function abstract in this paper we study the generalized marcum q function of order 0 real defined by q a b 1 a 1 b t e t 2 a 2 2 i 1 a t dt where a 0 b 0 and i stands for the modified bessel function of the first kind our aim is to improve and extend some recent results of wang to the generalized marcum q function in order to deduce some sharp lower and upper bounds in both cases b a and b a we give the best possible upper bound for q a b the key tools in our proofs are some monotonicity properties of certain functions involving the modified bessel function of the first kind these monotonicity properties are deduced from some results on modified bessel functions which have been used in wave mechanics and finite elasticity

inequalities for integrals of modified bessel functions and expressions involving them abstract simple inequalities are established for some integrals involving the modified bessel functions of the first and second kind in most cases we show that we obtain the best possible constant or that our bounds are tight in certain limits we apply these inequalities to obtain uniform bounds for several expressions involving integrals of modified bessel functions such expressions occur in stein s method for variance gamma approximation and the results obtained in this paper allow for technical advances in the method we also present some open problems that arise from this research

functional inequalities for modified struve functions in this paper by using a general result on the monotonicity of quotients of power series our aim is to prove some monotonicity and convexity results for the modified struve functions moreover as consequences of the above mentioned results we present some functional inequalities as well as lower and upper bounds for modified struve functions our main results complement and improve the results of joshi and nalwaya