some results on the apostol bernoulli and apostol euler polynomials the main object of this paper is to investigate the apostol bernoulli polynomials and the apostol euler polynomials we first establish two relationships between the generalized apostol bernoulli and apostol euler polynomials it can be found that many results obtained before are special cases of these two relationships moreover we have a study on the sums of products of the apostol bernoulli polynomials and of the apostol euler polynomials

some new results on products of apostol bernoulli and apostol euler polynomials abstract we perform a further investigation for the apostol bernoulli and apostol euler polynomials and numbers by making use of an elementary idea used by euler in the discovery of his famous pentagonal number theorem we establish some new formulae for products of an arbitrary number of apostol bernoulli and apostol euler polynomials and numbers these results are the corresponding generalizations of some known formulae including the higher order convolution ones discovered by agoh and dilcher 2014 5 on the classical bernoulli and euler polynomials

effective nonparametric estimation in the case of severely discretized data often economic data are discretized or rounded to some extent this paper proposes a regression and a density estimator that work especially well when discretization causes conventional kernel based estimators to behave poorly the estimator proposed here is a weighted average of neighboring frequency estimators and the weights are composed of cubic b splines interestingly we show that this estimator can have both a smaller bias and variance than frequency estimators as a means to obtain asymptotic normality and rates of convergence we assume that the discreteness becomes finer as the sample size increases

instrumental variables regression with weak instruments this paper develops asymptotic distribution theory for instrumental variable regression when the partial correlation between the instruments and a single included endogenous variable is weak here modeled as local to zero asymptotic representations are provided for various instrumental variable statistics including the two stage least squares tsls and limited information maximum likelihood liml estimators and their t statistics the asymptotic distributions are found to provide good approximations to sampling distributions with just 20 observations per instrument even in large samples tsls can be badly biased but liml is in many cases approximately median unbiased the theory suggests concrete quantitative guidelines for applied work these guidelines help to interpret angrist and krueger s 1991 estimates of the returns to education whereas tsls estimates with many instruments approach the ols estimate of 6 the more reliable liml and tsls estimates with fewer instruments fall between 8 and 10 with a typical confidence interval of 6 14

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

bandwidth selection classical or plug in bandwidth selection for procedures such as kernel density estimation and local regression have been widely studied over the past decade substantial evidence has been collected to establish superior performance of modern plug in methods in comparison to methods such as cross validation this has ranged from detailed analysis of rates of convergence to simulations to superior performance on real datasets in this work we take a detailed look at some of this evidence looking into the sources of differences our findings challenge the claimed superiority of plug in methods on several fronts first plug in methods are heavily dependent on arbitrary specification of pilot bandwidths and fail when this specification is wrong second the often quoted variability and undersmoothing of cross validation simply reflects the uncertainty of band width selection plug in methods reflect this uncertainty by oversmoothing and missing important features when given difficult problems third we look at asymptotic theory plug in methods use available curvature information in an inefficient manner resulting in inefficient estimates previous comparisons with classical approaches penalized the classical approaches for this inefficiency asymptotically the plug in based estimates are beaten by their own pilot estimates

pade approximation and apostol bernoulli and apostol euler polynomials using the pade approximation of the exponential function we obtain recurrence relations between apostol bernoulli and between apostol euler polynomials as applications we derive some new lacunary recurrence relations for bernoulli and euler polynomials with gap of length 4 and lacunary relations for bernoulli and euler numbers with gap of length 6

a set of finite order differential equations for the appell polynomials let r n x n 0 denote the set of appell polynomials which includes among others hermite bernoulli euler and genocchi polynomials in this paper by introducing the generalized factorization method for each k n we determine the differential operator l n k x n 0 such that l n k x r n x l n kr n x where l n k n k n k the special case k 1 reduces to the result obtained in m x he p e ricci differential equation of appell polynomials via the factorization method j comput appl math 139 2002 231 237 the differential equations for the hermite and bernoulli polynomials are exhibited for the case k 2

an explicit formula for the generalized bernoulli polynomials abstract the object of the present note is to prove a new explicit formula for the generalized bernoulli polynomials the main result 3 below provides an interesting extension of a representation for the generalized bernoulli numbers given recently by p g todorov c r acad sci paris ser i math 301 1985 665 666

nonparametric estimation of regression functions with both categorical and continuous data abstract in this paper we propose a method for nonparametric regression which admits continuous and categorical data in a natural manner using the method of kernels a data driven method of bandwidth selection is proposed and we establish the asymptotic normality of the estimator we also establish the rate of convergence of the cross validated smoothing parameters to their benchmark optimal smoothing parameters simulations suggest that the new estimator performs much better than the conventional nonparametric estimator in the presence of mixed data an empirical application to a widely used and publicly available dynamic panel of patent data demonstrates that the out of sample squared prediction error of our proposed estimator is only 14 20 of that obtained by some popular parametric approaches which have been used to model this data set

leverage effect in financial markets the retarded volatility model we investigate quantitatively the so called leverage effect which corresponds to a negative correlation between past returns and future volatility for individual stocks this correlation is moderate and decays over 50 days while for stock indices it is much stronger but decays faster for individual stocks the magnitude of this correlation has a universal value that can be rationalized in terms of a new retarded model which interpolates between a purely additive and a purely multiplicative stochastic process for stock indices a specific amplification phenomenon seems to be necessary to account for the observed amplitude of the effect

a new unified family of generalized apostol euler bernoulli and genocchi polynomials in this paper we introduce a new unified family of polynomials including unified apostol bernoulli euler and genocchi polynomials see luo and srivastava 2006 and a unified family of polynomials defined by ozden et al 2010 also by applying the mellin transformation for the generating function of the new unified family of polynomials we deduced a unified class of zeta function finally we derive some relationships between our new unified family of polynomials and some types of generalized polynomials

p adic distribution of the unification of the bernoulli euler and genocchi polynomials abstract the aim of this paper is to construct p adic distribution on x c p of the unification of the bernoulli euler and genocchi polynomials y n x k a b which is given by n k a b j dp n z p a b dp n p dp n n k a jb y n dp n j dp n k a dp n b where y n x k a b are defined by 1 1 we give some applications related to these functions and distribution

binomial convolution and transformations of appell polynomials abstract we show that the set a of appell sequences is an abelian group under the binomial convolution this is essentially equivalent to other approaches considered in the literature in particular the determinantal approach we also define a scale transformation t a a which is an isomorphism and a transformation r y a a based on expectations with respect to a random variable y using these tools we give explicit formulas for the determinantal form of appell sequences as well as for the formula of representation of powers and the srivastava pinter addition theorem various illustrative examples mainly referring to various generalizations of bernoulli and euler polynomials are discussed in detail

some new identities for the apostol bernoulli polynomials and the apostol genocchi polynomials in this paper we present a further investigation for the apostol bernoulli polynomials and the apostol genocchi polynomials by making use of the generating function methods and summation transform techniques we establish some new identities involving the products of the apostol bernoulli polynomials and the apostol genocchi polynomials many of the results presented here are the corresponding generalizations of some known formulas on the classical bernoulli polynomials and the classical genocchi polynomials

some relationships between the apostol bernoulli and apostol euler polynomials recently srivastava and pinter 1 investigated several interesting properties and relationships involving the classical as well as the generalized or higher order bernoulli and euler polynomials they also showed among other things that the main relationship proven earlier by cheon 2 can easily be put in a much more general setting the main object of the present sequel to these earlier works is to derive several general properties and relationships involving the apostol bernoulli and apostol euler polynomials some of these general results can indeed be suitably specialized in order to deduce the corresponding properties and relationships involving the generalized bernoulli and generalized euler polynomials other relationships associated with the stirling numbers of the second kind are also considered

certain results for the 2 variable apostol type and related polynomials in this article the 2 variable general polynomials are taken as base with apostol type polynomials to introduce a family of 2 variable apostol type polynomials these polynomials are framed within the context of monomiality principle and their properties are established certain summation formulae for these polynomials are also derived examples of some members belonging to this family are considered and numbers related to some mixed special polynomials are also explored

on the analogs of euler numbers and polynomials associated with p adic q integral on zp at q 1 abstract the purpose of this paper is to construct euler numbers and polynomials by using fermionic expression of p adic q integral at q 1 from these euler polynomials we derive the harmonic sums of higher order finally we investigate several interesting properties and relationships involving the classical as well as the generalized euler numbers and polynomials as an application we will treat p adic invariant integrals on z p involving trigonometric functions

several identities for the generalized apostol bernoulli polynomials the purpose of this paper is to give several symmetric identities on the generalized apostol bernoulli polynomials by applying the generating functions these results extend some known identities

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

multifractal random walks with fractional brownian motion via malliavin calculus we introduce a multifractal random walk mrw defined as a stochastic integral of an infinitely divisible noise with respect to a dependent fractional brownian motion using the techniques of the malliavin calculus we study the existence of this object and its properties we then propose a continuous time model for financial modeling that captures the main properties observed in the empirical data including the leverage effect we illustrate our result by numerical simulations

multifractal random walks as fractional wiener integrals multifractal random walks are defined as integrals of infinitely divisible stationary multifractal cascades with respect to fractional brownian motion their key properties are studied such as finiteness of moments and scaling with respect to the chosen values of the self similarity and infinite divisibility parameters the range of these parameters is larger than that considered previously in the literature and the cases of both exact and nonexact scale invariance are considered special attention is paid to various types of definitions of multifractal random walks the resulting random walks are of interest in modeling multifractal processes whose marginals exhibit stationarity and symmetry

on non scale invariant infinitely divisible cascades multiplicative processes multifractals and more recently also infinitely divisible cascades have seen increased popularity in a host of applications requiring versatile multiscale models ranging from hydrodynamic turbulence to computer network traffic from image processing to economics the methodologies prevalent as of today rely to a large extent on iterative schemes used to produce infinite detail and repetitive structure across scales while appealing due to their simplicity these constructions have limited applicability as they lead by default to power law progression of moments through scales to nonstationary increments and often to inherent log periodic scaling which favors an exponential set of scales this paper studies and develops a wide class of infinitely divisible cascades idc thereby establishing the first reported cases of controllable scaling of moments in non power law form embedded in the framework of idc these processes exhibit stationary increments and scaling over a continuous range of scales criteria for convergence further statistical properties as well as matlab routines are provided

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

some generalizations of the apostol bernoulli and apostol euler polynomials abstract the main object of this paper is to give analogous definitions of apostol type see t m apostol on the lerch zeta function pacific j math 1 1951 161 167 and h m srivastava some formulas for the bernoulli and euler polynomials at rational arguments math proc cambridge philos soc 129 2000 77 84 for the so called apostol bernoulli numbers and polynomials of higher order we establish their elementary properties derive several explicit representations for them in terms of the gaussian hypergeometric function and the hurwitz or generalized zeta function and deduce their special cases and applications which are shown here to lead to the corresponding results for the classical bernoulli numbers and polynomials of higher order

a brief survey of bandwidth selection for density estimation abstract there has been major progress in recent years in data based bandwidth selection for kernel density estimation some second generation methods including plug in and smoothed bootstrap techniques have been developed that are far superior to well known first generation methods such as rules of thumb least squares cross validation and biased cross validation we recommend a solve the equation plug in bandwidth selector as being most reliable in terms of overall performance this article is intended to provide easy accessibility to the main ideas for nonexperts

some generalizations of the apostol genocchi polynomials and the stirling numbers of the second kind abstract recently the authors introduced some generalizations of the apostol bernoulli polynomials and the apostol euler polynomials see q m luo h m srivastava j math anal appl 308 2005 290 302 and q m luo taiwanese j math 10 2006 917 925 the main object of this paper is to investigate an analogous generalization of the genocchi polynomials of higher order that is the so called apostol genocchi polynomials of higher order for these generalized apostol genocchi polynomials we establish several elementary properties provide some explicit relationships with the apostol bernoulli polynomials and the apostol euler polynomials and derive various explicit series representations in terms of the gaussian hypergeometric function and the hurwitz or generalized zeta function we also deduce their special cases and applications which are shown here to lead to the corresponding results for the genocchi and euler polynomials of higher order by introducing an analogue of the stirling numbers of the second kind that is the so called stirling numbers of the second kind we derive some basic properties and formulas and consider some interesting applications to the family of the apostol type polynomials furthermore we also correct an error in a previous paper q m luo h m srivastava comput math appl 51 2006 631 642 and pose two open problems on the subject of our investigation

discretized and interpolated kernel density estimates abstract the kernel estimate of a probability density function inherits its smoothness properties from the kernel density chosen by the investigator nevertheless for computational and especially graphical reasons exact kernel density estimates are often presented in piecewise constant discretized or piecewise linear interpolated form using the exact estimate only at some grid of points the asymptotic integrated mean squared error properties of these modifications are studied it is seen that discretization may adversely affect the order of magnitude of this risk criterion so care needs to be taken in practice to limit the degree of discretization employed this roughness effect essentially disappears when interpolation is used instead then it takes a remarkably coarse grid to result in more than a negligible deterioration in the estimate s performance data discretization prior to applying the kernel density estimation prescription is also addressed such prebinning does not affect the smoot

a unified presentation of the generating functions of the generalized bernoulli euler and genocchi polynomials the goal of this paper is to unify and extend the generating functions of the generalized bernoulli polynomials the generalized euler polynomials and the generalized genocchi polynomials associated with the positive real parameters a and b and the complex parameter b by using this generating function we derive recurrence relations and other properties for these polynomials by applying the mellin transformation to the generating function of the unification of bernoulli euler and genocchi polynomials we construct a unification of the zeta functions furthermore we give many properties and applications involving the functions and polynomials investigated in this paper

on kolmogorov s representation of functions of several variables by functions of one variable abstract this paper proposes a multivariate nonparametric regression estimator motivated by kolmogorov doklady 114 1957 679 we consider functions of the form k 1 t g k k 1 k x 1 k d k x d 1 t

scaling of the distribution of fluctuations of financial market indices up to more than one month we find that the distributions for dt 4d 1560 min are consistent with a power law asymptotic behavior characterized by an exponent a 3 well outside the stable levy regime 0 a 2 to test the robustness of the s p result we perform a parallel analysis on two other financial market indices database iv contains 3560 daily records of the nikkei index for the 14 year period 1984 1997 and database v contains 4649 daily records of the hang seng index for the 18 year period 1980 1997 we find estimates ofa consistent with those describing the distribution of s p 500 daily returns one possible reason for the scaling of these distributions is the long persistence of the autocorrelation function of the volatility for time scales longer than dt 3 4 d our results are consistent with a slow convergence to gaussian behavior s1063 651x 99 11211 x pacs number s 05 40 fb 05 45 tp 89 90 1n

the influence of rounding errors on some nonparametric estimators of a density and its derivatives all continuous data is rounded to some extent and this rounding introduces errors into the estimation of the parent density and its derivatives in this paper we examine the influence of rounding errors on the two most popular types of nonparametric density estimator those based on kernels and those based on generalized fourier series our results lead directly to several important conclusions which are very relevant to the practical problems of density estimation

elliptic extensions of the apostol bernoulli and apostol euler polynomials in this paper we investigate the elliptic analogues of the apostol bernoulli and apostol euler polynomials and obtain the closed expressions of sums of products for these elliptic type polynomials some interesting special cases are also shown

frequency polygons theory and application abstract in this article i investigate the theoretical properties and applications of the frequency polygon which is constructed by connecting with straight lines the mid bin values of a histogram for estimating an unknown probability density function using a random sample the frequency polygon is shown to dominate the histogram with respect to the criterion of integrated mean squared error achieving the same rate of convergence to zero of the integrated mean squared error as non negative kernel estimators data based algorithms for constructing frequency polygons are discussed and illustrated one is based on a histogram with bin width equal to 2 15sn 1 5 where s is an estimate of the standard deviation from a sample of size n another approach is based on the method of generalized crossvalidation the bivariate frequency polygon is also investigated comparisons are made between the frequency polygon and other density estimators examples with data in one and two dimensions are presented

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

a new approach to q genocchi numbers and their interpolation functions abstract the aim of this paper is to define generating functions of higher order q genocchi numbers by using a p adic q deformed fermionic integral on z p we give witt s type integral of these numbers furthermore we give the interpolation functions of these numbers

some series identities involving the generalized apostol type and related polynomials a unification and generalization of various apostol type polynomials was introduced and investigated recently by luo and srivastava q m luo h m srivastava some generalizations of the apostol genocchi polynomials and the stirling numbers of the second kind appl math comput 217 2011 5702 5728 in this paper we prove several symmetry identities for these generalized apostol type polynomials by using their generating functions as special cases and consequences of our results we obtain the corresponding symmetry identities for the apostol euler polynomials of higher order the apostol bernoulli polynomials of higher order and the apostol genocchi polynomials of higher order and also for another family of generalized apostol type polynomials which were investigated systematically by ozden et al h ozden y simsek h m srivastava a unified presentation of the generating functions of the generalized bernoulli euler and genocchi polynomials comput math appl 60 2010 2779 2787 we also derive several relations between the apostol type polynomials the generalized sum of integer powers and the generalized alternating sum it is shown how each of these results would extend the corresponding known identities

unified apostol bernoulli euler and genocchi polynomials in this paper we present a unified family of polynomials including not only the apostol bernoulli euler and genocchi polynomials but also a general family of polynomials suggested by ozden et al h ozden y simsek h m srivastava a unified presentation of the generating functions of the generalized bernoulli euler and genocchi polynomials comput math appl 60 10 2010 2779 2787 we obtain the explicit representation of this unified family in terms of a gaussian hypergeometric function some symmetry identities and multiplication formula are also given

some array type polynomials associated with special numbers and polynomials abstract the main objective in this paper is first to establish new identities for the stirling type numbers of the second kind the array type polynomials the apostol bernoulli polynomials and the apostol bernoulli numbers we then construct a delta operator and investigate various generating functions for the bell type numbers and for some new polynomials associated with the array type polynomials we also derive several other identities and relations for these polynomials and numbers

multifractal random walk we introduce a class of multifractal processes referred to as multifractal random walks mrws to our knowledge it is the first multifractal process with continuous dilation invariance properties and stationary increments mrws are very attractive alternative processes to classical cascadelike multifractal models since they do not involve any particular scale ratio the mrws are indexed by four parameters that are shown to control in a very direct way the multifractal spectrum and the correlation structure of the increments we briefly explain how in the same way one can build stationary multifractal processes or positive random measures

nonparametric and semiparametric estimation with discrete regressors this note is concerned with nonparametric and semiparametric inference in regression models where regressors are not continuous when regressors are discrete with finite support a mere average of those observations of the dependent variable with the same regressor value will yield a root n consistent conditional expectation estimate we show that sequences of weights constructed in this way are consistent in the sense of stone 1977 even when the discrete regressors have infinite support the results are applied to the estimation of semiparametric models

fourier expansions and integral representations for the apostol bernoulli and apostol euler polynomials we investigate fourier expansions for the apostol bernoulli and apostol euler polynomials using the lipschitz summation formula and obtain their integral representations we give some explicit formulas at rational arguments for these polynomials in terms of the hurwitz zeta function we also derive the integral representations for the classical bernoulli and euler polynomials and related known results

some q extensions of the apostol bernoulli and the apostol euler polynomials of order n and the multiple hurwitz zeta function abstract in this paper we first investigate several further interesting properties of the multiple hurwitz lerch zeta function n z s a which was introduced recently by choi et al j choi d s jang h m srivastava a generalization of the hurwitz lerch zeta function integral transform spec funct 19 2008 we then introduce and investigate some q extensions of the multiple hurwitz lerch zeta function n z s a the apostol bernoulli polynomials b k n x of order n and the apostol euler polynomials e k n x of order n relevant connections of the results presented here with those obtained in earlier works are also indicated precisely

some identities involving exponential functions and stirling numbers and applications guo and qi 2013 posed a problem asking to determine the coefficients a k i 1 for 1 i k such that 1 1 e t k 1 i 1 k a k i 1 1 e t 1 i 1 the authors answer this question alternatively by faa di bruno s formula unify the eight identities due to guo and qi to two identities involving two parameters and apply them to obtain an explicit expression for the apostol bernoulli numbers and the fubini numbers respectively