local linear regression smoothers and their minimax efficiencies in this paper we introduce a smooth version of local linear regression estimators and address their advantages the mse and mise of the estimators are computed explicitly it turns out that the local linear regression smoothers have nice sampling properties and high minimax efficiency they are not only efficient in rates but also nearly efficient in constant factors in the nonparametric regression context the asymptotic minimax lower bound is developed via the heuristic of the hardest onedimensional subproblem of donoho and liu connections of the minimax risk with the modulus of continuity are made the lower bound is also applicable for estimating conditional mean regression and conditional quantiles for both fixed and random design regression problems

reducing variance in univariate smoothing a variance reduction technique in nonparametric smoothing is proposed at each point of estimation form a linear combination of a preliminary estimator evaluated at nearby points with the coefficients specified so that the asymptotic bias remains unchanged the nearby points are chosen to maximize the variance reduction we study in detail the case of univariate local linear regression while the new estimator retains many advantages of the local linear estimator it has appealing asymptotic relative efficiencies bandwidth selection rules are available by a simple constant factor adjustment of those for local linear estimation a simulation study indicates that the finite sample relative efficiency often matches the asymptotic relative efficiency for moderate sample sizes this technique is very general and has a wide range of applications

space time codes for high data rate wireless communication performance criteria in the presence of channel estimation errors mobility and multiple paths space time coding is a bandwidth and power efficient method of communication over fading channels that realizes the benefits of multiple transmit antennas specific codes have been constructed using design criteria derived for quasi static flat rayleigh or rician fading where channel state information is available at the receiver it is evident that the practicality of space time codes will be greatly enhanced if the derived design criteria remain valid in the absence of perfect channel state information it is even more desirable that the design criteria not be unduly sensitive to frequency selectivity and to the doppler spread this paper presents a theoretical study of these issues beginning with the effect of channel estimation error here it is assumed that a channel estimator extracts fade coefficients at the receiver and for constellations with constant energy it is proved that in the absence of ideal channel state information the design criteria for space time codes is still valid the analysis also demonstrates that standard channel estimation techniques can be used in conjunction with space time codes provided that the number of transmit antennas is small we also derive the maximum likelihood detection metric in the presence of channel estimation errors next the effect of multiple paths on the performance of space time codes is studied for a slowly changing rayleigh channel it is proved that the presence of multiple paths does not decrease the diversity order guaranteed by the design criteria used to construct the space time codes similar results hold for rapid fading channels with or without multiple paths the conclusion is that the diversity order promised by space time coding is achieved under a variety of mobility conditions and environmental effects

improved estimators in nonparametric regression problems abstract linear estimators of multivariate means are considered generalizations of some well known theorems about admissibility of linear estimators are given the results then are applied to show that commonly used kernel type estimators in nonparametric regression problems can be constructively improved in a simple way an asymptotic result is described that gives a quantitative measure of the maximum improvement to be gained in certain situations a theoretical bound shows that gains are achievable in the relative risk of up to 58 6 rectangular kernel or 29 2 epanechnikov kernel some examples of smaller sample size are also investigated and these show relative risk gains ranging up to 18 in realistic settings

orthogonally additive polynomials on banach function algebras abstract for a banach function algebra a we consider the problem of representing a continuous d homogeneous polynomial p a x where x is an arbitrary banach space that satisfies the property p f g p f p g whenever f g a are such that supp f supp g we show that such a polynomial can be represented as p f t f d f a for some continuous linear map t a x for a variety of banach function algebras such as the algebra of continuous functions c 0 for any locally compact hausdorff space the algebra of lipschitz functions lip k for any compact metric space k and 0 1 the figa talamanca herz algebra a p g for some locally compact groups g and p 1 the algebras a c a b and b v c a b of absolutely continuous functions and of continuous functions of bounded variation on the interval a b in the case where a c n a b p can be represented as p f t n 1 n d f n 1 f n d where the sum is taken over n 1 n d z d with 0 n 1 n d n for appropriate continuous linear maps t n 1 n d c n n d a b x

finite sample variance of local polynomials analysis and solutions abstract fitting local polynomials in nonparametric regression has a number of advantages the attractive theoretical features are in a partial contradiction to variance properties for random design and to practical experience over a broad range of situations no upper bound can be given for the conditional variance the unconditional variance is infinite when using optimal weights with compact support properties are better for gaussian weights we analyze local polynomials for finite sample size both theoretically and numerically it turns out that difficulties arise in sparse regions in the realization of the design when the realization has locally a small variance and or a skew empirical distribution two small sample modifications of local polynomials are presented local increase of bandwidth in sparse regions of the design and local polynomial ridge regression both modifications combine a good finite sample behavior with the asymptotic advantages of local polynomials

multiuser detection for overloaded cdma systems multiuser detection for overloaded code division multiple access cdma systems in which the number of users is larger than the dimension of the signal space is of particular interest when bandwidth is at a premium in this paper certain fundamental questions are answered regarding the asymptotic forms and performance of suboptimum multiuser detectors for cases where the desired and or interfering signal subspaces are of reduced rank and or have a nontrivial intersection in the process two new suboptimum detectors are proposed that are especially well suited to overloaded systems namely the group pseudo decorrelator and the group minimum mean squared error mmse detector the former is seen to be the correct extension of the group decorrelator in the sense that it is the limiting form in the low noise regime of the group mmse detector pseudo decorrelation is also used as a feedforward filter in a new decision feedback scheme for the particular case of real valued modulation it is shown that the proposals of the so called improved linear also known as linear conjugate or widely linear detectors were more simply derived earlier using the idea of minimal sufficiency which we also apply to the new detectors of this paper

a quasi orthogonal space time block code it has been shown that a complex orthogonal design that provides full diversity and full transmission rate for a space time block code is not possible for more than two antennas previous attempts have been concentrated in generalizing orthogonal designs which provide space time block codes with full diversity and a high transmission rate we design rate one codes which are quasi orthogonal and provide partial diversity the decoder of the proposed codes works with pairs of transmitted symbols instead of single symbols

orthogonally additive and multiplicative maps between figa talamanca herz algebras abstract we study the structure of an n homogeneous polynomial p a p g 1 a p g 2 between figa talamanca herz algebras among other results we show that there exist a proper affine map y g 2 g 1 defined in an open coset y of g 2 such that p f y f y n f a p g 1 y y when the locally compact groups g 1 g 2 are amenable and p is p completely contractive orthogonally additive and multiplicative and preserves a right identity of a p g 1

local polynomial regresssion estimators in survey sampling estimation of finite population totals in the presence of auxiliary information is considered a class of estimators based on local polynomial regression is proposed like generalized regression estimators these estimators are weighted linear combinations of study variables in which the weights are calibrated to known control totals but the assumptions on the superpopulation model are considerably weaker the estimators are shown to be asymptotically design unbiased and consistent under mild assumptions a variance approximation based on taylor linearization is suggested and shown to be consistent for the design mean squared error of the estimators the estimators are robust in the sense of asymptotically attaining the godambe joshi lower bound to the anticipated variance simulation experiments indicate that the estimators are more efficient than regression estimators when the model regression function is incorrectly specified while being approximately as efficient when the parametric specification is correct

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

an analysis of the mimo sdma channel with space time orthogonal and quasi orthogonal user transmissions and efficient successive cancellation decoders we consider space time transceiver architectures for space division multiple access sdma fading channels with simultaneous transmissions from multiple users each user has up to four transmit antennas and employs a space time orthogonal or a quasi orthogonal design as an inner code at the multiple antenna receiver efficient successive group interference cancellation strategies based on zero forcing or minimum mean square error mmse filtering are employed in some fixed or channel dependent order these strategies are efficient in the sense that they exploit the special structure of the inner codes to yield much higher diversity orders than would be otherwise possible while at the same time preserving what we call the decoupling property of the constituent inner codes which enables the use of low complexity outer encoders decoders for each user motivated by the special structure of the effective channel matrix induced by the inner codes we obtain several new distribution results on the qr and eigenvalue decompositions of certain structured random matrices these results are the key to a comprehensive performance analysis of the proposed multiuser transceiver architectures including the characterization of diversity multiplexing tradeoff dmt curves and exact per user bit error rates bers without making simplifying assumptions about error propagation

the reconstruction approach from interpolation to regression this article introduces an interpolation based method called the reconstruction approach for nonparametric regression based on the fact that interpolation usually has negligible errors compared

interpolation methods for adapting to sparse design in nonparametric regression abstract we suggest interpolation methods for overcoming the problem of sparse design in local linear smoothing these methods are based on simple rules determined by the kernel and bandwidth for deciding when and where pseudo design points should be added to augment the original design sequence new ordinates for the added design points are computed by simple interpolation then local linear smoothing is applied directly to the expanded dataset the method is competitive with alternatives e g those involving ridge regression in terms of both simplicity and performance

on bias reduction in local linear smoothing summary the standard approach to local linear regression involves fitting a straight line segment to a curve in a symmetrical way in that the segment is fitted directly above a small region whose midpoint is the abscissa x at which we wish to estimate the curve in this paper we show that if the segment is fitted in a skew manner with its centre a little to the left or right of x then bias can be reduced by an order of magnitude without affecting the order of magnitude of variance the amount by which the centre should be shifted depends only on the kernel function and not at all on the unknown regression mean or on the design density the average of two similarly but oppositely shifted estimators has two orders of magnitude less bias again at the expense of a slight increase in variance this particular estimator may be viewed as a limiting form of a convex combination of three local linear estimators the two oppositely shifted estimators and the symmetric one which has two orders of magnitude less bias and depending on the kernel function less variance as well

locally weighted regression an approach to regression analysis by local fitting abstract locally weighted regression or loess is a way of estimating a regression surface through a multivariate smoothing procedure fitting a function of the independent variables locally and in a moving fashion analogous to how a moving average is computed for a time series with local fitting we can estimate a much wider class of regression surfaces than with the usual classes of parametric functions such as polynomials the goal of this article is to show through applications how loess can be used for three purposes data exploration diagnostic checking of parametric models and providing a nonparametric regression surface along the way the following methodology is introduced a a multivariate smoothing procedure that is an extension of univariate locally weighted regression b statistical procedures that are analogous to those used in the least squares fitting of parametric functions c several graphical methods that are useful tools for understanding loess estimates and checking the a

orthogonally additive polynomials on convolution algebras associated with a compact group abstract let g be a compact group let x be a banach space and let p l 1 g x be an orthogonally additive continuous n homogeneous polynomial then we show that there exists a unique continuous linear map l 1 g x such that p f f n f for each f l 1 g we also seek analogues of this result about l 1 g for various other convolution algebras including l p g for 1 p and c g

orthogonally additive polynomials on spaces of continuous functions we show that for every orthogonally additive homogeneous polynomial p on a space of continuous functions c k with values in a banach space y there exists a linear operator s c k y such that p f s f n this is the c k version of a related result of sundaresam for polynomials on l p spaces

integral representation of product factorable bilinear operators and summability of bilinear maps on c k spaces abstract we present a constructive technique to represent classes of bilinear operators that allow a factorization through a bilinear product providing a general version of the well known characterization of integral bilinear forms as elements of the dual of an injective tensor product we show that this general method fits with several known situations coming from different contexts harmonic analysis c algebras c k spaces operator theory polynomials providing a unified approach to the integral representation of a broad class of bilinear operators some examples and applications are also shown regarding for example operator spaces and summability properties of bilinear maps

methods for estimating a conditional distribution function abstract motivated by the problem of setting prediction intervals in time series analysis we suggest two new methods for conditional distribution estimation the first method is based on locally fitting a logistic model and is in the spirit of recent work on locally parametric techniques in density estimation it produces distribution estimators that may be of arbitrarily high order but nevertheless always lie between 0 and 1 the second method involves an adjusted form of the nadaraya watson estimator it preserves the bias and variance properties of a class of second order estimators introduced by yu and jones but has the added advantage of always being a distribution itself our methods also have application outside the time series setting for example to quantile estimation for independent data this problem motivated the work of yu and jones

orthogonally additive polynomials on fourier algebras abstract we show that an n homogeneous polynomial p on the fourier algebra a g of a locally compact group g can be represented in the form p f t f n f a g for some t in the group von neumann algebra vn g of g if and only if it is orthogonally additive and completely bounded

an effective bandwidth selector for local least squares regression abstract local least squares kernel regression provides an appealing solution to the nonparametric regression or scatterplot smoothing problem as demonstrated by fan for example the practical implementation of any scatterplot smoother is greatly enhanced by the availability of a reliable rule for automatic selection of the smoothing parameter in this article we apply the ideas of plug in bandwidth selection to develop strategies for choosing the smoothing parameter of local linear squares kernel estimators our results are applicable to odd degree local polynomial fits and can be extended to other settings such as derivative estimation and multiple nonparametric regression an implementation in the important case of local linear fits with univariate predictors is shown to perform well in practice a by product of our work is the development of a class of nonparametric variance estimators based on local least squares ideas and plug in rules for their implementation