pair correlation densities of inhomogeneous quadratic forms under explicit diophantine conditions on 2 w eprove that the local two point correlations of the sequence given by the values m 2 n 2 with m n 2 are those of a poisson process this partly confirms a conjecture of berry and tabor 2 on spectral statistics of quantized integrable systems and also establishes a particular case of the quantitative version of the oppenheim conjecture for inhomogeneous quadratic forms of signature 2 2 the proof uses theta sums and ratner s classification of measures invariant under unipotent flows

forecasting cause age specific mortality using two random processes mortality forecasts are critical information for assessing the health of a population and are necessary for making informed decisions about how best to direct health related resources and activities timeliness in making health statistics available is crucial to identify and address current health problems being motivated to meet these needs we propose a method to forecast the number of cause age specific deaths through a two random processes model unlike the previous methods the new method incorporates both cross sectional and longitudinal correlations into our model without a high dimensional problem a bootstrap confidence interval is presented to measure the validity of our model and to detect an unusual occurrence of deaths our data analysis demonstrates that our method gives promising results compared with the true final counts

on the design of mimo block fading channels with feedback link capacity constraint in this paper we propose a combined adaptive power control and beamforming framework for optimizing multiple input multiple output mimo link capacity in the presence of feedback link capacity constraint the feedback channel is used to carry channel state information only it is assumed to be noiseless and causal with a feedback capacity constraint in terms of maximum number of feedback bits per fading block we show that the hybrid design could achieve the optimal mimo link capacity and we derive a computationally efficient algorithm to search for the optimal design under a specific average power constraint finally we shall illustrate that a minimum mean square error spatial processor with a successive interference canceller at the receiver could be used to realize the optimal capacity we found that feedback effectively enhances the forward channel capacity for all signal to noise ratio snr values when the number of transmit antennas n sub t is larger than the number of receive antennas n sub r the snr gain with feedback is contributed by focusing transmission power on active eigenchannel and temporal power waterfilling the former factor contributed at most 10log sub 10 n sub t n sub r db snr gain when n sub t n sub r while the latter factor s snr gain is significant only for low snr values

mixing for the time changes of heisenberg nilflows we consider reparametrizations of heisenberg nilflows we show that if a heisenberg nilflow is uniquely ergodic all non trivial time changes within a dense subspace of smooth time changes are mixing equivalently in the language of special flows we consider special flows over linear skew shifts over an irrational rotation of the circle without assuming any diophantine condition on the frequency we define a dense class of smooth roof functions for which the corresponding special flows are mixing mixing is produced by a mechanism known as stretching of birkhoff sums the set of mixing timechanges or equivalently roof functions has countable codimension and can be explicitely described in terms of invariant distributions for the nilflow allowing to produce concrete examples of mixing time changes

quantization methods for equal gain transmission with finite rate feedback we consider the design and analysis of quantizers for equal gain transmission egt systems with finite rate feedback based communication in flat fading multiple input single output miso systems egt is a beamforming technique that maximizes the miso channel capacity when there is an equal power per antenna constraint at the transmitter and requires the feedback of t 1 phase angles when there are t antennas at the transmitter in this paper we contrast two popular approaches for quantizing the phase angles vector quantization vq and scalar quantization sq on the vq side using the capacity loss with respect to egt with perfect channel information at transmitter as performance metric we develop a criterion for designing the beamforming codebook for quantized egt q egt we also propose an iterative algorithm based on the well known generalized lloyd algorithm for computing the beamforming vector codebook on the analytical side we study the performance of q egt and derive closed form expressions for the performance in terms of capacity loss and outage probability in the case of i i d rayleigh flat fading channels on the sq side assuming uniform scalar quantization and i i d rayleigh flat fading channels we derive the high resolution performance of quantized egt and contrast the performance with that of vq we find that although both vq and sq achieve the same rate of convergence to the capacity with perfect feedback as the number of feedback bits b increases there exists a fixed gap between the two

a martingale estimating equation for a capture recapture experiment in discrete time we use martingale theory and a method of moments technique to derive a class of estimators for the size of a closed population in a capture recapture experiment in discrete time where the capture probabilities on the successive capture occasions are allowed to be different the same capture probability is assumed to apply to all animals in the population on the same occasion explicit expressions are given for these estimators and their associated standard errors which involve only simple computation an optimal estimator for the population size is found asymptotic results are readily obtained by an application of a martingale central limit theorem two examples are given to compare the results

the foundations of finite sample estimation in stochastic processes generally in the literature on stochastic processes estimation is investigated in terms of asymptotic properties in this paper we establish some finite sample optimal estimation let y1 y2 be a discrete stochastic process since we will be discussing results for samples of finite size n we restrict the process to r in the following the term parameter is used in the same broad sense as by godambe thompson 1984 let y be a class of probability distributions f on r and 0 0 f f e jz be a real parameter let hi be a real function of y 1 yi and 0 be such that

a vector valued almost sure invariance principle for hyperbolic dynamical systems we prove an almost sure invariance principle approximation by d dimensional brownian motion for vector valued holder observables of large classes of nonuniformly hyperbolic dynamical systems these systems include axiom a dieomorphisms and flows as well as systems modelled by young towers with moderate tail decay rates in particular the position variable of the planar periodic lorentz gas with finite horizon approximates a 2 dimensional brownian motion

lattice point problems and values of quadratic forms for d dimensional ellipsoids e with d 5 we show that the number of lattice points in re is approximated by the volume of re as r tends to infinity up to an error of order mathcal o r d 2 for general ellipsoids and up to an error of order o rd 2 for irrational ones the estimate refines earlier bounds of the same order for dimensions d 9 as an application a conjecture of davenport and lewis about the shrinking of gaps between large consecutive values of q m m d of positive definite irrational quadratic forms q of dimension d 5 is proved finally we provide explicit bounds for errors in terms of certain minkowski minima of convex bodies related to these quadratic forms

a unified parametric regression model for recapture studies with random removals in continuous time nous combinons la vraisemblance conditionnelle dans des processus de comptage avec l estimateur de horvitz thompson pour obtenir un estimateur de la taille de la population plus efficient que les estimateurs existants des suppressions aleatoires peuvent etre introduites dans le processus de recapture on presente des etudes de simulation pour determiner la performance des estimateurs proposes on donne des exemples sur une repartition d oiseaux et une petite etude de recapture de mammiferes

limit theorems for self similar tilings we study deviation of ergodic averages for dynamical systems given by self similar tilings on the plane and in higher dimensions the main object of our paper is a special family of finitely additive measures for our systems an asymptotic formula is given for ergodic integrals in terms of these finitely additive measures and as a corollary limit theorems are obtained for dynamical systems given by self similar tilings

an exact renormalization formula for gaussian exponential sums and applications in the present paper we derive a renormalization formula a la hardy littlewood for the gaussian exponential sums with an exact formula for the remainder term we use this formula to describe the typical growth of the gaussian exponential sums r dans cet article nous obtenons une formule de renormalisa tion a la hardy littlewood pour des sommes exponentielles gaussi ennes avec une formule exacte pour les termes de reste nous utilisons cette formule pour decrire la croissance typique de ces sommes

fellow s corner foundations of statistical inference based on numerical roots of robust pivot functions abstract fisher s pivot functions pfs continue to dominate statistical inference and bootstrap literature despite efron and hinkley and royall s attempts to inject robustness vinod uses godambe s pivot functions gpfs based on godambe durbin estimating functions efs to develop numerically computed gpf roots such gpf roots can fill a long standing need in the bootstrap literature for robust pivots proposition 1 proves that gpfs are more robust than other pfs recently heyde rigorously explains why confidence intervals cis from gpfs are the shortest and davison and hinkley explain why the double bootstrap d boot yields second order correct cis we briefly discuss realistic econometric simulations supporting gpf roots in double bootstraps

analysis of multiple antenna systems with finite rate channel information feedback over spatially correlated fading channels this paper employs a high resolution quantization framework to study the effects of finite rate quantization of the channel state information csi on the performance of miso systems over correlated fading channels the contributions of this paper are twofold first as an application of the general distortion analysis tight lower bounds on the capacity loss of correlated miso systems due to the finite rate channel quantization are provided closed form expressions for the capacity loss in high signal to noise ratio snr and low snr regimes are also provided and their analysis reveals that the capacity loss of correlated miso channels is related to that of i i d fading channels by a simple multiplicative factor which is given by the ratio of the geometric mean to the arithmetic mean of the eigenvalues of the channel covariance matrix second this paper extends the general asymptotic distortion analysis to the important practical problem of suboptimal quantizers resulting from mismatches in the distortion functions source statistics and quantization criteria as a specific application two types of mismatched miso csi quantizers are investigated quantizers whose codebooks are designed with minimum mean square error mmse criterion but the distortion measure is the ergodic capacity loss i e mismatched design criterion and quantizers with codebook designed with a mismatched channel covariance matrix i e mismatched statistics bounds on the channel capacity loss of the mismatched codebooks are provided and compared to that of the optimal quantizers finally numerical and simulation results are presented and they confirm the tightness of theoretical distortion bounds

analysis of multiple antenna systems with finite rate feedback using high resolution quantization theory this paper considers the development of a general framework for the analysis of transmit beamforming methods in multiple antenna systems with finite rate feedback inspired by the results of classical high resolution quantization theory the problem of finite rate quantized communication system is formulated as a general fixed rate vector quantization problem with side information available at the encoder or the quantizer but unavailable at the decoder the framework of the quantization problem is sufficiently general to include quantization schemes with general non mean squared distortion functions and constrained source vectors asymptotic distortion analysis of the proposed general quantization problem is provided by extending the vector version of the bennett s integral specifically tight lower and upper bounds of the average asymptotic distortion are proposed sufficient conditions for the achievability of the distortion bounds are also provided and related to corresponding classical fixed rate quantization problems the proposed general methodology provides a powerful analytical tool to study a wide range of finite rate feedback systems to illustrate the utility of the framework we consider the analysis of a finite rate feedback multiple input single output miso beamforming system over independent and identically distributed i i d rayleigh flat fading channels numerical and simulation results are presented that further confirm the accuracy of the analytical results

markov regression models for time series a quasi likelihood approach this paper discusses a quasi likelihood ql approach to regression analysis with time series data we consider a class of markov models referred to by cox 1981 scandinavian journal of statistics 8 93 115 as observation driven models in which the conditional means and variances given the past are explicit functions of past outcomes the class includes autoregressive and markov chain models for continuous and categorical observations as well as models for counts e g poisson and continuous outcomes with constant coefficient of variation e g gamma we focus on poisson and gamma data for illustration analogous to ql for independent observations large sample properties of the regression coefficients depend only on correct specification of the first conditional moment

marginal analyses of longitudinal data with an informative pattern of observations we consider solutions to generalized estimating equations with singular working correlation matrices of which the estimator of diggle et al 2007 is a special case we give explicit conditions for consistent estimation when the pattern of observations may be informative in such cases simulations reveal reduced bias and reduced mean squared error compared with existing alternatives a study of peritoneal dialysis is used to illustrate the methodology

equal gain transmission in multiple input multiple output wireless systems multiple input multiple output mimo wireless systems are of interest due to their ability to provide substantial gains in capacity and quality the paper proposes equal gain transmission egt to provide diversity advantage in mimo systems experiencing rayleigh fading the applications of egt with selection diversity combining equal gain combining and maximum ratio combining are addressed it is proven that systems using egt with any of these combining schemes achieve full diversity order when transmitting over a memoryless flat fading rayleigh matrix channel with independent entries since in practice full channel knowledge at the transmitter is difficult to realize a quantized version of egt is proposed an algorithm to construct a beamforming vector codebook that guarantees full diversity order is presented monte carlo simulation comparisons with various beamforming and combining systems illustrate the performance as a function of quantization

optimal nonparametric estimation for some semimartingale stochastic differential equations this paper discusses the extension of a result of godamble on parametric estimation for discrete time stochastic processes to nonparametric estimation for the continuous time case following hasminskii and ibragimov 1980 the nonparametric problem is formulated as a parametric one but with infinite dimensional parameter a general optimality criterion for estimating functions based on that of godambe ann math statist 31 1208 1211 1960 is formulated in the case where the parameter is an element of a banach space and the optimality of a generalized score function is proved under further conditions the sense in which this theory is applicable to martingale estimating functions for a in the semimartingale stochastic differential equation model dx t a t dr t dm t a is discussed it is shown that the nelson aalen estimate for the cumulative hazard function can be regarded as optimal in godambe s sense applications to diffusion models and an extended gamma process model are given also

vinogradov s mean value theorem via efficient congruencing we obtain estimates for vinogradov s integral that for the rst time approach those conjectured to be the best possible several applications of these new bounds are provided in particular the conjectured asymptotic formula in waring s problem holds for sums of s kth powers of natural numbers whenever s 2k 2 2k 3

estimation of the size of an open population from capture recapture data using weighted martingale methods summary a weighted martingale method akin to a moving average is proposed to allow the use of modified closed population methods in the estimation of the size of a smoothly changing open population when there are frequent capture occasions we concentrate here on modifications to martingale estimating functions for model mt but a wide range of closed population estimators may be modified in this fashion the method is motivated by and applied to weekly capture recapture data from the mai po bird sanctuary in hong kong simulations show that the weighted martingale estimator compared well with the jolly seber estimator when the conditions for the latter to be valid are met and it performed far better when individuals were allowed to leave and reenter the population expressions are derived for the asymptotic bias and variance of the estimator in an appendix

robust estimation through estimating equations summary utilizing the theory of estimating equations godambe 1960 godambe thompson 1978 this paper develops concepts of parameter defining function and effective parameter the paper provides theory and techniques for choosing from a given set of robust parameters one that is most effective or one that can most efficiently be estimated the word parameter is used in statistics in two rather distinct contexts in the data analytic context a parameter characterizes some interesting aspect of the distribution such as its mean its median or some percentile in the context of modern statistical theory founded by k pearson r a fisher and others the parameter is related to and derives its meaning from a specific probabilistic model in this second context the model is of central importance and one seeks to complete it or perfect it by estimating its unknown parameter in the first context there is considerable flexibility and a parameter can typically be defined in many ways the choice of a parameter definition involves the following considerations a how much does it tell about the interesting or practically important part of the distribution b how insensitive or robust it is with respect to the uninteresting part of the distribution c how best can it be estimated the two contexts described above may of course overlap in the present paper we develop a theory which emphasizes the data analytic meaning of the parameter the approach on the other hand is more in line with the modern statistical theory mentioned above than that of most other works on robust estimation in particular asymptotic minimaxity plays no role the theory is designed to incorporate questions a b and c raised above it is based on the theory of estimating equations godambe 1960 godambe thompson 1978 the new concepts employed are those of parameter defining function and effectiveness of a parameter these are defined in 2 1 within

on the transformation formulas of theta series here s is a nondegenerate rational symmetric matrix of signature q s s 3 x iy is a variable in the siegel half space hn of degree n u is a variable in the space cn of all complex n x q matrices runs over the set qq of all rational q x n matrices a is a c valued function on qn that is a finite linear combination of characteristic functions of cosets of qq modulo a z lattice a is a real matrix such that s ta diag lq 1 a and e c exp 2iric for c e c if q 1 s a 1 and a is suitably chosen the series g becomes a classical theta function of jacobi riemann if s is positive definite n 1 u 0 and q is even g is an elliptic modular form of weight q 2 studied by hecke also siegel introduced the series with u 0 in his theory of quadratic forms in si now g satisfies a transformation formula of the following type for every

the uniform central limit theorem for theta sums sommes gaussiennes sommes de kloosterman theoreme de limite centrale uniforme pour des series theta theoreme de limite centrale uniformes pour des sommes theta moments sur des intervalles arbitraires

on raghunathan s measure conjecture this paper represents the last in our three part series on raghunathan s measure conjecture see r1 r2 for parts i and ii more specifically let g be a real lie group all groups in this paper are assumed to be second countable with the lie algebra r a discrete subgroup of g and ir g r g the projection lwg rg the group g acts by right translations on r g x x g x e r g g e g let gt be a borel probability measure on r g define

feedback rate capacity loss tradeoff for limited feedback mimo systems multiple input multiple output mimo communication systems can provide large capacity gains over traditional single input single output siso systems and are expected to be a core technology of next generation wireless systems often these capacity gains are achievable only with some form of adaptive transmission in this paper we study the capacity loss defined as the rate loss in bits s hz of the mimo wireless system when the covariance matrix of the transmitted signal vector is designed using a low rate feedback channel for the mimo channel we find a bound on the ergodic capacity loss when random codebooks generated from the uniform distribution on the complex unit sphere are used to convey the second order statistics of the transmitted signal from the receiver to the transmitter in this case we find a closed form expression for the ergodic capacity loss as a function of the number of bits fed back at each channel realization these results show that the capacity loss decreases at least as o 2 sup b 2mmt 2 where b is the number of feedback bits m sub t is the number of transmit antennas and m min m sub r m sub t where m sub r is the number of receive antennas in the high snr regime we present a new bound on the capacity loss that is tighter than the previously derived bound and show that the capacity loss decreases exponentially as a function of the number of feedback bits

disorder renormalizability theta functions and cornu spirals abstract the partial sums of the trigonometrical series s e i n p for small mod 2 lead to distributions of points in the complex plane c composed of cornu like spirals for p 2 and p 1 the number n l of points in the l th spiral is o l e 1 where e 2 p p 1 if p 2 then n l 1 2 l 1 2 1 2 l 1 2 x denotes the greatest integer in x thus n l increases decreases with n if p 2 1 p p 1 p 2 and p not an integer if is rational for p 2 the number of points per spiral decreases to 1 and the points are distributed in a seemingly random fashion while for 1 p l and the orientation of the spirals changes in seemingly random fashion with l there is order if p is an integer if is rational the pattern is a pseudo periodic arrangement of spirals which depending on also may be composed of spirals if is a quadratic irrational and p 2 i e can be represented by a periodic continued fraction then the pattern is renormalizable in the second case the numbers s l of points between the mid points points of inflection of successive spirals form a beatty sequence 16 the proof of renormalizability depends upon hardy and littlewood s approximate functional equation for the theta function 6 similar behavior is exhibited by n dimensional generalizations of the sum s above

the approximate functional formula for the theta function and diophantine gauss sums we cc nsider the polygonal lines in the complex plane c whose nth vertex is defined by sy en o exp iw7rn2 with w c r where the prime means that the first and last terms in the sum are halved by introducing the discrete curvature of the polygonal line and by exploiting the similarity of segments of the line for small w to cornu spirals c spirals we prove the precise renormalization formula 1 e exp iav7rk2 x e exp i k2 w where n n w g the nearest integer to n w and 1 c 3 14 this formula which sharpens hardy and littlewood s approximate functional formula for the theta function generalizes to irrationals as ab diophantine inequality the well known sum formula of gauss the geometrica l meaning of the relation between the two limits is that the first sum is taken to a point of inflection of the corresponding c spirals the second sum replaces whole c spirals of the first by unit vectors times scale and phase factors the block renormalization procedure implied by this replacement is governed by the circle map 2 w 1 mod 2 w c 1 1 0 w whose orbits are analyzed by expressing w as an even continued fraction

correlated binary regression with covariates specific to each binary observation regression methods are considered for the analysis of correlated binary data when each binary observation may have its own covariates it is argued that binary response models that condition on some or all binary responses in a given block are useful for studying certain types of dependencies but not for the estimation of marginal response probabilities or pairwise correlations fully parametric approaches to these latter problems appear to be unduly complicated except in such special cases as the analysis of paired binary data hence a generalized estimating equation approach is advocated for inference on response probabilities and correlations illustrations involving both small and large block sizes are provided

capacity analysis of mimo systems using limited feedback transmit precoding schemes this paper employs a high resolution quantization framework to study the effects of finite rate feedback of the channel state information csi on the performance of multiple input multiple output mimo systems over independently and identically distributed i i d rayleigh flat fading channels the contributions of this paper are twofold first we extend the general distortion analysis of vector quantizers to deal with complex source variables necessary and sufficient conditions that guarantee a concise high resolution distortion analysis in the complex domain is presented second as an application of the proposed complex distortion analysis tight lower bounds on the capacity loss due to the finite rate channel quantization are provided for mimo systems employing a fixed number of equal power spatial beams based on the obtained closed form analytical results it is shown that the system capacity loss decreases exponentially as the ratio of the quantization rate to the total degrees of freedom of the channel state information to be quantized moreover mimo csi quantizers using mismatched codebooks that are only optimized for high signal to noise ratio snr and low snr regimes are also investigated to quantify the penalties incurred by the use of mismatched codebooks in addition the analysis is extended to deal with mimo systems using multi mode spatial multiplexing transmission schemes with finite rate csi feedback finally numerical and simulation results are presented which confirm the tightness of the derived theoretical distortion bounds

voronoi regions of lattices second moments of polytopes and quantization if a point is picked at random inside a regular simplex octahedron 600 cell or other polytope what is its average squared distance from the centroid in n dimensional space what is the average squared distance of a random point from the closest point of the lattice a n or d n e n a n ast or d n ast the answers are given here together with a description of the voronoi or nearest neighbor regions of these lattices the results have applications to quantization and to the design of signals for the gaussian channel for example a quantizer based on the eight dimensional lattice e8 has a mean squared error per symbol of 0 0717 cdots when applied to uniformly distributed data compared with 0 08333 cdots for the best one dimensional quantizer

design and analysis of transmit beamforming based on limited rate feedback this paper deals with design and performance analysis of transmit beamformers for multiple input multiple output mimo systems based on bandwidth limited information that is fed back from the receiver to the transmitter by casting the design of transmit beamforming based on limited rate feedback as an equivalent sphere vector quantization svq problem multiantenna beamformed transmissions through independent and identically distributed i i d rayleigh fading channels are first considered the rate distortion function of the vector source is upper bounded and the operational rate distortion performance achieved by the generalized lloyd s algorithm is lower bounded although different in nature the two bounds yield asymptotically equivalent performance analysis results the average signal to noise ratio snr performance is also quantified finally beamformer codebook designs are studied for correlated rayleigh fading channels and a low complexity codebook design that achieves near optimal performance is derived

almost sure invariance principle for dynamical systems by spectral methods we prove the almost sure invariance principle for stationary d valued random processes with very precise dimension independent error terms solely under a strong assumption concerning the characteristic functions of these processes this assumption is easy to check for large classes of dynamical systems or markov chains using strong or weak spectral perturbation arguments

the distribution of free path lengths in the periodic lorentz gas and related lattice point problems the periodic lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers and is one of the fundamental models for chaotic diffusion in the present paper we investigate the boltzmann grad limit where the radius of each scatterer tends to zero and prove the existence of a limiting distribution for the free path length we also discuss related problems such as the statistical distribution of directions of lattice points that are visible from a fixed position

capture recapture when time and behavioral response affect capture probabilities summary we consider a capture recapture model in which capture probabilities vary with time and with behavioral response two inference procedures are developed under the assumption that recapture probabilities bear a constant relationship to initial capture probabilities these two procedures are the maximum likelihood method both unconditional and conditional types are discussed and an approach based on optimal estimating functions the population size estimators derived from the two procedures are shown to be asymptotically equivalent when population size is large enough the performance and relative merits of various population size estimators for finite cases are discussed the bootstrap method is suggested for constructing a variance estimator and confidence interval an example of the deer mouse analyzed in otis et al 1978 wildlife monographs62 93 is given for illustration

limited feedback unitary precoding for orthogonal space time block codes orthogonal space time block codes ostbcs are a class of easily decoded space time codes that achieve full diversity order in rayleigh fading channels ostbcs exist only for certain numbers of transmit antennas and do not provide array gain like diversity techniques that exploit transmit channel information when channel state information is available at the transmitter though precoding the space time codeword can be used to support different numbers of transmit antennas and to improve array gain unfortunately transmitters in many wireless systems have no knowledge about current channel conditions this motivates limited feedback precoding methods such as channel quantization or antenna subset selection this paper investigates a limited feedback approach that uses a codebook of precoding matrices known a priori to both the transmitter and receiver the receiver chooses a matrix from the codebook based on current channel conditions and conveys the optimal codebook matrix to the transmitter over an error free zero delay feedback channel a criterion for choosing the optimal precoding matrix in the codebook is proposed that relates directly to minimizing the probability of symbol error of the precoded system low average distortion codebooks are derived based on the optimal codeword selection criterion the resulting design is found to relate to the famous applied mathematics problem of subspace packing in the grassmann manifold codebooks designed by this method are proven to provide full diversity order in rayleigh fading channels monte carlo simulations show that limited feedback precoding performs better than antenna subset selection

transmit beamforming in multiple antenna systems with finite rate feedback a vq based approach this paper investigates quantization methods for feeding back the channel information through a low rate feedback channel in the context of multiple input single output miso systems we propose a new quantizer design criterion for capacity maximization and develop the corresponding iterative vector quantization vq design algorithm the criterion is based on maximizing the mean squared weighted inner product mswip between the optimum and the quantized beamforming vector the performance of systems with quantized beamforming is analyzed for the independent fading case this requires finding the density of the squared inner product between the optimum and the quantized beamforming vector which is obtained by considering a simple approximation of the quantization cell the approximate density function is used to lower bound the capacity loss due to quantization the outage probability and the bit error probability the resulting expressions provide insight into the dependence of the performance of transmit beamforming miso systems on the number of transmit antennas and feedback rate computer simulations support the analytical results and indicate that the lower bounds are quite tight

incomplete higher order gauss sums abstract we consider the classical incomplete higher order gauss sums s m b j 0 b exp 2 ij m n 0 b where n is large in 1976 lehmer analyzed the beautiful spirals appearing in the directed graph s m 0 s m 1 s m 2 in the complex plane for m 2 in the process he obtained sharp uniform estimates for the sums s2 b sullivan and zannier expanded on lehmer s work in 1992 by giving an asymptotic formula for s2 b valid for b 1 the directed graphs still exhibit spirals but with more complicated behavior than in the quadratic case we analyze this behavior and in the process obtain sharp asymptotic estimates for certain incomplete gauss sums sm b

projected partial likelihood and its application to longitudinal data summary an estimating equation which we call the projected partial score is introduced for longitudinal data analysis the estimating equation is obtained by projecting the partial likelihood score function onto the vector space spanned by a class of conditionally linear estimating equations we demonstrate that removing certain terms from the projection of the full likelihood score does not alter important inferential properties of the estimating equation and doing so is advantageous in handling missing data and time varying covariates within a prequential frame of reference it is shown that the estimating equation is optimal among the largest collection of estimating equations determined by the conditional moments furthermore the method possesses similar properties to generalised estimating equations in particular the correct conditional variance specification is necessary for efficiency but not for asymptotic consistency and distribution theory

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12

grassmannian beamforming for multiple input multiple output wireless systems transmit beamforming and receive combining are simple methods for exploiting the significant diversity that is available in multiple input multiple output mimo wireless systems unfortunately optimal performance requires either complete channel knowledge or knowledge of the optimal beamforming vector both are hard to realize in this article a quantized maximum signal to noise ratio snr beamforming technique is proposed where the receiver only sends the label of the best beamforming vector in a predetermined codebook to the transmitter by using the distribution of the optimal beamforming vector in independent and identically distributed rayleigh fading matrix channels the codebook design problem is solved and related to the problem of grassmannian line packing the proposed design criterion is flexible enough to allow for side constraints on the codebook vectors bounds on the codebook size are derived to guarantee full diversity order results on the density of grassmannian line packings are derived and used to develop bounds on the codebook size given a capacity or snr loss monte carlo simulations are presented that compare the probability of error for different quantization strategies

design and analysis of mimo spatial multiplexing systems with quantized feedback this paper investigates the problem of transmit beamforming in multiple antenna spatial multiplexing sm systems employing a finite rate feedback channel assuming a fixed number of spatial channels and equal power allocation we propose a new criterion for designing the codebook of beamforming matrices that is based on minimizing an approximation to the capacity loss resulting from the limited rate in the feedback channel using the criterion we develop an iterative design algorithm that converges to a locally optimum codebook under the independent identically distributed channel and high signal to noise ratio snr assumption the effect on channel capacity of the finite bit representation of the beamforming matrix is analyzed central to this analysis is the complex multivariate beta distribution and tractable approximations to the voronoi regions associated with the code points furthermore to compensate for the degradation due to the equal power allocation assumption we propose a multimode sm transmission strategy wherein the number of data streams is determined based on the average snr this approach is shown to allow for effective utilization of the feedback bits resulting in a practical and efficient multiple input multiple output system design

simple capture recapture models permitting unequal catchability and variable sampling effort we consider two capture recapture models that imply that the logit of the probability of capture is an additive function of an animal catchability parameter and a parameter reflecting the sampling effort the models are special cases of the rasch model and satisfy the property of quasi symmetry one model is log linear and the other is a latent class model for the log linear model point and interval estimates of the population size are easily obtained using standard software such as glim

efficiency of martingale methods in recapture studies summary the asymptotic variances of various estimators of the population size from a recapture study are calculated the estimation methods compared are maximum likelihood and a class of natural martingale estimators with both time variation and behavioural response it is found that i the martingale methods appropriate for time variation or behavioural response are fully efficient ii there is no loss of efficiency at all in allowing for time variation iii there is large loss of efficiency in allowing for behavioural response iv there is a natural martingale method that allows for both time variation and behavioural response

on the uniform equidistribution of long closed horocycles it is well known that on any given hyperbolic surface of finite area a closed horocycle of length becomes asymptotically equidistributed as in this paper we prove that any subsegment of length greater than 1 2 e of such a closed horocycle also becomes equidistributed as the exponent 1 2 e is the best possible and improves upon a recent result by hejhal he3 we give two proofs of the above result our second proof leads to explicit information on the rate of convergence we also prove a result on the asymptotic joint equidistribution of a finite number of distinct subsegments having equal length proportional to

statistical inference procedure for a hypergeometric model for capture recapture experiments abstract we use martingale theory and a method of moments technique to derive a class of estimators for a multihypergeometric model for a capture recapture experiment certain choices of weight function lead to some standard estimators explicit expression for the estimators and their associated standard errors of the parameter population size is obtained optimal estimator among the class of estimators is derived which gives the tightest confidence interval for the population size three examples are given a simulation study is given to assess the performance of the proposed estimators with other existing ones

an asymptotic approximation for incomplete gauss sums ii the classical incomplete gauss email protected j 0m 1exp email protected p n 1 1 is studied for large n an expansion is derived for this sum when p is an integer that holds uniformly for 1 1 numerical results are given to demonstrate the accuracy of the various approximations

spectral form factors of rectangle billiards the berry tabor conjecture asserts that local statistical measures of the eigenvalues j of a generic integrable quantum system coincide with those of a poisson process we prove that in the case of a rectangle billiard with random ratio of sides the sum behaves for random and n large like a random walk in the complex plane with a non gaussian limit distribution the expectation value of the distribution is zero its variance which is essentially the average pair correlation function is one in accordance with the berry tabor conjecture but all higher moments 4 diverge the proof of the existence of the limit distribution uses the mixing property of a dynamical system defined on a product of hyperbolic surfaces the berry tabor conjecture and the existence of the limit distribution for a fixed generic rectangle are related to an equidistribution conjecture for long horocycles on this product space

limit theorems for translation flows the aim of this paper is to obtain an asymptotic expansion for ergodic integrals of translation flows on flat surfaces of higher genus theorem 1 and to give a limit theorem for these flows theorem 2

gaps in n mod 1 and ergodic theory cut the unit circle s r z at the points 1 2 n where x x mod 1 and let j1 jn denote the complementary intervals or gaps that remain we show that in contrast to the case of random points whose gaps are exponentially distributed the lengths ji n are governed by an explicit piecewise real analytic distribution f t dt with phase transitions at t 1 2 and t 2 the gap distribution is related to the probability p t that a random unimodular lattice translate r meets a fixed triangle st of area t in fact p t f t the proof uses ergodic theory on the universal elliptic curve e sl2 r r 2 sl2 z z and ratner s theorem on unipotent invariant measures

estimating the number of species a review how many kinds are there suppose that a population is partitioned into c classes in many situations interest focuses not on estimation of the relative sizes of the classes but on estimation of c itself for example biologists and ecologists may be interested in estimating the number of species in a population of plants or animals numismatists may be concemed with estimating the number of dies used to produce an ancient coin issue and linguists may be interested in estimating the size of an author s vocabulary in this article we review the problem of statistical estimation of c many approaches have been proposed some purely data analytic and others based in sampling theory in the latter case numerous variations have been considered the population may be finite or infinite if finite samples may be taken with replacement multinomial sampling or without replacement hypergeometric sampling or by bernoulli sampling if infinite sampling may be multinomial or bernoulli or the sample may be th

unified maximum likelihood estimates for closed capture recapture models using mixtures summary agresti 1994 biometrics50 494 500 and norris and pollock 1996a biometrics52 639 649 suggested using methods of finite mixtures to partition the animals in a closed capture recapture experiment into two or more groups with relatively homogeneous capture probabilities this enabled them to fit the models mh mbh norris and pollock and mth agresti of otis et al 1978 wildlife monographs62 1 135 in this article finite mixture partitions of animals and or samples are used to give a unified linear logistic framework for fitting all eight models of otis et al by maximum likelihood likelihood ratio tests are available for model comparisons for many data sets a simple dichotomy of animals is enough to substantially correct for heterogeneity induced bias in the estimation of population size although there is the option of fitting more than two groups if the data warrant it

an inference procedure for a simple epidemic model a martingale approach this paper studies various estimators the infection rate of the simple stochastic epidemic model using the theory of counting processes martingales and stochastic integrals a class of estimators for the infection rate have been obtained which includes the maximum likelihood estimator explicit expressions are given for these estimators which involve only simple computation in the case of incomplete observation of the epidemic estimators are available by choosing an appropriate weight function asymptotic results are given asymptotic efficiencies among these estimators are considered

hazards regression analysis for length biased data summary length biased sampling is frequently a convenient technique for the collection of positive valued or lifetime data estimation procedures based on pseudo likelihood are developed under the proportional hazards model for length biased data bias adjusted risk set sampling is used for the construction of the pseudo likelihood and the risk set sampling is replicated to improve estimation performance the average of the resulting likelihood estimators is taken as the estimator for the regression coefficients although the replication procedure is expected to improve estimation accuracy when the sample size is small or moderate it does not increase the asymptotic efficiency for estimating the regression coefficient an estimator for the baseline survival function is also presented it reduces to the maximum likelihood estimator when the regression parameter in the proportional hazards model is zero statistical properties of the proposed estimation procedures are developed examples are presented to illustrate the methods

an inference procedure for a capture and recapture experiment with time dependent capture probabilities summary estimates for the size of a closed population are given for multiple recapture studies in discrete time the estimates are derived by a method of moments for martingales an estimate and associated standard error of the population size are derived for a homogeneous population when the capture probabilities are permitted to be time dependent in the discrete case corresponding results are obtained for the continuous case as well explicit expressions are given for these estimates and their standard errors only simple computation is involved

logarithm laws for flows on homogeneous spaces in this paper we generalize and sharpen d sullivan s logarithm law for geodesics by specifying conditions on a sequence of subsets at t n of a homogeneous space g g a semisimple lie group an irreducible lattice and a sequence of elements ft of g under which t n ftx at is infinite for a e x g the main tool is exponential decay of correlation coefficients of smooth functions ong besides the general higher rank version of sullivan s result as a consequence we obtain a new proof of the classical khinchin groshev theorem on simultaneous diophantine approximation and settle a conjecture recently made by m skriganov

on beamforming with finite rate feedback in multiple antenna systems we study a multiple antenna system where the transmitter is equipped with quantized information about instantaneous channel realizations assuming that the transmitter uses the quantized information for beamforming we derive a universal lower bound on the outage probability for any finite set of beamformers the universal lower bound provides a concise characterization of the gain with each additional bit of feedback information regarding the channel using the bound it is shown that finite information systems approach the perfect information case as t 1 2 sup b t 1 where b is the number of feedback bits and t is the number of transmit antennas the geometrical bounding technique used in the proof of the lower bound also leads to a design criterion for good beamformers whose outage performance approaches the lower bound the design criterion minimizes the maximum inner product between any two beamforming vectors in the beamformer codebook and is equivalent to the problem of designing unitary space time codes under certain conditions finally we show that good beamformers are good packings of two dimensional subspaces in a 2t dimensional real grassmannian manifold with chordal distance as the metric