hadamard s finite part concept in dimension n 2 distributional definition regularization forms and distributional derivatives the aim of this paper is to apply the hadamard finite part concept to singular pseudo functions such as a theta r beta log j r where r 2 x 1 2 ldots x n 2 in r n j in n beta leq n and a is a function depending on theta om r for these specific pseudo functions both regularizations forms and distributional derivatives frac partial p partial x 1 p 1 ldots partial x n p n a theta r beta log j r where p p 1 ldots p n are positive integers with p p 1 ldots p n are investigated whenever possible the results are compared with those obtained by estrada kanwal 1985 1989

sensitivity analysis for antenna near field imaging although imaging and inverse scattering problems have been thoroughly studied during the last century there is only a partial understanding of these complex problems most of the efforts have been placed on the development of efficient inversion algorithms and mathematical uniqueness results in comparison there are very few results and a limited knowledge about the information content in the inversion data in this paper we provide a mathematical framework for sensitivity analysis of antenna near field imaging problems based on the multipole expansion of the electromagnetic field and the fisher information to quantify the quality of data by exploiting this framework a fundamental relation for accuracy and resolution is formulated based on the crameacuter rao bound crb the sensitivity analysis is illustrated using a relevant example with cylindrical measurement data

the beta model maximum likelihood cramer rao bounds and hypothesis testing we study the maximum likelihood estimator in a setting where the dependent variable is a random graph and covariates are available on a graph level the model generalizes the well known beta model for random graphs by replacing the constant model parameters with regression functions cramer rao bounds are derived for special cases of the undirected beta model the directed beta model and the covariate based beta model the corresponding maximum likelihood estimators are compared with the bounds by means of simulations moreover examples are given on how to use the presented maximum likelihood estimators to test for directionality and significance finally the applicability of the model is demonstrated using temporal social network data describing communication among healthcare workers

joint estimation of porosity and saturation using stochastic rock physics modeling sediment porosity and saturation affect bulk modulus shearmodulus anddensity consequently estimatinghydrocarbon saturation and reservoir porosity from seismic data is a joint estimation problem uncertainty in porosity will lead toerrorsinsaturationprediction andviceversa porosityand saturation can be jointly estimated using stochastic rockphysicsmodelingandformalbayesianestimationmethodology knowledge of shear impedance reduces the uncertainty inporosityandthusalsoreducesuncertaintyinsaturationestimation this study investigates joint estimation of porosity and saturation by using rock physics stochastic modeling and bayesian estimation theory to derive saturation and porosity maps of expected pay sands in the field example the uncertainty in porosity quantified by the standard deviation stdassociatedwiththeposteriorprobabilitydensityfunctionpdf derivedfrominversionofseismicdataismuchless than the uncertainty in the derived saturation for a typical case the std associated with saturation is 24 while porositystdisabout1 34porosityunitsgivenseismic derived inversionattributeswithreasonableaccuracy comparisonof these numbers with prior estimates showed that inversion of seismic data decreased the uncertainty in porosity to 15 of the prior uncertainty while saturation uncertainty was only reduced to 92 of the prior uncertainty although these resultsmayvaryfromonelocationtoanother themethodology isgeneralandcanbeappliedtootherlocations

mse bounds with affine bias dominating the cramer rao bound in continuation to an earlier work we further develop bounds on the mean squared error mse when estimating a deterministic parameter vector thetas0 in a given estimation problem as well as estimators that achieve the optimal performance the traditional cramer rao cr type bounds provide benchmarks on the variance of any estimator of thetas0 under suitable regularity conditions while requiring a priori specification of a desired bias gradient to circumvent the need to choose the bias which is impractical in many applications it was suggested in our earlier work to directly treat the mse which is the sum of the variance and the squared norm of the bias while previously we developed mse bounds assuming a linear bias vector here we study in the same spirit affine bias vectors we demonstrate through several examples that allowing for an affine transformation can often improve the performance significantly over a linear approach using convex optimization tools we show that in many cases we can choose an affine bias that results in an mse bound that is smaller than the unbiased cr bound for all values of thetas0 furthermore we explicitly construct estimators that achieve these bounds in cases where an efficient estimator exists by performing an affine transformation of the standard maximum likelihood ml estimator this leads to estimators that result in a smaller mse than ml for all possible values of thetas0

exploring estimator bias variance tradeoffs using the uniform cr bound we introduce a plane which we call the delta sigma plane that is indexed by the norm of the estimator bias gradient and the variance of the estimator the norm of the bias gradient is related to the maximum variation in the estimator bias function over a neighborhood of parameter space using a uniform cramer rao cr bound on estimator variance a delta sigma tradeoff curve is specified that defines an unachievable region of the delta sigma plane for a specified statistical model in order to place an estimator on this plane for comparison with the delta sigma tradeoff curve the estimator variance bias gradient and bias gradient norm must be evaluated we present a simple and accurate method for experimentally determining the bias gradient norm based on applying a bootstrap estimator to a sample mean constructed from the gradient of the log likelihood we demonstrate the methods developed in this paper for linear gaussian and nonlinear poisson inverse problems

limits of sensing temporal concentration changes by single cells berg and purcell biophys j 20 193 1977 calculated how the accuracy of concentration sensing by single celled organisms is limited by noise from the small number of counted molecules here we generalize their results to the sensing of concentration ramps which is often the biologically relevant situation e g during bacterial chemotaxis we calculate lower bounds on the uncertainty of ramp sensing by three measurement devices a single receptor an absorbing sphere and a monitoring sphere we contrast two strategies simple linear regression of the input signal versus maximum likelihood estimation and show that the latter can be twice as accurate as the former finally we consider biological implementations of these two strategies and identify possible signatures that maximum likelihood estimation is implemented by real biological systems

mean and variance of implicitly defined biased estimators such as penalized maximum likelihood applications to tomography many estimators in signal processing problems are defined implicitly as the maximum of some objective function examples of implicitly defined estimators include maximum likelihood penalized likelihood maximum a posteriori and nonlinear least squares estimation for such estimators exact analytical expressions for the mean and variance are usually unavailable therefore investigators usually resort to numerical simulations to examine the properties of the mean and variance of such estimators this paper describes approximate expressions for the mean and variance of implicitly defined estimators of unconstrained continuous parameters we derive the approximations using the implicit function theorem the taylor expansion and the chain rule the expressions are defined solely in terms of the partial derivatives of whatever objective function one uses for estimation as illustrations we demonstrate that the approximations work well in two tomographic imaging applications with poisson statistics we also describe a plug in approximation that provides a remarkably accurate estimate of variability even from a single noisy poisson sinogram measurement the approximations should be useful in a wide range of estimation problems

ridge regression biased estimation for nonorthogonal problems in multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory if not incorrect if the prediction vectors are not orthogonal proposed is an estimation procedure based on adding small positive quantities to the diagonal of x x introduced is the ridge trace a method for showing in two dimensions the effects of nonorthogonality it is then shown how to augment x x to obtain biased estimates with smaller mean square error

robust linear channel estimation methods for per subcarrier transmit antenna selection the linear minimum mean squared error mse channel estimator for systems employing per subcarrier transmit antenna selection is developed frequency domain correlations after the selection process are shown to be approximated well using a simple function which makes near optimal channel estimation practically possible however the resultant estimators are not robust to errors in the assumed model in terms of the antenna to subcarrier assignment used at transmission as we motivate both analytically and via simulations this is an issue when the channels for conveying this information to the receiver are severely constrained we present windowed channel estimations and new robust estimation algorithms either individually or in combination as solutions to counter this sensitivity the robust estimators are developed based on the principles of 1 minimising the worst case mse over the set of possible models and 2 minimising the expected mse over the set of possible models the latter estimator is preferred due to the lower implementation complexity and better mse performance we also prove theoretical asymptotic in signal to noise ratio performances of the two estimators used with the proposed correlation model simulation results illustrate the performance gains and improved robustness offered by the developed estimators

active learning applied to rcs computations with nonuniform sampling using different objective functions an active learning framework is introduced to reduce the number of frequencies and angles one must consider for wideband monostatic scattering computations or measurements this method is used to optimally select those frequencies and angles that would be most informative resulting in nonuniform sampling and often a reduced number of points vis agrave vis uniform sampling in this paper we focus on jointly two dimensional optimal sampling in frequency and incident angle thetas for monostatic scattering the method consists of two basic steps one step involves estimation of model parameters using a least square ls algorithm the next step is to optimally choose the next point frequency and thetas for analysis by the computational model or experiment this new point is selected with the goal of reducing uncertainty in the parametric model quantified via the fisher information matrix iterating these two steps a sequence of numerical computations or measurements are performed each at the most informative point for learning the parameters of the associated simpler parametric model this idea is demonstrated here in the context of reducing the number of points frequencies and orientations at which a computational model must be employed and in order to avoid repeatedly gathering samplings at the edge of the input space an alternative objective function is applied which makes the actively selected points closer to a region of interest

penalized maximum likelihood image reconstruction using space alternating generalized em algorithms most expectation maximization em type algorithms for penalized maximum likelihood image reconstruction converge slowly particularly when one incorporates additive background effects such as scatter random coincidences dark current or cosmic radiation in addition regularizing smoothness penalties or priors introduce parameter coupling rendering intractable the m steps of most em type algorithms this paper presents space alternating generalized em sage algorithms for image reconstruction which update the parameters sequentially using a sequence of small hidden data spaces rather than simultaneously using one large complete data space the sequential update decouples the m step so the maximization can typically be performed analytically we introduce new hidden data spaces that are less informative than the conventional complete data space for poisson data and that yield significant improvements in convergence rate this acceleration is due to statistical considerations not numerical overrelaxation methods so monotonic increases in the objective function are guaranteed we provide a general global convergence proof for sage methods with nonnegativity constraints

a chaos based symmetric image encryption scheme using a bit level permutation in recent years a variety of chaos based digital image encryption algorithms have been suggested most of these algorithms implement permutations and diffusions at the pixel level by considering the pixel as the smallest atomic element of an image in fact a permutation at the bit level not only changes the position of the pixel but also alters its value here we propose an image cryptosystem employing the arnold cat map for bit level permutation and the logistic map for diffusion simulations have been carried out and analyzed in detail demonstrating the superior security and high efficiency of our cryptosystem

calculation of the fisher information matrix for multidimensional data sets data sets that are acquired in many practical systems can be described as the output of a multidimensional linear separable denominator system with gaussian measurement noise an important example is nuclear magnetic resonance nmr spectroscopy in nmr spectroscopy high accuracy parameter estimation is of central importance a classical result on the crame spl acute r rao lower bound states that the inverse of the fisher information matrix fim provides a lower bound for the covariance of any unbiased estimator of the parameter vector the calculation of the fim is therefore of central importance for an assessment of the accuracy with which parameters can be estimated it is shown how the fim can be expressed using the matrices that determine the system that generates the data set for uniformly sampled data it is shown how the fim can be expressed through the solutions of lyapunov equations the novel techniques are demonstrated with an example arising from nmr spectroscopy

retinal vessel centerline extraction using multiscale matched filters confidence and edge measures motivated by the goals of improving detection of low contrast and narrow vessels and eliminating false detections at nonvascular structures a new technique is presented for extracting vessels in retinal images the core of the technique is a new likelihood ratio test that combines matched filter responses confidence measures and vessel boundary measures matched filter responses are derived in scale space to extract vessels of widely varying widths a vessel confidence measure is defined as a projection of a vector formed from a normalized pixel neighborhood onto a normalized ideal vessel profile vessel boundary measures and associated confidences are computed at potential vessel boundaries combined these responses form a six dimensional measurement vector at each pixel a training technique is used to develop a mapping of this vector to a likelihood ratio that measures the vesselness at each pixel results comparing this vesselness measure to matched filters alone and to measures based on the hessian of intensities show substantial improvements both qualitatively and quantitatively the hessian can be used in place of the matched filter to obtain similar but less substantial improvements or to steer the matched filter by preselecting kernel orientations finally the new vesselness likelihood ratio is embedded into a vessel tracing framework resulting in an efficient and effective vessel centerline extraction algorithm

quasi ml period estimation from incomplete timing data given a noisy sequence of possibly shifted integer multiples of a certain period it is often of interest to accurately estimate the period with known integer regressors the problem is classical linear regression in many applications however the regressors are unknown integers and only loose bounds on the period are available examples include hop period and timing estimation wherein hops may be missed at the output of the frequency discriminator or the emitter may hop out of band pulse repetition interval pri analysis and passive rotating beam radio scanning we study several pertinent period estimators our emphasis is on a quasi maximum likelihood approach developed herein and an earlier method based on the fourier transform of a dirac delta train representation of the data surprisingly both are capable of attaining the clairvoyant crame spl acute r rao bound at moderate signal to noise ratios snrs even for short e g 10 samples we carefully address parameter identifiability issues and corroborate our findings with extensive simulations

averaging maximum penalized likelihood and bayesian estimation for improving gaussian mixture probability density estimates we apply the idea of averaging ensembles of estimators to probability density estimation in particular we use gaussian mixture models which are important components in many neural network applications we investigate the performance of averaging using three data sets for comparison we employ two traditional regularization approaches i e a maximum penalized likelihood approach and a bayesian approach in the maximum penalized likelihood approach we use penalty functions derived from conjugate bayesian priors such that an expectation maximization em algorithm can be used for training in all experiments the maximum penalized likelihood approach and averaging improved performance considerably if compared to a maximum likelihood approach in two of the experiments the maximum penalized likelihood approach outperformed averaging in one experiment averaging was clearly superior our conclusion is that maximum penalized likelihood gives good results if the penalty term in the cost function is appropriate for the particular problem if this is not the case averaging is superior since it shows greater robustness by not relying on any particular prior assumption the bayesian approach worked very well on a low dimensional toy problem but failed to give good performance in higher dimensional problems

iterative cdma multiuser receiver with soft decision directed channel estimation we develop an iterative code division multiple access cdma receiver for multipath block fading channels in the uplink the mean square error mse of the channel estimates is reduced iteratively by using soft data estimates as additional pilots the multiuser detector performs parallel interference cancellation pic with subsequent linear filtering we consider a single user matched filter sumf and develop a low complexity linear minimum mse lmmse filter for intersymbol interference isi channels that allows for 33 more users than the sumf we investigate two choices of training sequences for initial channel estimation the first is a random pilot symbol sequence modulated by a short spreading sequence rand and the second is a perfect root of unity sequence prus used as long sounding chip sequence we observe that the rand scheme approaches the single user bound up to 2 db and in case of prus up to 1 db at a target bit error rate ber of 10 3 further several channel estimators are developed that use various combinations of soft decision symbols and statistical a priori knowledge about the channel these are the approximated least squares als approximated linear mmse almmse and linear mmse lmmse estimators the iterative estimators show an improvement of 2 5 and 5 db with prus and rand pilots respectively over estimators that do not exploit soft feedback symbols at a ber of 10 3 the als is sacrificing less than 0 6 db to obtain a ber of 10 3 in a system at load 1 5 when compared with the almmse and lmmse schemes the load is understood as the ratio of the number of users to the spreading factor this paper shows that significant improvements in system capacity can be achieved in multipath scenarios with low complexity choices of multiuser detectors and channel estimators

analytic sequential weiss weinstein bounds in this paper we explore a sequential bayesian bound for state space models focusing on hybrid continuous and discrete random states we provide an analytic recursion for the sequential weiss weinstein sww bound for linear state space models with solutions for gaussian uniform and exponential distributions as derived as well as for a combination of these we compare the sww bound for discretized states with the corresponding bound for the continuous states the sww bound is contrasted with the sequential cramer rao bound for gaussian distributions practical issues of sww bounds are discussed and numerical simulation results provide insights into their behavior

em based recursive estimation of channel parameters recursive online expectation maximization em algorithm along with stochastic approximation is employed in this paper to estimate unknown time invariant variant parameters the impulse response of a linear system channel is modeled as an unknown deterministic vector process and as a gaussian vector process with unknown stochastic characteristics using these models which are embedded in white or colored gaussian noise different types of recursive least squares rls kalman filtering and smoothing and combined rls and kalman type algorithms are derived directly from the recursive em algorithm the estimation of unknown parameters also generates new recursive algorithms for situations such as additive colored noise modeled by an autoregressive process the recursive em algorithm is shown as a powerful tool which unifies the derivations of many adaptive estimation methods

bounds on maximum likelihood ratio part ii application to antenna array detection estimation with imperfect wavefront coherence the multiple hypothesis testing problem of the detection estimation of an unknown number of independent gaussian point sources is adequately addressed by likelihood ratio lr maximization over the set of admissible covariance matrix models we introduce nonasymptotic lower and upper bounds for the maximum lr since lr optimization is generally a nonconvex multiextremal problem any practical solution could now be tested against these bounds enabling a high probability of recognizing nonoptimal solutions we demonstrate that in many applications the lower bound is quite tight with approximate maximum likelihood ml techniques often unable to approach this bound the introduced lower bound analysis is shown to be very efficient in determining whether or not performance breakdown has occurred for subspace based direction of arrival doa estimation techniques we also demonstrate that by proper lr maximization we can extend the range of signal to noise ratio snr values and or number of data samples wherein accurate parameter estimates are produced yet when the snr and or sample size falls below a certain limit for a given scenario we show that ml estimation suffers from a discontinuity in the parameter estimates a phenomenon that cannot be eliminated within the ml paradigm

on the non singular traction bie in elasticity the work reported herein develops a generalized traction bie formulation which involves only weakly singular integrals in the three dimensional problem or totally regular integrals in the two dimensional problem the first step deals with the terms in the somigliana displacement identity and then the derivatives of these terms the only conditions required for the existence of the traction bie and the related somigliana stress identity are weak continuity of the in plane derivatives of the surface displacements and of the surface tractions it is shown that the cauchy principal value cpv interpretations so commonly used in bie developments are unnecessary the formulation is established not only at a smooth boundary point but also at a corner point the extension of the non singular formulation to discontinuous boundary tractions and tangential derivatives of the boundary displacements applicable to a generalized problem statement as well as the usual bem implementations is also shown in the demonstrated formulation the source points are located directly at the boundary nodes and non conformal elements are not needed

on minimum variance unbiased estimation of clock offset in a two way message exchange mechanism for many applications distributed networks require the local clocks of the constituent nodes to run close to an agreed upon notion of time most of the widely used clock synchronization algorithms in such systems employ the sender receiver protocol based on a two way timing message exchange paradigm maximum likelihood estimator mle of the clock offset based on the timing message exchanges between two clocks was derived in d r jeske on maximum likelihood estimation of clock offset ieee trans commun vol 53 pp 53 54 jan 2005 when the fixed delays are symmetric and the variable delays in each direction assume an exponential distribution with an unknown mean herein the best linear unbiased estimate using order statistics blue os of the clock offset between two nodes is derived assuming both symmetric and asymmetric exponential network delays respectively the rao blackwell lehmann scheffe theorem is then exploited to obtain the minimum variance unbiased estimate mvue for the clock offset which it is shown to coincide with the blue os in addition it is found that the mvue of the clock offset in the presence of symmetric network delays also coincides with the mle finally in the presence of asymmetric network delays although the mle is biased it is shown to achieve lesser mean square error mse than the mvue in the region around the point where the bidirectional network link delays are symmetric and hence its merit as the most versatile estimator is fairly justified

on the evaluation of the polyanskiy poor verdu converse bound for finite block length coding in awgn a tight converse bound to the channel coding rate in the finite block length regime and under additive white gaussian noise conditions was recently proposed by polyanskiy poor verdu ppv the bound is a generalization of a number of other classical results and it was also claimed to be equivalent to shannon s 1959 cone packing bound in this paper we investigate methods for a reliable numerical evaluation of the bound which is troublesome even for not too large values of the block length n by compactly expressing the polyanskiy poor and verdu ppv converse bound in terms of non central chi squared distributions and by evaluating those through an integral expression and a corresponding series expansion which exploit a method proposed by temme as a result a robust evaluation method and new insights on the bound s asymptotics as well as new approximate expressions are obtained

the computation of potentials near and on the boundary by an extraction technique for boundary element methods here we present the two dimensional version of a bootstrapping algorithm for the computation of arbitrary derivatives of the cauchy data and for the extraction of potentials near and on the boundary for the laplacian if corresponding boundary integral equations are used the method employs the derivatives of the green representation formula in terms of cauchy singular or weakly singular integrals and also their compositions with derivatives it allows the robust and superconvergent numerical evaluation of potentials and their derivatives up to and on the boundary curve furthermore we determine a criterion whether to use the traditional green representation formula or our extraction technique for the point evaluation of the potentials in the domain

class of widely linear complex kalman filters recently a class of widely linear augmented complex valued kalman filters kfs that make use of augmented complex statistics have been proposed for sequential state space estimation of the generality of complex signals this was achieved in the context of neural network training and has allowed for a unified treatment of both second order circular and noncircular signals that is both those with rotation invariant and rotation dependent distributions in this paper we revisit the augmented complex kf augmented complex extended kf and augmented complex unscented kf in a more general context and analyze their performances for different degrees of noncircularity of input and the state and measurement noises for rigor a theoretical bound for the performance advantage of widely linear kfs over their strictly linear counterparts is provided the analysis also addresses the duality with bivariate real valued kfs together with several issues of implementation simulations using both synthetic and real world proper and improper signals support the analysis

uplink synchronization in ofdma spectrum sharing systems spectrum sharing employs dynamic allocation of frequency resources for a more efficient use of the radio spectrum despite its appealing features this technology inevitably complicates the synchronization task which must be accomplished in an environment that may involve interference this paper considers the uplink of an orthogonal frequency division multiple access ofdma based spectrum sharing system and provides solutions for estimating the frequency and timing errors of multiple unsynchronized users in doing so we exploit suitably designed training blocks and apply maximum likelihood methods after modeling the interference power on each subcarrier as a random variable with an inverse gamma distribution the resulting frequency estimator turns out to be the extension to a multiuser scenario of a scheme that has previously been proposed for single user spectrum sharing systems timing recovery is more challenging and leads to a complete search over a multidimensional domain to overcome such a difficulty two alternative approaches are proposed the first one relies on a simplifying assumption about the interference distribution in the frequency domain while the second scheme operates in an iterative fashion according to the expectation maximization algorithm

representation of mutual information via input estimates a relationship between information theory and estimation theory was recently shown for the gaussian channel relating the derivative of mutual information with the minimum mean square error this paper generalizes the link between information theory and estimation theory to arbitrary channels giving representations of the derivative of mutual information as a function of the conditional marginal input distributions given the outputs we illustrate the use of this representation in the efficient numerical computation of the mutual information achieved by inputs such as specific codes or natural language

statistical properties of linear prediction analysis underlying the challenge of formant bandwidth estimation formant bandwidth estimation is often observed to be more challenging than the estimation of formant center frequencies due to the presence of multiple glottal pulses within a period and short closed phase durations this study explores inherently different statistical properties between linear prediction lp based estimates of formant frequencies and their corresponding bandwidths that may be explained in part by the statistical bounds on the variances of estimated lp coefficients a theoretical analysis of the cramer rao bounds on lp estimator variance indicates that the accuracy of bandwidth estimation is approximately twice as low as that of center frequency estimation monte carlo simulations of all pole vowels with stochastic and mixed source excitation demonstrate that the distributions of estimated lp coefficients exhibit expectedly different variances for each coefficient transforming the lp coefficients to formant parameters results in variances of bandwidth estimates being typically larger than the variances of respective center frequency estimates depending on vowel type and fundamental frequency these results provide additional evidence underlying the challenge of formant bandwidth estimation due to inherent statistical properties of lp based speech analysis

minimum variance in biased estimation with singular fisher information matrix this paper extends the work of y c eldar ldquominimum variance in biased estimation bounds and asymptotically optimal estimators rdquo in ieee trans signal process vol 52 pp 1915 1929 jul 2004 which deals with only nonsingular fisher information matrix in order to guarantee the uniform cramer rao bound to be a finite lower bound and also to have a feasible solution to the optimization problem in the work of y c eldar it is proved that the norms of bias gradient matrices of all biased estimators must have a nonzero exact lower bound which mainly depends on the rank of the singular fisher information matrix the smaller the rank of the singular fisher information matrix is the larger the lower bound of norms of bias gradient matrices of all biased estimators is for a specific frobenius norm the exact lower bound is simply the difference between the parameter dimension and the rank of the singular fisher information matrix

the risk unbiased cramer rao bound for non bayesian multivariate parameter estimation how accurately can one estimate a deterministic parameter subject to other unknown deterministic model nuisance parameters the most popular answer to this question is given by the cramer rao bound crb the main assumption behind the derivation of the crb is local unbiased estimation of all model parameters the foundations of this paper rely on doubting this assumption generally in multivariate parameter estimation each parameter in its turn can be treated as a single parameter of interest whereas the other model parameters are treated as nuisance as their misknowledge interferes with the estimation of the parameter of interest this approach is utilized in this paper to provide a fresh look at deterministic parameter estimation a new cramer rao cr type bound is derived without assuming unbiased estimation of the nuisance parameters rather than that we apply lehmann s concept of unbiasedness for a risk that measures the distance between the estimator and the locally best unbiased estimator which assumes perfect knowledge of the nuisance parameters the proposed risk unbiased crb rucrb is proven to be asymptotically attainable by the maximum likelihood estimator while being tighter than the conventional crb furthermore simulations verify the asymptotic achievability of the rucrb by the maximum likelihood estimator for an array processing problem

distributed snr estimation with power constrained signaling over gaussian multiple access channels a sensor network is used for distributed signal to noise ratio snr estimation in a single time snapshot sensors observe a signal embedded in noise and each observation is phase modulated using a constant modulus scheme and transmitted over a gaussian multiple access channel to a fusion center at the fusion center the mean and variance are estimated jointly using an asymptotically minimum variance estimator it is shown that this joint estimator decouples into simple individual estimators of the mean and the variance the constant modulus phase modulation scheme ensures a fixed transmit power robust estimation across several sensing noise distributions as well as an snr estimate that requires a single set of transmissions from the sensors to the fusion center the estimators are evaluated in terms of asymptotic variance which are then used to evaluate the performance of the snr estimator with gaussian and cauchy sensing noise distributions in the cases of total transmit power constraint as well as a per sensor power constraint for each sensing noise distribution the optimal phase transmission parameters are also determined the asymptotic relative efficiency of the estimators is evaluated it is shown that among the noise distributions considered the estimators are asymptotically efficient only when the noise distribution is gaussian simulation results corroborate analytical results

improvement of tdoa position fixing using the likelihood curvature a conventional multilateration is a two step process where in the first step the time difference of arrivals tdoa of a signal at multiple sensors are estimated and in the second step these tdoas are used in some position fixing technique to estimate the location of an emitter several estimators have been proposed over the years for the estimation of the tdoas many techniques have been proposed for position fixing as well much of the research on position fixing has been focused on obtaining a simplified closed form solution for the unknown deterministic signal model stein had derived the maximum likelihood estimator mle for the tdoa between two sensors which is the peak location of the cross correlation function since the asymptotic variance of an mle approaches the cramer rao lower bound crlb which is the inverse of the negative of the expected value of the curvature of the log likelihood function using this as a motivation we propose a weighted least squares type position fixing technique where the weights are computed from the curvature of the log likelihood function

provably convertible multi authenticated encryption scheme for generalized group communications in 2008 wu et al proposed a convertible multi authenticated encryption cmae scheme based on discrete logarithms to improve the computational efficiency of wu et al s scheme tsai proposed another cmae scheme without using message redundancy these cmae schemes however might be inadequate for group oriented applications in the same year chang presented a cmae scheme using message redundancy for group communications published in information sciences 178 17 2008 3426 3434 in his scheme multiple signers of the same group can cooperate with each other to generate a valid authenticated encryption signature for a designated verifying group with access structured multiple verifiers in this paper we propose a new convertible multi authenticated encryption scheme without using message redundancy for generalized group communications multiple signers of the signing group can cooperate with each other to generate a valid authenticated encryption signature for a verifying group with access structured multiple verifiers the verifiers in the same access structure can cooperatively recover and verify the signing group s signature in case of a later dispute any participant verifier can convert the authenticated encryption signature into an ordinary one as compared with previous works our proposed scheme is more efficient in terms of computational efforts and communication overheads

a lower bound on the bayesian mse based on the optimal bias function a lower bound on the minimum mean squared error mse in a bayesian estimation problem is proposed in this paper this bound utilizes a well known connection to the deterministic estimation setting using the prior distribution the bias function which minimizes the cramer rao bound can be determined resulting in a lower bound on the bayesian mse the bound is developed for the general case of a vector parameter with an arbitrary probability distribution and is shown to be asymptotically tight in both the high and low signal to noise ratio snr regimes a numerical study demonstrates several cases in which the proposed technique is both simpler to compute and tighter than alternative methods

concurrent zero knowledge concurrent executions of a zero knowledge protocol by a single prover with one or more verifiers may leak information and may not be zero knowledge in toto in this article we study the problem of maintaining zero knowledge we introduce the notion of an timing constraint for any two processors p1 and p2 if p1 measures elapsed time on its local clock and p2 measures elapsed time on its local clock and p2 starts after p1 does then p2 will finish after p1 does we show that if the adversary is constrained by an assumption then there exist four round almost concurrent zero knowledge interactive proofs and perfect concurrent zero knowledge arguments for every language in np we also address the more specific problem of deniable authentication for which we propose several particularly efficient solutions deniable authentication is of independent interest even in the sequential case our concurrent solutions yield sequential solutions without recourse to timing that is in the standard model

on the extraction technique in boundary integral equations in this paper we develop and analyze a bootstrapping algorithm for the extraction of potentials and arbitrary derivatives of the cauchy data of regular three dimensional second order elliptic boundary value problems in connection with corresponding boundary integral equations the method rests on the derivatives of the generalized green s representation formula which are expressed in terms of singular boundary integrals as hadamard s finite parts their regularization together with asymptotic pseudohomogeneous kernel expansions yields a constructive method for obtaining generalized jump relations these expansions are obtained via composition of taylor expansions of the local surface representation the density functions differential operators and the fundamental solution of the original problem together with the use of local polar coordinates in the parameter domain for boundary integral equations obtained by the direct method this method allows the recursive numerical extraction of potentials and their derivatives near and up to the boundary surface

penalized maximum likelihood estimation of covariance matrices with linear structure in this paper a space alternating generalized expectation maximization sage algorithm is presented for the numerical computation of maximum likelihood ml and penalized ml pml estimates of the parameters of covariance matrices with linear structure for complex gaussian processes by using a less informative hidden data space and a sequential parameter update scheme a sage based algorithm is derived for which convergence of the likelihood is demonstrated to be significantly faster than that of an em based algorithm that has been previously proposed in addition the sage procedure is shown to easily accommodate penalty functions and a sage based algorithm is derived and demonstrated for forming pml estimates with a quadratic smoothness penalty

robust rls in the presence of correlated noise using outlier sparsity relative to batch alternatives the recursive least squares rls algorithm has well appreciated merits of reduced complexity and storage requirements for online processing of stationary signals and also for tracking slowly varying nonstationary signals however rls is challenged when in addition to noise outliers are also present in the data due to e g impulsive disturbances existing robust rls approaches are resilient to outliers but require the nominal noise to be white an assumption that may not hold in e g sensor networks where neighboring sensors are affected by correlated ambient noise prewhitening with the known noise covariance is not a viable option because it spreads the outliers to noncontaminated measurements which leads to inefficient utilization of the available data and unsatisfactory performance in this correspondence a robust rls algorithm is developed capable of handling outliers and correlated noise simultaneously in the proposed method outliers are treated as nuisance variables and estimated jointly with the wanted parameters identifiability is ensured by exploiting the sparsity of outliers which is effected via regularizing the least squares ls criterion with the l1 norm of the outlier vectors this leads to an optimization problem whose solution yields the robust rls estimates for low complexity real time operation a suboptimal online algorithm is proposed which entails closed form updates per time step in the spirit of rls simulations demonstrate the effectiveness and improved performance of the proposed method in comparison with the nonrobust rls and its state of the art robust renditions

location estimation of a random signal source based on correlated sensor observations the problem of location estimation of a source of random signals using a network of sensors is considered a novel maximum likelihood estimation mle based approach using copula functions is proposed the measurements received at the sensors are often spatially correlated and characterized by a multivariate distribution using the theory of copulas the joint parametric density of sensor observations joint likelihood is approximated assuming only the knowledge of the marginal likelihood functions of the sensor observations the problem of selecting the best copula function to model the joint likelihood is approached as one of model selection and a model fusion strategy is used to reduce the effect of selection bias an example involving source localization of a poisson source is presented to illustrate the proposed approach and demonstrate its performance

number theoretic constructions of efficient pseudo random functions we describe efficient constructions for various cryptographic primitives in private key as well as public key cryptography our main results are two new constructions of pseudo random functions we prove the pseudo randomness of one construction under the assumption that factoring blum integers is hard while the other construction is pseudo random if the decisional version of the diffie hellman assumption holds computing the value of our functions at any given point involves two subset products this is much more efficient than previous proposals furthermore these functions have the advantage of being in tc0 the class of functions computable by constant depth circuits consisting of a polynomial number of threshold gates this fact has several interesting applications the simple algebraic structure of the functions implies additional features such as a zero knowledge proof for statements of the form y e fs x and y n fs x given a commitment to a key s of a pseudo random function fs

image reconstruction and restoration overview of common estimation structures and problems developments in the theory of image reconstruction and restoration over the past 20 or 30 years are outlined particular attention is paid to common estimation structures and to practical problems not properly solved yet the problem of image reconstruction and restoration is first formulated some of the current regularization approaches used to solve the problem are then described the concepts of a priori information and compound criterion are introduced a bayesian interpretation of the regularization techniques is given which clarifies the role of the tuning parameters and indicates how they could be estimated the practical aspects of computing the solution first when the hyperparameters are known and second when they must be estimated are then considered conclusions are drawn and points that still need to be investigated are outlined

robust precoding with limited feedback design based on precoding mse for mu miso systems for the separation of the signals in the vector broadcast channel bc some information about the channel state is necessary at the transmitter in many cases this channel state information csi must be fed back from the receivers to the transmitter we jointly design the channel estimators and the quantizers at the receivers together with the precoder at the transmitter based on a precoder centric criterion i e the minimization of a mean square error mse metric appropriate for the precoder design this is in contrast to our previous works where the quantizer design was based on a csi mse metric i e based on the minimization of the mse between the true channel and the channel recovered by the transmitter using a feedback channel interestingly the estimators resulting from this joint formulation are independent of the used codebook the codebook entries are the employed precoders therefore each receiver feeds back the index of a set of precoders and the intersection of the sets gives the appropriate precoder since the quantizers of the different receivers have to work separately the metric for the computation of the partition cells cannot be expressed as a simple squared error depending on the quantizer output the proposed system based on a joint optimization clearly outperforms previous designs with separate optimization of feedback and precoding

method of constructing elliptic curve authenticated encryption scheme an elliptic curve authenticated encryption scheme with message recovery is proposed and its security is also analyzed in accordance with this scheme the more general signature equation of the proposed scheme is presented accordingly the method of constructing elliptic curve authenticated encryption scheme is described including the method of choosing the parameters of signature equation then the two forgery attacks with known plaintext is discussed respectively the results indicate that some schemes can resist the imitating attacks and some cannot

auditory motivated level crossing approach to instantaneous frequency estimation we address the problem of estimating the instantaneous frequency if of a phase signal using its level crossing lc information based on front end auditory processing motivation we show that the problem of if estimation using lc information can be cast in the framework of estimation from irregularly sampled data the formulation has the generality of estimating different types of if without the need for a quasistationary assumption we consider two types of if polynomial and bandlimited we use polynomial interpolating functions for the former and for the latter we propose a novel line plus sum of sines model the model parameters are estimated by linear regression considering the noisy case lc data for different levels is analyzed and methods for combining different estimators from lcs are discussed theoretical and extensive simulation results show that the performance of the zero crossing zc based if estimator and the level crossing based if estimator with smaller level values is better than those obtained with higher level values or their combinations the new technique reaches the crame spl acute r rao bound crb roughly above 4 db signal to noise ratio snr and its performance does not deteriorate rapidly with mismatch in the if order compared with the other techniques in the literature

maximum likelihood least squares and penalized least squares for pet the em algorithm is the basic approach used to maximize the log likelihood objective function for the reconstruction problem in positron emission tomography pet the em algorithm is a scaled steepest ascent algorithm that elegantly handles the nonnegativity constraints of the problem it is shown that the same scaled steepest descent algorithm can be applied to the least squares merit function and that it can be accelerated using the conjugate gradient approach the experiments suggest that one can cut the computation by about a factor of 3 by using this technique the results are applied to various penalized least squares functions which might be used to produce a smoother image

subspace partitioning for target detection and identification detection of a given target or set of targets from observed data is a problem countered in many applications regardless of the algorithm selected detection performance can be severely degraded when the subspace defined by the target data set is singular or ill conditioned high correlations between target components and their linear combinations lead to false positives and misidentifications especially for subspace based detectors in this paper we propose a subspace partitioning scheme that allows for detection to be performed in a number of better conditioned subspaces instead of the original subspace the proposed technique is applied to raman spectroscopic data analysis through both simulation and experimental results we demonstrate the improvement in the overall detection performance when using the proposed subspace partitioning scheme in conjunction with several subspace detection methods that are commonly used in practice

minimum variance in biased estimation bounds and asymptotically optimal estimators we develop a uniform cramer rao lower bound ucrlb on the total variance of any estimator of an unknown vector of parameters with bias gradient matrix whose norm is bounded by a constant we consider both the frobenius norm and the spectral norm of the bias gradient matrix leading to two corresponding lower bounds we then develop optimal estimators that achieve these lower bounds in the case in which the measurements are related to the unknown parameters through a linear gaussian model tikhonov regularization is shown to achieve the ucrlb when the frobenius norm is considered and the shrunken estimator is shown to achieve the ucrlb when the spectral norm is considered for more general models the penalized maximum likelihood pml estimator with a suitable penalizing function is shown to asymptotically achieve the ucrlb to establish the asymptotic optimality of the pml estimator we first develop the asymptotic mean and variance of the pml estimator for any choice of penalizing function satisfying certain regularity constraints and then derive a general condition on the penalizing function under which the resulting pml estimator asymptotically achieves the ucrlb this then implies that from all linear and nonlinear estimators with bias gradient whose norm is bounded by a constant the proposed pml estimator asymptotically results in the smallest possible variance

a convertible multi authenticated encryption scheme for group communications recently wu et al proposed a convertible multi authenticated encryption cmae scheme which allows a signing group with multiple signers to generate a multi authenticated ciphertext signature on the chosen message so that only a designated verifier can recover and verify the message in case of later dispute the verifier can convert the multi authenticated ciphertext signature into an ordinary one that can be verified by anyone in this study a cmae scheme for group communications is proposed this is presented by first reviewing the concepts of group oriented encryption schemes and the merits of wu et al s scheme this shows that not only can a multi authenticated ciphertext signature be generated by a signing group but also the message can be recovered and verified by a verifying group with multiple verifiers the security of the proposed scheme is based solely on the ddh problem which provides higher security confidence than using the cdh problem in wu et al s cmae scheme

low complexity lmmse based mimo ofdm channel estimation via angle domain processing in many situations multiple input multiple output with orthogonal frequency division multiplexing mimo ofdm channels tend to be spatially correlated due for example to limited scattering prior knowledge of this channel spatial correlation and the channel frequency correlation can be exploited by using the linear minimum mean square error lmmse technique however the complexity of the 2 d lmmse technique which fully utilizes both the channel spatial and frequency correlation is quite high to solve this problem this paper presents and analyzes several low complexity suboptimal approximated lmmse channel estimation techniques in the angle domain where the channel model lends itself to a physical interpretation the choice of angle domain techniques is largely dependent on the extent of channel stochastic information e g channel correlation or power that is available to the receiver nevertheless all the proposed angle domain techniques have much lower complexity compared to the 2 d lmmse technique further all the angle domain techniques improve over the conventional least square ls technique for all the typical mimo ofdm models under consideration more importantly our simulation results show that the angle domain quasi 1 d ql d lmmse technique can achieve similar performance compared to the 2 d lmmse technique for all typical mimo ofdm models with significantly lower complexity

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

optimality analysis of sensor target localization geometries the problem of target localization involves estimating the position of a target from multiple noisy sensor measurements it is well known that the relative sensor target geometry can significantly affect the performance of any particular localization algorithm the localization performance can be explicitly characterized by certain measures for example by the cramer rao lower bound which is equal to the inverse fisher information matrix on the estimator variance in addition the cramer rao lower bound is commonly used to generate a so called uncertainty ellipse which characterizes the spatial variance distribution of an efficient estimate i e an estimate which achieves the lower bound the aim of this work is to identify those relative sensor target geometries which result in a measure of the uncertainty ellipse being minimized deeming such sensor target geometries to be optimal with respect to the chosen measure the optimal sensor target geometries for range only time of arrival based and bearing only localization are identified and studied in this work the optimal geometries for an arbitrary number of sensors are identified and it is shown that an optimal sensor target configuration is not in general unique the importance of understanding the influence of the sensor target geometry on the potential localization performance is highlighted via formal analytical results and a number of illustrative examples

signcryption from randomness recoverable public key encryption we propose a new generic construction for signcryption and show that it is secure under the security models which are comparable to the security against adaptive chosen ciphertext attacks for public key encryption and the existential unforgeability against chosen message attacks for signature in particular the security models also capture the notion of insider security the generic construction relies on the existence of a special class of efficient public key encryption schemes which allow the encryption randomness to be recovered during decryption we also propose two efficient instantiations for the generic construction and show that one of them has less message expansion and yields smaller ciphertext when compared with all the existing signcryption schemes

minimum bias multiple taper spectral estimation two families of orthonormal tapers are proposed for multitaper spectral analysis minimum bias tapers and sinusoidal tapers spl upsisup k where spl upsisub nsup k spl radic 2 n 1 sin spl pi kn n 1 and n is the number of points the resulting sinusoidal multitaper spectral estimate is s spl circ f 1 2k n 1 spl sigmasub j 1sup k y f j 2n 2 y f j 2n 2 sup 2 where y f is the fourier transform of the stationary time series s f is the spectral density and k is the number of tapers for fixed j the sinusoidal tapers converge to the minimum bias tapers like 1 n since the sinusoidal tapers have analytic expressions no numerical eigenvalue decomposition is necessary both the minimum bias and sinusoidal tapers have no additional parameter for the spectral bandwidth the bandwidth of the jth taper is simply 1 n centered about the frequencies spl plusmn j 2n 2 thus the bandwidth of the multitaper spectral estimate can be adjusted locally by simply adding or deleting tapers the band limited spectral concentration spl intsub wsup w v f sup 2 df of both the minimum bias and sinusoidal tapers is very close to the optimal concentration achieved by the slepian 1978 tapers in contrast the slepian tapers can have the local bias spl intsub 1 2 sup 1 2 f sup 2 v f sup 2 df much larger than of the minimum bias tapers and the sinusoidal tapers

on time harmonic elastic wave analysis by the boundary element method for moderate to high frequencies abstract certain inherent difficulties in obtaining accurate boundary element solutions to problems of elastodynamics at moderate to high frequencies are examined some procedures for overcoming or compensating for these difficulties are described in addition results demonstrating the effectiveness of these procedures are presented finally the use of these procedures in solving transient problems via fourier transform inversion is illustrated

the total variance of a periodogram based spectral estimate of a stochastic process with spectral uncertainty and its application to classifier design the variance of a spectral estimate of a stochastic process is essential to the formulation and performance of a spectral classifier the overall variance of a spectral estimate originates from two sources the within class spectral uncertainty and the variance introduced in the spectral estimation procedure in this paper we derive the total variance of a periodogram based spectral estimate under some assumptions using this result we formulate a linear spectral classifier based on fisher s separability metric the classifier is used to classify two oceanographic processes ocean convection versus internal waves

fundamental limits of linear equalizers diversity capacity and complexity linear equalizers les have been widely adopted for practical systems due to their low computational complexity however it is also well known that les provide inferior performance relative to a maximum likelihood equalizer mle or other near mles because les usually cannot collect the diversity order enabled by the transmitter and at the same time they lose mutual information more importantly unlike mle or near mles the performance of les has not been well quantified this hinders more general applications of les in wireless systems in this paper we reveal a fundamental link between a channel parameter orthogonality deficiency od of the channel matrix and the diversity and capacity of les we identify that when the od of channel matrix has an upper bound strictly less than 1 the same diversity order as that of mles is collected by les and the outage capacity loss relative to mles is also a constant over signal to noise ratio snr these results can be applied to designing a framework for hybrid equalizers furthermore by studying the statistical properties of the od and comparing the complexity of different equalizers we show that hybrid equalizers can trade off between performance and complexity by tuning the channel matrix od the theoretical analysis is corroborated by computer simulations

on posterior distributions for signals in gaussian noise with unknown covariance matrix a bayesian approach to estimate parameters of signals embedded in complex gaussian noise with unknown color is presented the study specifically focuses on a bayesian treatment of the unknown noise covariance matrix making up a nuisance parameter in such problems by integrating out uncertainties regarding the noise color an enhanced ability to estimate both the signal parameters as well as properties of the error is exploited several noninformative priors for the covariance matrix such as the reference prior the jeffreys prior and modifications to this are considered some of the priors result in analytical solutions whereas others demand numerical approximations in the linear signal model connections are made between the standard adaptive maximum likelihood aml estimate and a bayesian solution using the jeffreys prior with adjustments to the jeffreys prior correspondence to the regularized solution is also established this in turn enables a formal treatment of the regularization parameter simulations indicate that significant improvements compared to the aml estimator can be obtained by considering both the derived regularized solutions as well as the one obtained using the reference prior the simulations also indicate the possibility of enhancing the predictions of properties of the error as uncertainties in the noise color are acknowledged

on the computation of the derivatives of potentials on a boundary by using boundary integral equations this work presents a new recursion scheme to compute the cartesian derivatives of potentials on the smooth surface of a connected solid the advocated strategy solely appeals to boundary integral equations and a very few informations regarding the surface geometry the whole algorithm is carefully tested against analytical solutions both for interior and exterior problems by implementing a collocation points method

a novel location penalized maximum likelihood estimator for bearing only target localization in this paper we present a location penalized maximum likelihood lpml estimator for bearing only target localization we develop a new penalized maximum likelihood cost function by transforming the variables of target position and bearings the new penalized likelihood function can also be recognized as a posterior distribution under a bayesian framework by penalizing a prior we give analysis of the asymptotic properties and show that both traditional bearing maximum likelihood tbml and lpml estimators are asymptotically efficient estimators to compare the performances of the tbml and lpml estimators we analyze the cramer rao lower bound crlb of the two estimators and show that the bound of the lpml estimator is lower than that of the tbml estimator extensive simulations are performed it is observed that the new lpml algorithm consistently outperforms other well known algorithms field experiments are also conducted by applying this method to localize a vehicle using real world data acquired by an acoustic array sensor network the new lpml algorithm demonstrates superior performance in all the field experiments

estimation of flows in flow networks abstract let g be a directed graph with an unknown flow on each edge such that the following flow conservation constraint is maintained except for sources and sinks the sum of flows into a node equals the sum of flows going out of a node given a noisy measurement of the flow on each edge the problem we address which we call the m ost p robable f low e stimation problem mpfe is to estimate the most probable assignment of flow for every edge such that the flow conservation constraint is maintained we provide an algorithm called y mpfe for solving the mpfe problem when the measurement error is gaussian g aussian mpfe the algorithm works in o e v 2 when the underlying undirected graph of g is a 2 connected planar graph and in o e v when it is a 2 connected serial parallel graph or a tree this result is applicable to any m inimum c ost f low problem for which the cost function is e x e e 2 for edge e where e and e are constants and x e is the flow on edge e we show that for all topologies the g aussian mpfe s precision for each edge is analogous to the equivalent resistance measured in series to this edge in an electrical network built by replacing every edge with a resistor reflecting the measurement s precision on that edge

hybrid fm polynomial phase signal modeling parameter estimation and cramer rao bounds parameter estimation for a class of nonstationary signal models is addressed the class contains combination of a polynomial phase signal pps and a frequency modulated fm component of the sinusoidal or hyperbolic type such signals appear in radar and sonar applications involving moving targets with vibrating or rotating components a novel approach is proposed that allows us to decouple estimation of the fm parameters from those of the pps relying on properties of the multilag high order ambiguity function ml haf the accuracy achievable by any unbiased estimator of the hybrid fm pps parameters is investigated by means of the cramer rao lower bounds crlbs both exact and large sample approximate expressions of the bounds are derived and compared with the performance of the proposed methods based on monte carlo simulations

the ring authenticated encryption scheme how to provide a clue wisely though cryptography is being used more and more widely in reality it seems that there exists no scheme or a concatenation of some existing schemes that could deal soundly with such practical situations as providing a clue where the provider of the clue may want to reserve his beneficial rights while keeping his identity secret to address this problem inspired by the two notions of the ring signature and the authenticated encryption signature we propose a new type of authenticated encryption scheme which we call the ring authenticated encryption scheme which can enable any member of a group of persons to provide a clue to some designated recipient wisely

new results on deterministic cramer rao bounds for real and complex parameters the cramer rao bounds crb is a lower bound of great interest for system analysis and design in the asymptotic region high signal to noise ratio snr and or large number of snapshots as it is simple to calculate and it is usually possible to obtain closed form expressions the first part of the paper is a generalization to complex parameters of the barankin rationale for deriving mse lower bounds that is the minimization of a norm under a set of linear constraints with the norm minimization approach the study of fisher information matrix fim singularity constrained crb and regularity conditions become straightforward corollaries of the derivation the second part provides new results useful for system analysis and design a general reparameterization inequality the equivalence between reparameterization and equality constraints and an explicit relationship between parameters unidentifiability and fim singularity

uniformly improving the cramer rao bound and maximum likelihood estimation an important aspect of estimation theory is characterizing the best achievable performance in a given estimation problem as well as determining estimators that achieve the optimal performance the traditional cramer rao type bounds provide benchmarks on the variance of any estimator of a deterministic parameter vector under suitable regularity conditions while requiring a priori specification of a desired bias gradient in applications it is often not clear how to choose the required bias a direct measure of the estimation error that takes both the variance and the bias into account is the mean squared error mse which is the sum of the variance and the squared norm of the bias here we develop bounds on the mse in estimating a deterministic parameter vector x0 over all bias vectors that are linear in x0 which includes the traditional unbiased estimation as a special case in some settings it is possible to minimize the mse over all linear bias vectors more generally direct minimization is not possible since the optimal solution depends on the unknown x0 nonetheless we show that in many cases we can find bias vectors that result in an mse bound that is smaller than the cramer rao lower bound crlb for all values of x0 furthermore we explicitly construct estimators that achieve these bounds in cases where an efficient estimator exists by performing a simple linear transformation on the standard maximum likelihood ml estimator this leads to estimators that result in a smaller mse than the ml approach for all possible values of x0

low complexity methods for estimation after parameter selection statistical inference of multiple parameters often involves a preliminary parameter selection stage the selection stage has an impact on subsequent estimation for example by introducing a selection bias the post selection maximum likelihood psml estimator is shown to reduce the selection bias and the post selection mean squared error psmse compared with conventional estimators such as the maximum likelihood ml estimator however the computational complexity of the psml is usually high due to the multi dimensional exhaustive search for a global maximum of the post selection log likelihood psll function moreover the psll involves the probability of selection that in general does not have an analytical form in this paper we develop new low complexity post selection estimation methods for a two stage estimation after parameter selection architecture the methods are based on implementing the iterative maximization by parts mbp approach which is based on the decomposition of the psll function into easily optimized and complicated parts the proposed second best psml method applies the mbp psml algorithm with a pairwise probability of selection between the two highest ranked parameters w r t the selection rule the proposed sa psml method is based on using stochastic approximation sa and monte carlo integrations to obtain a non parametric estimation of the gradient of the probability of selection and then applying the mbp psml algorithm on this approximation for low complexity performance analysis we develop the empirical post selection cram acute text e r rao type lower bound simulations demonstrate that the proposed post selection estimation methods are tractable and reduce both the bias and the psmse compared with the ml estimator while only requiring moderate computational complexity