maximum likelihood estimation of a structured covariance matrix with a condition number constraint in this paper we deal with the problem of estimating the disturbance covariance matrix for radar signal processing applications when a limited number of training data is present we determine the maximum likelihood ml estimator of the covariance matrix starting from a set of secondary data assuming a special covariance structure i e the sum of a positive semi definite matrix plus a term proportional to the identity and a condition number upper bound constraint we show that the formulated constrained optimization problem falls within the class of maxdet problems and develop an efficient procedure for its solution in closed form remarkably the computational complexity of the algorithm is of the same order as the eigenvalue decomposition of the sample covariance matrix at the analysis stage we assess the performance of the proposed algorithm in terms of achievable signal to interference plus noise ratio sinr both for a spatial and a doppler processing the results show that interesting sinr improvements with respect to some existing covariance matrix estimation techniques can be achieved

on hamiltonian structure of integrable equations under the group and matrix reductions the hamiltonian structure of a differential equations integrable by means of an arbitrary order linear spectral problem under reductions to classical lie algebras bn cn dn and also of b equations associated with the matrix analogue of the linear zakharov shabat linear problem is analysed

complex elliptically symmetric distributions survey new results and applications complex elliptically symmetric ces distributions have been widely used in various engineering applications for which non gaussian models are needed in this overview circular ces distributions are surveyed some new results are derived and their applications e g in radar and array signal processing are discussed and illustrated with theoretical examples simulations and analysis of real radar data the maximum likelihood ml estimator of the scatter matrix parameter is derived and general conditions for its existence and uniqueness and for convergence of the iterative fixed point algorithm are established specific ml estimators for several ces distributions that are widely used in the signal processing literature are discussed in depth including the complex t distribution k distribution the generalized gaussian distribution and the closely related angular central gaussian distribution a generalization of ml estimators the m estimators of the scatter matrix are also discussed and asymptotic analysis is provided applications of ces distributions and the adaptive signal processors based on ml and m estimators of the scatter matrix are illustrated in radar detection problems and in array signal processing applications for direction of arrival doa estimation and beamforming furthermore experimental validation of the usefulness of ces distributions for modelling real radar data is given

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

subspace based adaptive generalized likelihood ratio detection subspace based adaptive detection performance is examined for the generalized likelihood ratio detector based on wilks spl lambda statistic the problem considered here is detecting the presence of one or more signals of known shape embedded in gaussian distributed noise with unknown covariance structure the data is mapped into a subspace prior to detection the probability of false alarm is independent of the subspace transformation and depends only on subspace dimension the probability of detection depends on the subspace transformation through a nonadaptive signal to noise ratio snr parameter subspace processing results in an snr loss that tends to decrease performance and a gain in statistical stability that tends to increase performance it is shown that the statistical stability effect dominates the snr loss for short data records and subspace detectors can require substantially less snr than full space detectors for equivalent performance a method for designing the subspace transformation to minimize the snr loss is proposed and illustrated through simulations

multivariate generalized gaussian distribution convexity and graphical models we consider covariance estimation in the multivariate generalized gaussian distribution mggd and elliptically symmetric es distribution the maximum likelihood optimization associated with this problem is non convex yet it has been proved that its global solution can be often computed via simple fixed point iterations our first contribution is a new analysis of this likelihood based on geodesic convexity that requires weaker assumptions our second contribution is a generalized framework for structured covariance estimation under sparsity constraints we show that the optimizations can be formulated as convex minimization as long the mggd shape parameter is larger than half and the sparsity pattern is chordal these include for example maximum likelihood estimation of banded inverse covariances in multivariate laplace distributions which are associated with time varying autoregressive processes

hamiltonian structure of the general integrable equations under reductions the general form of the nonlinear evolution equations integrable by the linear matrix spectral problem and their hamiltonian structures are presented for potentials with nonzero constant asymptotic values the reduction problem is discussed within the framework of the akns method a method is proposed for the calculation of the hamiltonian structures for the integrable equations under reductions matrix zn and n reductions and some other reductions are considered as examples general forms of the corresponding reduced equations and their hamiltonian structures are found

a constrained l1 minimization approach to sparse precision matrix estimation this article proposes a constrained l1 minimization method for estimating a sparse inverse covariance matrix based on a sample of n iid p variate random variables the resulting estimator is shown to have a number of desirable properties in particular the rate of convergence between the estimator and the true s sparse precision matrix under the spectral norm is when the population distribution has either exponential type tails or polynomial type tails we present convergence rates under the elementwise l norm and frobenius norm in addition we consider graphical model selection the procedure is easily implemented by linear programming numerical performance of the estimator is investigated using both simulated and real data in particular the procedure is applied to analyze a breast cancer dataset and is found to perform favorably compared with existing methods

sparsistency and rates of convergence in large covariance matrix estimation this paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions here sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probability tending to one depending on the case of applications sparsity priori may occur on the covariance matrix its inverse or its cholesky decomposition we study these three sparsity exploration problems under a unified framework with a general penalty function we show that the rates of convergence for these problems under the frobenius norm are of order s n log p n n 1 2 where s n is the number of nonzero elements p n is the size of the covariance matrix and n is the sample size this explicitly spells out the contribution of high dimensionality is merely of a logarithmic factor the conditions on the rate with which the tuning parameter n goes to 0 have been made explicit and compared under different penalties as a result for the l 1 penalty to guarantee the sparsistency and optimal rate of convergence the number of nonzero elements should be small sn o pn at most among o pn2 parameters for estimating sparse covariance or correlation matrix sparse precision or inverse correlation matrix or sparse cholesky factor where sn is the number of the nonzero elements on the off diagonal entries on the other hand using the scad or hard thresholding penalty functions there is no such a restriction

on recursive estimation for time varying autoregressive processes this paper focuses on recursive estimation of time varying autoregressive processes in a nonparametric setting the stability of the model is revisited and uniform results are provided when the time varying autoregressive parameters belong to appropriate smoothness classes an adequate normalization for the correction term used in the recursive estimation procedure allows for very mild assumptions on the innovations distributions the rate of convergence of the pointwise estimates is shown to be minimax in lipschitz classes for 0 1 for 1 2 this property no longer holds this can be seen by using an asymptotic expansion of the estimation error a bias reduction method is then proposed for recovering the minimax rate

unified framework to regularized covariance estimation in scaled gaussian models we consider regularized covariance estimation in scaled gaussian settings e g elliptical distributions compound gaussian processes and spherically invariant random vectors asymptotically in the number of samples the classical maximum likelihood ml estimate is optimal under different criteria and can be efficiently computed even though the optimization is nonconvex we propose a unified framework for regularizing this estimate in order to improve its finite sample performance our approach is based on the discovery of hidden convexity within the ml objective we begin by restricting the attention to diagonal covariance matrices using a simple change of variables we transform the problem into a convex optimization that can be efficiently solved we then extend this idea to nondiagonal matrices using convexity on the manifold of positive definite matrices we regularize the problem using appropriately convex penalties these allow for shrinkage towards the identity matrix shrinkage towards a diagonal matrix shrinkage towards a given positive definite matrix and regularization of the condition number we demonstrate the advantages of these estimators using numerical simulations

semiparametric model building for regression models with time varying parameters abstract this paper considers the problem of semiparametric model building for linear regression models with potentially time varying coefficients by allowing the response variable and explanatory variables be jointly a nonstationary process the proposed methods are widely applicable to nonstationary and dependent observations we propose a local linear shrinkage method that can simultaneously achieve parameter estimation and variable selection its selection consistency and the favorable oracle property are established due to the fear of losing efficiency an information criterion is further proposed for distinguishing between time varying and time constant components numerical examples are presented to illustrate the proposed methods

improved subspace estimation for multivariate observations of high dimension the deterministic signals case we consider the problem of subspace estimation in situations where the number of available snapshots and the observation dimension are comparable in magnitude in this context traditional subspace methods tend to fail because the eigenvectors of the sample correlation matrix are heavily biased with respect to the true ones it has recently been suggested that this situation where the sample size is small compared to the observation dimension can be very accurately modeled by considering the asymptotic regime where the observation dimension m and the number of snapshots n converge to at the same rate using large random matrix theory results it can be shown that traditional subspace estimates are not consistent in this asymptotic regime furthermore new consistent subspace estimate can be proposed which outperform the standard subspace methods for realistic values of m and n the work carried out so far in this area has always been based on the assumption that the observations are random independent and identically distributed in the time domain the goal of this paper is to propose new consistent subspace estimators for the case where the source signals are modelled as unknown deterministic signals in practice this allows to use the proposed approach regardless of the statistical properties of the source signals in order to construct the proposed estimators new technical results concerning the almost sure location of the eigenvalues of sample covariance matrices of information plus noise complex gaussian models are established these results are believed to be of independent interest

local inference for locally stationary time series based on the empirical spectral measure abstract the time varying empirical spectral measure plays a major role in the treatment of inference problems for locally stationary processes the properties of the empirical spectral measure and related statistics are studied both when its index function is fixed or when dependent on the sample size in particular we prove a general central limit theorem several applications and examples are given including semiparametric whittle estimation local least squares estimation and spectral density estimation

geodesic convexity and covariance estimation geodesic convexity is a generalization of classical convexity which guarantees that all local minima of g convex functions are globally optimal we consider g convex functions with positive definite matrix variables and prove that kronecker products and logarithms of determinants are g convex we apply these results to two modern covariance estimation problems robust estimation in scaled gaussian distributions and kronecker structured models maximum likelihood estimation in these settings involves non convex minimizations we show that these problems are in fact g convex this leads to straight forward analysis allows the use of standard optimization methods and paves the road to various extensions via additional g convex regularization

a bayesian approach to adaptive detection in nonhomogeneous environments we consider the adaptive detection of a signal of interest embedded in colored noise when the environment is nonhomogeneous i e when the training samples used for adaptation do not share the same covariance matrix as the vector under test a bayesian framework is proposed where the covariance matrices of the primary and the secondary data are assumed to be random with some appropriate joint distribution the prior distributions of these matrices require a rough knowledge about the environment this provides a flexible yet simple knowledge aided model where the degree of nonhomogeneity can be tuned through some scalar variables within this framework an approximate generalized likelihood ratio test is formulated accordingly two bayesian versions of the adaptive matched filter are presented where the conventional maximum likelihood estimate of the primary data covariance matrix is replaced either by its minimum mean square error estimate or by its maximum a posteriori estimate two detectors require generating samples distributed according to the joint posterior distribution of primary and secondary data covariance matrices this is achieved through the use of a gibbs sampling strategy numerical simulations illustrate the performances of these detectors and compare them with those of the conventional adaptive matched filter

the joint graphical lasso for inverse covariance estimation across multiple classes we consider the problem of estimating multiple related gaussian graphical models from a high dimensional data set with observations belonging to distinct classes we propose the joint graphical lasso which borrows strength across the classes in order to estimate multiple graphical models that share certain characteristics such as the locations or weights of nonzero edges our approach is based upon maximizing a penalized log likelihood we employ generalized fused lasso or group lasso penalties and implement a fast admm algorithm to solve the corresponding convex optimization problems the performance of the proposed method is illustrated through simulated and real data examples

high dimensional covariance matrix estimation using a factor model abstract high dimensionality comparable to sample size is common in many statistical problems we examine covariance matrix estimation in the asymptotic framework that the dimensionality p tends to as the sample size n increases motivated by the arbitrage pricing theory in finance a multi factor model is employed to reduce dimensionality and to estimate the covariance matrix the factors are observable and the number of factors k is allowed to grow with p we investigate the impact of p and k on the performance of the model based covariance matrix estimator under mild assumptions we have established convergence rates and asymptotic normality of the model based estimator its performance is compared with that of the sample covariance matrix we identify situations under which the factor approach increases performance substantially or marginally the impacts of covariance matrix estimation on optimal portfolio allocation and portfolio risk assessment are studied the asymptotic results are supported by a thorough simulation study

factorization of operators ii we extend the methods of a previous paper j math phys 21 2508 1980 factorizing the general scalar third order differential operator and obtain a miura transformation for the boussinesq equation we give a general factorized eigenvalue problem we also give a hamiltonian structure associated with the factorized eigenvalue problem we derive several isospectral flows some of klein gordon type

the polynomial spectral problem of arbitrary order a general form of the integrable equations and backlund transformations the generalisation of the akns method ablowitz et al 1974 to the matrix polynomial spectral problem of arbitrary order is given both the general form of the integrable partial differential equations and their backlund transformations are described

linear integral transformations and hierarchies of integrable nonlinear evolution equations abstract integrable hierarchies of nonlinear evolution equations are investigated on the basis of linear integral equations these are riemann hilbert type of integral transformations which leave invariant an infinite sequence of ordinary differential matrix equations of increasing order in an indefinite parameter k the potential matrices in these equations obey a set of nonlinear recursion relations leading to a heirarchy of nonlinear partial differential equations in decreasing order the same equations give rise to a reciprocal hierarchy associated with heisenberg ferromagnet type of equations central in the treatment is an embedding of the hierarchy into an infinite matrix structure which is constructed on the basis of the integral equations in terms of this infinite matrix structure the equations governing the hierarchies become quite simple furthermore it leads in a straightforward way to various generalizations such as to other types of linear spectral problems multicomponent system and lattice equations generalizations to equations associated with noncommuting flows follow as a direct consequence of the treatment finally some results on conserved densities and the hamiltonian structure are briefly discussed

recursive estimation of a drifted autoregressive parameter suppose the x 0 x n are observations of a one dimensional stochastic dynamic process described by autoregression equations when the autoregressive parameter is drifted with time i e it is some function oftime 0 n on with k k n the function t is assumed to belong a priori to a predetermined nonparametric class of functions satisfying the lipschitz smoothness condition at each time point t those observations are accessible which have been obtained during the preceding time interval a recursive algorithm is proposed to estimate t under some conditions on the model we derive the rate of convergence of the proposed estimator when the frequency of observations n tends to infinity

eigen inference for energy estimation of multiple sources in this paper a new method is introduced to blindly estimate the transmit power of multiple signal sources in multiantenna fading channels when the number of sensing devices and the number of available samples are sufficiently large compared to the number of sources recent advances in the field of large dimensional random matrix theory are used that result in a simple and computationally efficient consistent estimator of the power of each source a criterion to determine the minimum number of sensors and the minimum number of samples required to achieve source separation is then introduced simulations are performed that corroborate the theoretical claims and show that the proposed power estimator largely outperforms alternative power inference techniques

stability of recursive stochastic tracking algorithms first the paper gives a stability study for the random linear equation x n 1 i a n x n it is shown that for a quite general class of random matrices a n of interest the stability of such a vector equation can be guaranteed by that of a corresponding scalar linear equation for which various results are given without requiring stationary or mixing conditions then these results are applied to the main topic of the paper i e to the estimation of time varying parameters in linear stochastic systems giving a unified stability condition for various tracking algorithms including the standard kalman filter least mean squares and least squares with forgetting factor

fluctuations of spiked random matrix models and failure diagnosis in sensor networks in this paper the joint fluctuations of the extreme eigenvalues and eigenvectors of a large dimensional sample covariance matrix are analyzed when the associated population covariance matrix is a finite rank perturbation of the identity matrix corresponding to the so called spiked model in random matrix theory the asymptotic fluctuations as the matrix size grows large are shown to be intimately linked with matrices from the gaussian unitary ensemble when the spiked population eigenvalues have unit multiplicity the fluctuations follow a central limit theorem this result is used to develop an original framework for the detection and diagnosis of local failures in large sensor networks for known or unknown failure magnitude

hypercomplex correlation techniques for vector images correlation techniques have been applied to almost every area of signal processing over the past century yet their use has in general been limited to scalar signals while there have been implementations in multichannel applications these can be characterized as a combination of single channel processes true vector correlation techniques with global and interchannel measures have only recently been demonstrated and are still in their infancy by comparison this paper describes our work on vector correlation based on the use of hypercomplex fourier transforms and presents for the first time a unified theory behind the information contained in the peak of a vector correlation response by using example applications for color images we also demonstrate some of the practical implications together with our latest results

model selection and estimation in the gaussian graphical model summary we propose penalized likelihood methods for estimating the concentration matrix in the gaussian graphical model the methods lead to a sparse and shrinkage estimator of the concentration matrix that is positive definite and thus conduct model selection and estimation simultaneously the implementation of the methods is nontrivial because of the positive definite constraint on the concentration matrix but we show that the computation can be done effectively by taking advantage of the efficient maxdet algorithm developed in convex optimization we propose a bic type criterion for the selection of the tuning parameter in the penalized likelihood methods the connection between our methods and existing methods is illustrated simulations and real examples demonstrate the competitive performance of the new methods

tracy widom limit for the largest eigenvalue of a large class of complex sample covariance matrices we consider the asymptotic fluctuation behavior of the largest eigenvalue of certain sample covariance matrices in the asymptotic regime where both dimensions of the corresponding data matrix go to infinity more precisely let x be an n x p matrix and let its rows be i i d complex normal vectors with mean 0 and covariance p we show that for a large class of covariance matrices p the largest eigenvalue of x x is asymptotically distributed after recentering and rescaling as the tracy widom distribution that appears in the study of the gaussian unitary ensemble we give explicit formulas for the centering and scaling sequences that are easy to implement and involve only the spectral distribution of the population covariance n and p the main theorem applies to a number of covariance models found in applications for example well behaved toeplitz matrices as well as covariance matrices whose spectral distribution is a sum of atoms under some conditions on the mass of the atoms are among the models the theorem can handle generalizations of the theorem to certain spiked versions of our models and a s results about the largest eigenvalue are given we also discuss a simple corollary that does not require normality of the entries of the data matrix and some consequences for applications in multivariate statistics

some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size this paper analyzes whether standard covariance matrix tests work when dimensionality is large and in particular larger than sample size in the latter case the singularity of the sample covariance matrix makes likelihood ratio tests degenerate but other tests based on quadratic forms of sample covariance matrix eigenvalues remain well defined we study the consistency property and limiting distribution of these tests as dimensionality and sample size go to infinity together with their ratio converging to a finite nonzero limit we find that the existing test for sphericity is robust against high dimensionality but not the test for equality of the covariance matrix to a given matrix for the latter test we develop a new correction to the existing test statistic that makes it robust against high dimensionality

covariance regularized regression and classification for high dimensional problems in recent years many methods have been developed for regression in high dimensional settings we propose covariance regularized regression a family of methods that use a shrunken estimate of the inverse covariance matrix of the features in order to achieve superior prediction an estimate of the inverse covariance matrix is obtained by maximizing its log likelihood under a multivariate normal model subject to a constraint on its elements this estimate is then used to estimate coefficients for the regression of the response onto the features we show that ridge regression the lasso and the elastic net are special cases of covariance regularized regression and we demonstrate that certain previously unexplored forms of covariance regularized regression can outperform existing methods in a range of situations the covariance regularized regression framework is extended to generalized linear models and linear discriminant analysis and is used to analyze gene expression data sets with multiple class and survival outcomes

time varying autoregressive tvar models for multiple radar observations we consider the adaptive radar problem where the properties of the nonstationary clutter signals can be estimated using multiple observations of radar returns from a number of sufficiently homogeneous range azimuth resolution cells we derive a method for approximating an arbitrary hermitian covariance matrix by a time varying autoregressive model of order m tvar m that is based on the dym gohberg band matrix extension technique which gives the unique tvar m model for any nondegenerate covariance matrix we demonstrate that the dym gohberg transformation of the sample covariance matrix gives the maximum likelihood ml estimate of the tvar m covariance matrix we introduce an example of tvar m clutter modeling for high frequency over the horizon radar that demonstrates its practical importance

a comparison between hmlp and hrbf for attitude control in this paper the problem of controlling the attitude of a rigid body such as a spacecraft in three dimensional space is approached by introducing two new control strategies developed in hypercomplex algebra the proposed approaches are based on two parallel controllers both derived in quaternion algebra the first is a feedback controller of the proportional derivative pd type while the second is a feedforward controller which is implemented either by means of a hypercomplex multilayer perceptron hmlp neural network or by means of a hypercomplex radial basis function hrbf neural network several simulations show the performance of the two approaches the results are also compared with a classical pd controller and with an adaptive controller showing the improvements obtained by using neural networks especially when an external disturbance acts on the rigid body in particular the hmlp network gave better results when considering trajectories not presented during the learning phase

semiparametric estimation by model selection for locally stationary processes over recent decades increasingly more attention has been paid to the problem of how to fit a parametric model of time series with time varying parameters a typical example is given by autoregressive models with time varying parameters we propose a procedure to fit such time varying models to general non stationary processes the estimator is a maximum whittle likelihood estimator on sieves the results do not assume that the observed process belongs to a specific class of time varying parametric models we discuss in more detail the fitting of time varying ar p processes for which we treat the problem of the selection of the order p and we propose an iterative algorithm for the computation of the estimator a comparison with model selection by akaike s information criterion is provided through simulations

yang baxterization of algebraic solutions of reflection equation for su 1 1 q model a quadratic relation between the central elements of reflection algebra for su 1 1 q model is obtained yang baxterization of algebraic solutions to reflection equation for su 1 1 q model is presented in terms of this relation

group symmetric robust covariance estimation in this paper we consider tyler s robust covariance rm m estimator under group symmetry constraints we assume that the covariance matrix is invariant to the conjugation action of a unitary matrix group referred to as group symmetry examples of group symmetric structures include circulant perhermitian and proper quaternion matrices we introduce a group symmetric version of tyler s estimator styler and provide an iterative fixed point algorithm to compute it the classical results claim that at least n p 1 sample points in general position are necessary to ensure the existence and uniqueness of tyler s estimator where p is the ambient dimension we show that the styler requires significantly less samples in some groups even two samples are enough to guarantee its existence and uniqueness in addition in the case of elliptical populations we provide high probability bounds on the error of the styler these too quantify the advantage of exploiting the symmetry structure finally these theoretical results are supported by numerical simulations

time varying autoregressive moving average models for covariance estimation we consider large scale covariance estimation using a small number of samples in applications where there is a natural ordering between the random variables the two classical approaches to this problem rely on banded covariance and banded inverse covariance structures corresponding to time varying moving average ma and autoregressive ar models respectively motivated by this analogy to spectral estimation and the well known modeling power of autoregressive moving average arma processes we propose a novel time varying arma covariance structure similarly to known results in the context of ar and ma we address the completion of an arma covariance matrix from its main band and its estimation based on random samples finally we examine the advantages of our proposed methods using numerical experiments

covariance matrix estimation with heterogeneous samples we consider the problem of estimating the covariance matrix mp of an observation vector using heterogeneous training samples i e samples whose covariance matrices are not exactly mp more precisely we assume that the training samples can be clustered into k groups each one containing lk snapshots sharing the same covariance matrix mk furthermore a bayesian approach is proposed in which the matrices mk are assumed to be random with some prior distribution we consider two different assumptions for mp in a fully bayesian framework mp is assumed to be random with a given prior distribution under this assumption we derive the minimum mean square error mmse estimator of mp which is implemented using a gibbs sampling strategy moreover a simpler scheme based on a weighted sample covariance matrix scm is also considered the weights minimizing the mean square error mse of the estimated covariance matrix are derived furthermore we consider estimators based on colored or diagonal loading of the weighted scm and we determine theoretically the optimal level of loading finally in order to relax the a priori assumptions about the covariance matrix mp the second part of the paper assumes that this matrix is deterministic and derives its maximum likelihood estimator numerical simulations are presented to illustrate the performance of the different estimation schemes

the general uq sl 2 invariant xxz integrable quantum spin chain the boundary conditions are constructed for the general xxz quantum integrable spin chain such that the resulting system is invariant under the quantum algebra uq sl 2 action the result generalizes that due to pasquier and saleur 1990 obtained for spin 1 2 and spin 1

radar detection of distributed targets in homogeneous interference whose inverse covariance structure is defined via unitary invariant functions in this paper we deal with the problem of detecting an extended target embedded in homogeneous gaussian interference with unknown but structured covariance matrix we model the possible target echo from each range bin under test as a deterministic signal with an unknown scaling factor accounting for the target response at the design stage we exploit some a priori knowledge about the operating environment enforcing the inverse interference plus noise covariance matrix to belong to a set described via unitary invariant continuous functions hence we derive the constrained maximum likelihood ml estimates of the unknown parameters under both the h0 and h1 hypotheses and design the generalized likelihood ratio test glrt for the considered decision problem at the analysis stage we assess the performance of the devised glrt for some covariance matrix uncertainty sets of practical relevance both for spatial and doppler processing the results highlight that correct use of the a priori knowledge can lead to a detection performance quite close to the optimum receiver which supposes the perfect knowledge of the interference plus noise covariance matrix

on the structure of equations integrable by the arbitrary order linear spectral problem the general form of partial differential equations integrable by the arbitrary order linear spectral problem is found the groups of backlund transformations corresponding to these equations are constructed it is shown that partial differential equations of the class under study are hamiltonian ones some reductions of general equations are considered in particular the hamiltonian structure of the generalisations of the sine gordon equation to the groups gl n su n and so n at arbitrary n is proved

joint estimation of inverse covariance matrices lying in an unknown subspace we consider the problem of joint estimation of inverse covariance matrices lying in an unknown subspace of the linear space of symmetric matrices we perform the estimation using groups of measurements with different covariances assuming the inverse covariances span a low dimensional subspace our aim is to determine this subspace and to exploit this knowledge in order to improve the estimation we develop a novel optimization algorithm discovering and exploiting the underlying low dimensional subspace we provide a computationally efficient algorithm and derive a tight upper performance bound numerical simulations on synthetic and real world data are presented to illustrate the performance benefits of the algorithm

a distribution free m estimator of multivariate scatter the existence and uniqueness of a limiting form of a huber type m estimator of multivariate scatter is established under certain conditions on the observed sample these conditions hold with probability one when sampling randomly from a continuous multivariate distribution the existence of the estimator is proven by showing that it is the limiting point of a specific algorithm hence the proof is constructive for continuous populations the estimator of multivariate scatter is shown to be strongly consistent and asymptotically normal an important property of the estimator is that its asymptotic distribution is distribution free with respect to the class of continuous elliptically distributed populations this distribution free property also holds for the finite sample size distribution when the location parameter is known in addition the estimator is the most robust estimator of the scatter matrix of an elliptical distribution in the sense of minimizing the maximum asymptotic variance

joint estimation of multiple dependent gaussian graphical models with applications to mouse genomics gaussian graphical models are widely used to represent conditional dependence among random variables in this paper we propose a novel estimator for data arising from a group of gaussian graphical models that are themselves dependent a motivating example is that of modeling gene expression collected on multiple tissues from the same individual here the multivariate outcome is affected by dependencies acting not only at the level of the specific tissues but also at the level of the whole body existing methods that assume independence among graphs are not applicable in this case to estimate multiple dependent graphs we decompose the problem into two graphical layers the systemic layer which affects all outcomes and thereby induces cross graph dependence and the category specific layer which represents graph specific variation we propose a graphical em technique that estimates both layers jointly establish estimation consistency and selection sparsistency of the proposed estimator and confirm by simulation that the em method is superior to a simple one step method we apply our technique to mouse genomics data and obtain biologically plausible results

multiple matrix gaussian graphs estimation matrix valued data where the sampling unit is a matrix consisting of rows and columns of measurements are emerging in numerous scientific and business applications matrix gaussian graphical model is a useful tool to characterize the conditional dependence structure of rows and columns in this article we employ nonconvex penalization to tackle the estimation of multiple graphs from matrix valued data under a matrix normal distribution we propose a highly efficient nonconvex optimization algorithm that can scale up for graphs with hundreds of nodes we establish the asymptotic properties of the estimator which requires less stringent conditions and has a sharper probability error bound than existing results we demonstrate the efficacy of our proposed method through both simulations and real functional magnetic resonance imaging analyses

knowledge aided covariance matrix estimation and adaptive detection in compound gaussian noise let x be a signal to be sparsely decomposed over a redundant dictionary a i e a sparse coefficient vector s has to be found such that x as it is known that this problem is inherently unstable against

joint covariance estimation with mutual linear structure we consider the problem of joint estimation of structured covariance matrices assuming the structure is unknown estimation is achieved using heterogeneous training sets namely given groups of measurements coming from centered populations with different covariances our aim is to determine the mutual structure of these covariance matrices and estimate them supposing that the covariances span a low dimensional affine subspace in the space of symmetric matrices we develop a new efficient algorithm discovering the structure and using it to improve the estimation our technique is based on the application of principal component analysis in the matrix space we also derive an upper performance bound of the proposed algorithm in the gaussian scenario and compare it with the cramer rao lower bound numerical simulations are presented to illustrate the performance benefits of the proposed method

joint estimation of multiple graphical models gaussian graphical models explore dependence relationships between random variables through the estimation of the corresponding inverse covariance matrices in this paper we develop an estimator for such models appropriate for data from several graphical models that share the same variables and some of the dependence structure in this setting estimating a single graphical model would mask the underlying heterogeneity while estimating separate models for each category does not take advantage of the common structure we propose a method that jointly estimates the graphical models corresponding to the different categories present in the data aiming to preserve the common structure while allowing for differences between the categories this is achieved through a hierarchical penalty that targets the removal of common zeros in the inverse covariance matrices across categories we establish the asymptotic consistency and sparsity of the proposed estimator in the high dimensional case and illustrate its performance on a number of simulated networks an application to learning semantic connections between terms from webpages collected from computer science departments is included

statistical analysis for the angular central gaussian distribution on the sphere summary the angular central gaussian distribution is an alternative to the bingham distribution for modeling antipodal symmetric directional data in this paper the statistical theory for the angular central gaussian model is presented some topics treated are maximum likelihood estimation of the parameters testing for uniformity and circularity and principal components analysis comparisons to methods based upon the sample second moments are made via an example

partial correlation estimation by joint sparse regression models this article features online supplementary material in this article we propose a computationally efficient approach space sparse partial correlation estimation for selecting nonzero partial correlations under the high dimension low sample size setting this method assumes the overall sparsity of the partial correlation matrix and employs sparse regression techniques for model fitting we illustrate the performance of space by extensive simulation studies it is shown that space performs well in both nonzero partial correlation selection and the identification of hub variables and also outperforms two existing methods we then apply space to a microarray breast cancer dataset and identify a set of hub genes that may provide important insights on genetic regulatory networks finally we prove that under a set of suitable assumptions the proposed procedure is asymptotically consistent in terms of model selection and parameter estimation

robust shrinkage estimation of high dimensional covariance matrices we address high dimensional covariance estimation for elliptical distributed samples which are also known as spherically invariant random vectors sirv or compound gaussian processes specifically we consider shrinkage methods that are suitable for high dimensional problems with a small number of samples large p small n we start from a classical robust covariance estimator tyler 1987 which is distribution free within the family of elliptical distribution but inapplicable when n p using a shrinkage coefficient we regularize tyler s fixed point iterations we prove that for all n and p the proposed fixed point iterations converge to a unique limit regardless of the initial condition next we propose a simple closed form and data dependent choice for the shrinkage coefficient which is based on a minimum mean squared error framework simulations demonstrate that the proposed method achieves low estimation error and is robust to heavy tailed samples finally as a real world application we demonstrate the performance of the proposed technique in the context of activity intrusion detection using a wireless sensor network

algebraic structures related to reflection equations quadratic algebras related to the reflection equations are introduced they are quantum group comodule algebras the quantum group fq gl 2 is taken as the example the properties of the algebras centre representation realizations real forms fusion procedure etc as well as the generalizations are discussed

on the linear convergence of the alternating direction method of multipliers we analyze the convergence rate of the alternating direction method of multipliers admm for minimizing the sum of two or more nonsmooth convex separable functions subject to linear constraints previous analysis of the admm typically assumes that the objective function is the sum of only two convex functions defined on two separable blocks of variables even though the algorithm works well in numerical experiments for three or more blocks moreover there has been no rate of convergence analysis for the admm without strong convexity in the objective function in this paper we establish the global r linear convergence of the admm for minimizing the sum of any number of convex separable functions assuming that a certain error bound condition holds true and the dual stepsize is sufficiently small such an error bound condition is satisfied for example when the feasible set is a compact polyhedron and the objective function consists of a smooth strictly convex function composed with a linear mapping and a nonsmooth ell 1 l1 regularizer this result implies the linear convergence of the admm for contemporary applications such as lasso without assuming strong convexity of the objective function

on the use of isospectral eigenvalue problems for obtaining hereditary symmetries for hamiltonian systems we present an algorithmic method for obtaining an hereditary symmetry the generalized squared eigenfunction operator from a given isospectral eigenvalue problem this method is applied to the n n eigenvalue problem considered by ablowitz and haberman and to the eigenvalue problem considered by alonso the relevant hamiltonian formulations are also determined finally an alternative method is presented in the case two evolution equations are related by a miura type transformation and their hamiltonian formulations are known

model selection through sparse maximum likelihood estimation for multivariate gaussian or binary data we consider the problem of estimating the parameters of a gaussian or binary distribution in such a way that the resulting undirected graphical model is sparse our approach is to solve a maximum likelihood problem with an added l1 norm penalty term the problem as formulated is convex but the memory requirements and complexity of existing interior point methods are prohibitive for problems with more than tens of nodes we present two new algorithms for solving problems with at least a thousand nodes in the gaussian case our first algorithm uses block coordinate descent and can be interpreted as recursive l1 norm penalized regression our second algorithm based on nesterov s first order method yields a complexity estimate with a better dependence on problem size than existing interior point methods using a log determinant relaxation of the log partition function wainwright and jordan 2006 we show that these same algorithms can be used to solve an approximate sparse maximum likelihood problem for the binary case we test our algorithms on synthetic data as well as on gene expression and senate voting records data

modified subspace algorithms for doa estimation with large arrays this paper proposes the use of a new generalized asymptotic paradigm in order to analyze the performance of subspace based direction of arrival doa estimation in array signal processing applications instead of assuming that the number of samples is high whereas the number of sensors antennas remains fixed the asymptotic situation analyzed herein assumes that both quantities tend to infinity at the same rate this asymptotic situation provides a more accurate description of a potential situation where these two quantities are finite and hence comparable in magnitude it is first shown that both music and ssmusic are inconsistent when the number of antennas sensors increases without bound at the same rate as the sample size this is done by analyzing and deriving closed form expressions for the two corresponding asymptotic cost functions by examining these asymptotic cost functions one can establish the minimum number of samples per antenna needed to resolve closely spaced sources in this asymptotic regime next two alternative estimators are constructed that are strongly consistent in the new asymptotic situation i e they provide consistent doa estimates not only when the number of snapshots goes to infinity but also when the number of sensors antennas increases without bound at the same rate these estimators are inspired by the theory of g estimation and are therefore referred to as g music and g ssmusic respectively simulations show that the proposed algorithms outperform their traditional counterparts in finite sample size situations although they still present certain limitations

on a perturbation approach for the analysis of stochastic tracking algorithms in this paper a perturbation expansion technique is introduced to decompose the tracking error of a general adaptive tracking algorithm in a linear regression model this method results in a tracking error bound and tight approximate expressions for the moments of the tracking error these expressions allow the evaluation both qualitatively and quantitatively of the impact of several factors on the tracking error performance

aggregation of predictors for non stationary sub linear processes and online adaptive forecasting of time varying autoregressive processes in this work we study the problem of aggregating a finite number of predictors for non stationary sub linear processes we provide oracle inequalities relying essentially on three ingredients 1 a uniform bound of the ell 1 norm of the time varying sub linear coefficients 2 a lipschitz assumption on the predictors and 3 moment conditions on the noise appearing in the linear representation two kinds of aggregations are considered giving rise to different moment conditions on the noise and more or less sharp oracle inequalities we apply this approach for deriving an adaptive predictor for locally stationary time varying autoregressive tvar processes it is obtained by aggregating a finite number of well chosen predictors each of them enjoying an optimal minimax convergence rate under specific smoothness conditions on the tvar coefficients we show that the obtained aggregated predictor achieves a minimax rate while adapting to the unknown smoothness to prove this result a lower bound is established for the minimax rate of the prediction risk for the tvar process numerical experiments complete this study an important feature of this approach is that the aggregated predictor can be computed recursively and is thus applicable in an online prediction context

a new parametric glrt for multichannel adaptive signal detection a parametric generalized likelihood ratio test glrt for multichannel signal detection in spatially and temporally colored disturbance was recently introduced by modeling the disturbance as a multichannel autoregressive ar process the detector however involves a highly nonlinear maximum likelihood estimation procedure which was solved via a two dimensional iterative search method initialized by a suboptimal estimator in this paper we present a simplified glrt along with a new estimator for the problem both the estimator and the glrt are derived in closed form at considerably lower complexity with adequate training data the new glrt achieves a similar detection performance as the original one however for the more interesting case of limited training the original glrt may become inferior due to poor initialization because of its simpler form the new glrt also offers additional insight into the parametric multichannel signal detection problem the performance of the proposed detector is assessed using both a simulated dataset which was generated using multichannel ar models and the kassper dataset a widely used dataset with challenging heterogeneous effects found in real world environments

regularized discriminant analysis abstract linear and quadratic discriminant analysis are considered in the small sample high dimensional setting alternatives to the usual maximum likelihood plug in estimates for the covariance matrices are proposed these alternatives are characterized by two parameters the values of which are customized to individual situations by jointly minimizing a sample based estimate of future misclassification risk computationally fast implementations are presented and the efficacy of the approach is examined through simulation studies and application to data these studies indicate that in many circumstances dramatic gains in classification accuracy can be achieved

performance analysis of tyler s covariance estimator this paper analyzes the performance of tyler s m estimator of the scatter matrix in elliptical populations we focus on the non asymptotic setting and derive estimation error bounds depending on the number of samples n and the dimension p we show that under mild conditions the squared frobenius norm of the error of the inverse estimator decays like p 2 n with high probability

yang baxterization of reflection equations for r with two distinct eigenvalues a condition of yang baxterization of reflection equations for r with two distinct eigenvalues is given constant solutions to reflection equations for r related to fundamental representations of slq n and slq 1 1 and for r of the eight vertex model are yang baxterized

nonparametric regression for locally stationary time series in this paper we study nonparametric models allowing for locally stationary regressors and a regression function that changes smoothly over time these models are a natural extension of time series models with time varying coecients we introduce a kernel based method to estimate the time varying regression function and provide asymptotic theory for our estimates moreover we show that the main conditions of the theory are satised for a large class of nonlinear autoregressive processes with a time varying regression function finally we examine structured models where the regression function splits up into time varying additive components as will be seen estimation in these models does not suer from the curse of dimensionality we complement the technical analysis of the paper by an application to nancial data

boundary conditions for integrable quantum systems a new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering r matrix method the method proposed allows the author to treat open quantum chains with appropriate boundary terms in the hamiltonian the general considerations are applied to the xxz and xyz models the nonlinear schrodinger equation and toda chain

better subset regression using the nonnegative garrote a new method called the nonnegative nn garrote is proposed for doing subset regression it both shrinks and zeroes coefficients in tests on real and simulated data it produces lower prediction error than ordinary subset selection it is also compared to ridge regression if the regression equations generated by a procedure do not change drastically with small changes in the data the procedure is called stable subset selection is unstable ridge is very stable and the nn garrote is intermediate simulation results illustrate the effects of instability on prediction error

regularized estimation of large covariance matrices this paper considers estimating a covariance matrix of p variables from n observations by either banding the sample covariance matrix or estimating a banded version of the inverse of the covariance we show that these estimates are consistent in the operator norm as long as logp 2 n 0 and obtain explicit rates the results are uniform over some fairly natural well conditioned families of covariance matrices we also introduce an analogue of the gaussian white noise model and show that if the population covariance is embeddable in that model and well conditioned then the banded approximations produce consistent estimates of the eigenvalues and associated eigenvectors of the covariance matrix the results can be extended to smooth versions of banding and to non gaussian distributions with su ciently short tails a resampling approach is proposed for choosing the banding parameter in practice this approach is illustrated numerically on both simulated and real data

inference of time varying regression models we consider parameter estimation hypothesis testing and variable selection for partially time varying coefficient models our asymptotic theory has the useful feature that it can allow dependent nonstationary error and covariate processes with a two stage method the parametric component can be estimated with a n 1 2 convergence rate a simulation assisted hypothesis testing procedure is proposed for testing significance and parameter constancy we further propose an information criterion that can consistently select the true set of significant predictors our method is applied to autoregressive models with time varying coefficients simulation results and a real data application are provided

embedding nonnegative definite toeplitz matrices in nonnegative definite circulant matrices with application to covariance estimation the class of nonnegative definite toeplitz matrices that can be embedded in nonnegative definite circulant matrices of a larger size is characterized an equivalent characterization in terms of the spectrum of the underlying process is also presented together with the corresponding extremal processes it is shown that a given finite duration sequence rho can be extended to be the covariance of a periodic stationary processes whenever the toeplitz matrix r generated by this sequence is strictly positive definite the sequence rho 1 cos alpha cos 2 alpha with alpha pi irrational which has a unique nonperiodic extension as a covariance sequence demonstrates that the strictness is needed a simple constructive proof supplies a bound on the abovementioned period in terms of the minimal eigenvalue of r it also yields under the same conditions an extension of rho to covariances that eventually decay to zero for the maximum likelihood estimate of the covariance of a stationary gaussian process the extension length required for using the estimate maximize iterative algorithm is determined

critical behaviour in aperiodic systems the author considers layered systems with aperiodically modulated couplings and study the effect of inhomogeneity on the critical behaviour using scaling arguments a relevance irrelevance type criterion is formulated and for relevant inhomogeneities the scaling form of singular quantities is determined the author has performed exact calculations for the surface magnetization of aperiodic quantum ising chains and for a 1 1 dimensional directed polymer in a longitudinal aperiodic environment the results for both systems are in accord with scaling considerations for a marginal aperiodicity the critical exponents are found to be non universal

joint estimation of multiple precision matrices with common structures estimation of inverse covariance matrices known as precision matrices is important in various areas of statistical analysis in this article we consider estimation of multiple precision matrices sharing some common structures in this setting estimating each precision matrix separately can be suboptimal as it ignores potential common structures this article proposes a new approach to parameterize each precision matrix as a sum of common and unique components and estimate multiple precision matrices in a constrained l1 minimization framework we establish both estimation and selection consistency of the proposed estimator in the high dimensional setting the proposed estimator achieves a faster convergence rate for the common structure in certain cases our numerical examples demonstrate that our new estimator can perform better than several existing methods in terms of the entropy loss and frobenius loss an application to a glioblastoma cancer data set reveals some interesting gene networks across multiple cancer subtypes

covariance estimation in decomposable gaussian graphical models graphical models are a framework for representing and exploiting prior conditional independence structures within distributions using graphs in the gaussian case these models are directly related to the sparsity of the inverse covariance concentration matrix and allow for improved covariance estimation with lower computational complexity we consider concentration estimation with the mean squared error mse as the objective in a special type of model known as decomposable this model includes for example the well known banded structure and other cases encountered in practice our first contribution is the derivation and analysis of the minimum variance unbiased estimator mvue in decomposable graphical models we provide a simple closed form solution to the mvue and compare it with the classical maximum likelihood estimator mle in terms of performance and complexity next we extend the celebrated stein s unbiased risk estimate sure to graphical models using sure we prove that the mse of the mvue is always smaller or equal to that of the biased mle and that the mvue itself is dominated by other approaches in addition we propose the use of sure as a constructive mechanism for deriving new covariance estimators similarly to the classical mle all of our proposed estimators have simple closed form solutions but result in a significant reduction in mse

smallest singular value of a random rectangular matrix we prove an optimal estimate of the smallest singular value of a random sub gaussian matrix valid for all dimensions for an nn matrix a with inde pendent and identically distributed sub gaussian entries the smallest singular value of a is at least of the order p np n 1 with high probability a sharp

discrete fractal dimensions of the ranges of random walks in mathbb z d associate with random conductances let x x t t ge 0 be a continuous time random walk in an environment of i i d random conductances mu e in 1 infty e in e d where e d is the set of nonoriented nearest neighbor bonds on the euclidean lattice mathbb z d and d 3 let rm r x in mathbb z d x t x rm for some t ge 0 be the range of x it is proved that for almost every realization of the environment dimh r dimp r 2 almost surely where dimh and dimp denote respectively the discrete hausdorff and packing dimension furthermore given any set a subseteq mathbb z d a criterion for a to be hit by x t for arbitrarily large t 0 is given in terms of dimh a similar results for bouchoud s trap model in mathbb z d d 3 are also proven

on the structure of nonlinear evolution equations integrable by the z2 graded quadratic bundle the general form of nonlinear evolution equations and their backlund transformations connected with the quadratic in the spectral parameter z2 graded arbitrary order linear matrix spectral problem is found the hamiltonian structure of the integrable equations is discussed the infinite family of poisson brackets which corresponds to the class of equations under consideration is given relativistic invariant integrable equations are considered the explicit forms of elementary and soliton backlund transformations are found a nonlinear superposition principle is obtained

reduction technique for matrix nonlinear evolution equations solvable by the spectral transform the main purpose of this paper is to describe a technique of reduction whereby from the class of evolution equations for matrices of order n solvable via the spectral transform associated to the matrix linear schrodinger eigenvalue problem one derives subclasses of nonlinear evolution equations involving less than n2 fields to illustrate the method from the equations for matrices of order 2 two subclasses of equations for 2 fields rather than 4 are obtained the first class coincides or rather includes that solvable via the spectral transform associated to the generalized zakharov shabat spectral problem further reduction to nonlinear evolution equations for a single field reproduces a number of well known equations but also yields a novel one highly nonlinear the second class also yields highly nonlinear equations some examples are given including another novel evolution equation for a single field

on the asymptotics of maronna s robust pca the eigenvalue decomposition evd parameters of the second order statistics are ubiquitous in statistical analysis and signal processing notably the evd of the m estimators of the scatter matrix is a popular choice to perform robust probabilistic pca or other dimension reduction related applications towards the goal of characterizing this process this paper derives new asymptotics for the evd parameters i e eigenvalues eigenvectors and principal subspace of m estimators in the context of complex elliptically symmetric distributions first their gaussian asymptotic distribution is obtained by extending standard results on the sample covariance matrix in the gaussian context second their convergence towards the evd parameters of a gaussian core wishart equivalent is derived this second result represents the main contribution in the sense that it quantifies when it is acceptable to directly rely on well established results on the evd of wishart distributed matrix for characterizing the evd of m estimators finally some examples intrinsic bias analysis rank estimation and low rank adaptive filtering illustrate where the obtained results can be leveraged

cramer rao lower bounds on covariance matrix estimation for complex elliptically symmetric distributions this paper introduces the cramer rao lower bounds crlbs for the scatter matrix of complex elliptically symmetric distributions and compares them to the performance of the constrained ml estimators in the particular cases of complex gaussian generalized gaussian gg and t distributed observation vectors numerical results confirm the goodness of the ml estimators and the advantage of taking into proper account a constraint on the matrix trace for small data size the work is completed with the comparison with the performance of tyler s matrix estimator that shows a very robust behavior in almost all the analyzed cases and with the crlbs for the complex angular elliptical distributions whose tyler s estimator is the ml one

optimal rates of convergence for estimating toeplitz covariance matrices toeplitz covariance matrices are used in the analysis of stationary stochastic processes and a wide range of applications including radar imaging target detection speech recognition and communications systems in this paper we consider optimal estimation of large toeplitz covariance matrices and establish the minimax rate of convergence for two commonly used parameter spaces under the spectral norm the properties of the tapering and banding estimators are studied in detail and are used to obtain the minimax upper bound the results also reveal a fundamental difference between the tapering and banding estimators over certain parameter spaces the minimax lower bound is derived through a novel construction of a more informative experiment for which the minimax lower bound is obtained through an equivalent gaussian scale model and through a careful selection of a finite collection of least favorable parameters in addition optimal rate of convergence for estimating the inverse of a toeplitz covariance matrix is also established

covariance structure maximum likelihood estimates in compound gaussian noise existence and algorithm analysis recently a new adaptive scheme conte 1995 gini 1997 has been introduced for covariance structure matrix estimation in the context of adaptive radar detection under non gaussian noise this latter has been modeled by compound gaussian noise which is the product c of the square root of a positive unknown variable tau deterministic or random and an independent gaussian vector x c radictaux because of the implicit algebraic structure of the equation to solve we called the corresponding solution the fixed point fp estimate when tau is assumed deterministic and unknown the fp is the exact maximum likelihood ml estimate of the noise covariance structure while when tau is a positive random variable the fp is an approximate maximum likelihood aml this estimate has been already used for its excellent statistical properties without proofs of its existence and uniqueness the major contribution of this paper is to fill these gaps our derivation is based on some likelihood functions general properties like homogeneity and can be easily adapted to other recursive contexts moreover the corresponding iterative algorithm used for the fp estimate practical determination is also analyzed and we show the convergence of this recursive scheme ensured whatever the initialization