stepwise signal extraction via marginal likelihood this article studies the estimation of a stepwise signal to determine the number and locations of change points of the stepwise signal we formulate a maximum marginal likelihood estimator which can be computed with a quadratic cost using dynamic programming we carry out an extensive investigation on the choice of the prior distribution and study the asymptotic properties of the maximum marginal likelihood estimator we propose to treat each possible set of change points equally and adopt an empirical bayes approach to specify the prior distribution of segment parameters a detailed simulation study is performed to compare the effectiveness of this method with other existing methods we demonstrate our method on single molecule enzyme reaction data and on dna array comparative genomic hybridization cgh data our study shows that this method is applicable to a wide range of models and offers appealing results in practice supplementary materials for this article are available online

tightness of products of random matrices and stability of linear stochastic systems soit n la distribution d un produit de n matrices aleatoire i i d on etudie la compacite et la convergence de la suite n n on l applique aux equations differentielles stochastiques lineaires

a dependent data extension of wald s identity and its application to sequential test performance computation the development of an exponential transformation of measure not restricted to independent identically distributed data is presented for a class of semi markov processes known as markov additive ma process the ma measure transformation is applied to obtain a new form of the wald identity to demonstrate the utility of this extension to ma processes the application of sequential tests with markov data to error probability performance analysis is investigated in particular it is demonstrated how previous work dealing with robust error probability evaluation in the presence of uncertain serial dependency can be applied to sequential tests

a dimension formula for bernoulli convolutions we present a dynamical approach to the study of the singularity of infinitely convolved bernoulli measuresv for the golden section we introducev as the transverse measure of the maximum entropy measure on the repelling set invariant for contracting maps of the square the fat baker s transformation our approach strongly relies on the markov structure of the underlying dynamical system indeed if golden mean the fat baker s transformation has a very simple markov coding the ambiguity of order two of this coding which appears when projecting on the line due to passages for the central overlapping zone can be expressed by means of products of matrices of order two this product has a markov distribution inherited by the markov structure of the map the dimension of the projected measure is therefore associated to the growth of this product our dimension formula appears in a natural way as a version of the furstenberg guivarch formula our technique provides an explicit dimension formula and most important provides a formalism well suited for the multifractal analysis of this measure as we will show in a forthcoming paper

arbitrarily varying channels with constrained inputs and states random coding theorems are proved for discrete memoryless arbitrarily varying channels avcs with constraints on the transmitted codewords and channel state sequences two types of constraints are considered peak i e required for each n length sequence almost surely and average over the message set or over an ensemble for peak constraints on the codewords and on the channel state sequences the avc is shown to have a strong random coding capacity if the codewords and or the channel state sequences are constrained in the average sense the avcs do not possess strong capacities only epsilon capacities are shown to exist

optimal simultaneous detection and estimation under a false alarm constraint this paper addresses the problem of finite sample simultaneous detection and estimation which arises when estimation of signal parameters is desired but signal presence is uncertain in general a joint detection and estimation algorithm cannot simultaneously achieve optimal detection and optimal estimation performance we develop a multihypothesis testing framework for studying the tradeoffs between detection and parameter estimation classification for a finite discrete parameter set our multihypothesis testing problem is based on the worst case detection and worst case classification error probabilities of the class of joint detection and classification algorithms which are subject to a false alarm constraint this framework leads to the evaluation of greatest lower bounds on the worst case decision error probabilities and a construction of decision rules which achieve these lower bounds for illustration we apply these methods to signal detection order selection and signal classification for a multicomponent signal in noise model for two or fewer signals an snr of 3 db and signal space dimension of n 10 numerical results are obtained which establish the existence of fundamental tradeoffs between three performance criteria probability of signal detection probability of correct order selection and probability of correct classification furthermore based on numerical performance comparisons between our optimal decision rule and other suboptimal penalty function methods we observe that rissanen s 1978 order selection penalty method is nearly min max optimal in some nonasymptotic regimes

asymptotic optimality of invariant sequential probability ratio tests it is well known that wald s sprt for testing simple hypotheses based on i i d observations minimizes the expected sample size both under the null and under the alternative hypotheses among all tests with the same or smaller error probabilities and with finite expected sample sizes under the two hypotheses in this paper it is shown that this optimum property can be extended at least asymptotically as the error probabilities tend to 0 to invariant sprts like the sequential t test the savage sethuraman sequential rank order test etc in fact not only do these invariant sprts asymptotically minimize the expected sample size but they also asymptotically minimize all the moments of the sample size distribution among all invariant tests with the same or smaller error probabilities modifications of these invariant sprts to asymptotically minimize the moments of the sample size at an intermediate parameter are also considered

asymptotic normality of the maximum likelihood estimator for general hidden markov models hidden markov models hmms have during the last decade become a widespread tool for modeling sequences of dependent random variables inference for such models is usually based on the maximum likelihood estimator mle and consistency of the mle for general hmms was recently proved by leroux in this paper we show that under mild conditions the mle is also asymptotically normal and prove that the observed information matrix is a consistent estimator of the fisher information

on the minimum time function and the minimum energy problem a nonlinear case abstract for a semilinear control system in general banach spaces we prove results on exact and approximate controllability regularity of the minimum time function connections between the minimum time function and the minimum energy

information bounds and quick detection of parameter changes in stochastic systems by using information theoretic bounds and sequential hypothesis testing theory this paper provides a new approach to optimal detection of abrupt changes in stochastic systems this approach not only generalizes previous work in the literature on optimal detection far beyond the relatively simple models treated but also suggests alternative performance criteria which are more tractable and more appropriate for general stochastic systems in addition it leads to detection rules which have manageable computational complexity for on line implementation and yet are nearly optimal under the different performance criteria considered

sequential detection of markov targets with trajectory estimation the problem of detection and possible estimation of a signal generated by a dynamic system when a variable number of noisy measurements can be taken is here considered assuming a markov evolution of the system in particular the pair signal observation forms a hidden markov model hmm a sequential procedure is proposed wherein the detection part is a sequential probability ratio test sprt and the estimation part relies upon a maximum a posteriori probability map criterion gated by the detection stage the parameter to be estimated is the trajectory of the state evolution of the system itself a thorough analysis of the asymptotic behavior of the test in this new scenario is given and sufficient conditions for its asymptotic optimality are stated i e for almost sure minimization of the stopping time and for first order minimization of any moment of its distribution an application to radar surveillance problems is also examined

abstract dynamic programming models under commutativity conditions the unifying purpose of the abstract dynamic programming models is to find sufficient conditions on the recursive definition of the objective function that guarantee the validity of the dynamic programming iteration this paper presents backward forward and backward forward models that weaken previous sufficient conditions and that include but are not restricted to optimization problems the backward forward model is devoted to the simultaneous solution of a collection of interrelated sequential problems based on the independent computation of a cost to arrive function and a cost to go function several extremization and nonextremization problems illustrate the applicability of the proposed models

decision theoretic optimality of the cusum procedure suppose x 1 x 2 are independent random variables such that for some unknown each of x 1 x 1 is distributed according to f 0 while x x 1 are all distributed according to f 1 we prove a result of moustakides that claims that the cusum procedures are optimal in the sense of lorden we do that by proving that the procedures are bayes for some stochastic mechanism of generating

asymptotic minimax robust quickest change detection for dependent stochastic processes with parametric uncertainty in this paper we consider the problem of quickly detecting an unknown change in the conditional densities of a dependent stochastic process in contrast to the existing quickest change detection approaches for dependent stochastic processes we propose minimax robust versions of the popular lorden pollak and bayesian criteria for when there is uncertainty about the parameter of the post change conditional densities under an information theoretic pythagorean inequality condition on the uncertainty set of possible post change parameters we identify asymptotic minimax robust solutions to our lorden pollak and bayesian problems finally through simulation examples we illustrate that asymptotically minimax robust rules can provide detection performance comparable to the popular but more computationally expensive generalized likelihood ratio rule

universal classification for hidden markov models binary hypotheses testing using empirically observed statistics is studied in the neyman pearson formulation for the hidden markov model hmm an asymptotically optimal decision rule is proposed and compared to the generalized likelihood ratio test glrt which has been shown in earlier studies to be asymptotically optimal for simpler parametric families the proof of the main theorem is provided the result can be applied to several types of hmms commonly used in speech recognition and communication applications several applications are demonstrated

optimal stopping times for detecting changes in distributions it is shown that page s stopping time is optimal for the detection of changes in distributions in a well defined sense this work is a generalization of an existing result where it was shown that page s stopping time is optimal asymptotically

asymptotic optimality of mixture rules for detecting changes in general stochastic models the paper addresses a sequential changepoint detection problem for a general stochastic model assuming that the observed data may be non i i d i e dependent and non identically distributed and prior distribution of the change point is arbitrary tartakovsky and veeravalli 2005 baron and tartakovsky 2006 and more recently tartakovsky 2017 developed a general asymptotic theory of changepoint detection for non i i d stochastic models assuming certain stability of the log likelihood ratio process in the case of simple hypotheses when both pre change and post change models are completely specified however in most applications the post change distribution is not completely known in the present paper we generalize previous results to the case of parametric uncertainty assuming that the parameter of the post change distribution is unknown we introduce two detection rules based on mixtures the mixture shiryaev rule and the mixture shiryaev roberts rule and study their asymptotic properties in the bayesian context in particular we provide sufficient conditions under which these rules are first order asymptotically optimal minimizing moments of the delay to detection as the probability of false alarm approaches zero

laguerre polynomials and the inverse laplace transform using discrete data abstract we consider the problem of finding a function defined on 0 from a countable set of values of its laplace transform the problem is severely ill posed we shall use the expansion of the function in a series of laguerre polynomials to convert the problem in an analytic interpolation problem then using the coefficients of lagrange polynomials we shall construct a stable approximation solution error estimate is given numerical results are produced

input output properties of the page hinkley detector we consider the stochastic input output properties of a simple non linear dynamical system the so called page hinkley detector playing a key role in change detection and also in queuing theory we show that for l mixing inputs with negative expectation the output process of this system is l mixing the result is applied to get an upper bound for the false alarm rate the proof is then adapted to get a similar result for the case of random i i d inputs possible extensions and open problems are given in the discussion

on the multifractal analysis of bernoulli convolutions i large deviation results we show how the formalism developed in a previous paper allows us to exhibit the multifractal nature of the infinitely convolved bernoulli measures for the golden mean in this first part we establish some large deviation results for random products of matrices using perturbation theory of quasicompact operators

space time code design for mimo detection based on kullback leibler divergence the focus of the paper is on the design of space time codes for a general multiple input multiple output detection problem when multiple observations are available at the receiver the figure of merit used for optimization purposes is the convex combination of the kullback leibler divergences between the densities of the observations under the two hypotheses and different system constraints are considered this approach permits to control the average sample number i e the time for taking a decision in a sequential probability ratio test and to asymptotically minimize the probability of miss in a likelihood ratio test the solutions offer an interesting insight in the optimal transmit policies encapsulated in the rank of the code matrix which rules the amount of diversity to be generated as well as in the power allocation policy along the active eigenmodes a study of the region of achievable divergence pairs whose availability permits optimization of a wide range of merit figures is also undertaken a set of numerical results is finally given in order to analyze and discuss the performance and validate the theoretical results

large deviations for products of random two dimensional matrices we establish large deviation type estimates for i i d products of two dimensional random matrices with finitely supported probability distribution the estimates are stable under perturbations and require no irreducibility assumptions in consequence we obtain a uniform local modulus of continuity for the corresponding lyapunov exponent regarded as a function of the support of the distribution this in turn has consequences on the modulus of continuity of the integrated density of states and on the localization properties of random jacobi operators

universal decoding for finite state channels universal decoding procedures for finite state channels are discussed although the channel statistics are not known universal decoding can achieve an error probability with an error exponent that for large enough block length or constraint length in case of convolutional codes is equal to the random coding error exponent associated with the optimal maximum likelihood decoding procedure for the given channel the same approach is applied to sequential decoding yielding a universal sequential decoding procedure with a cutoff rate and an error exponent that are equal to those achieved by the classical sequential decoding procedure

localization for off diagonal disorder and for continuous schrodinger operators we extend the proof of localization by delyon levy and souillard to accommodate the anderson model with off diagonal disorder and the continuous schrodinger equation with a random potential

central limit theorem in negative curvature we prove a central limit theorem for the distance of the brownian point on the universal cover of a compact negatively curved riemannian manifold the technical point is a contraction property for the leafwise brownian motion along the stable foliation

the inverse laplace transform and analytic pseudo differential operators abstract by comparing the laplace transform l with the differential operatord we obtain a formula for the inverse laplace transform l 1 1 v 1 cos d v l wherevis a unitary transformation operator this helps us obtain an explicit spectral representation of l some applications of the above relation are discussed

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

theoremes limite pour les systemes lineaires a coefficients markoviens nous montrons un theoreme central limite et une estimee des grandes deviations pour les solutions d equations differentielles stochastiques lineaires a coefficients markoviens ces resultats sont obtenus dans le cadre general des processus markoviens multiplicatifs ils s appliquent alors par exemple au flot derive d un flot stochastique sur une variete compacte et aux produits de matrices aleatoires en dependance markovienne

asymptotic efficiencies of truncated sequential tests truncation of a sequential test with constant boundaries is considered for the problem of testing a location hypothesis f x theta 0 versus f x theta 1 a test design procedure is developed by using bounds for the error probabilities under the hypothesis and alternative by viewing the truncated sequential test as a mixture of a sequential probability ratio test and a fixed sample size test its boundaries and truncation point can be obtained once the degree of mixture is specified asymptotically correct approximations for the operating characteristic function and the average sample number function of the resulting test are derived numerical results show that an appropriately designed truncated sequential test performs favorably as compared to both the fixed sample size test and the sequential probability ratio test with the same error probabilities the average sample number function of the truncated test is uniformly smaller than that of the fixed sample size test and the truncated test maintains average sample sizes under the hypothesis and the alternative that are close to those optimum values achieved by wald s sequential probability ratio test moreover the truncated test is more favorable than the sequential probability ratio test in the sense that is has smaller average sample size when the actual location parameter is between theta 0 and theta 1 this behavior becomes more pronounced as the error probabilities become smaller implying that the truncated sequential test becomes more favorable as the error probabilities become smaller

markovian structures in biological sequence alignments abstract the alignment of multiple homologous biopolymer sequences is crucial in research on protein modeling and engineering molecular evolution and prediction in terms of both gene function and gene product structure in this article we provide a coherent view of the two recent models used for multiple sequence alignment the hidden markov model hmm and the block based motif model to develop a set of new algorithms that have both the sensitivity of the block based model and the flexibility of the hmm in particular we decompose the standard hmm into two components the insertion component which is captured by the so called propagation model and the deletion component which is described by a deletion vector such a decomposition serves as a basis for rational compromise between biological specificity and model flexibility furthermore we introduce a bayesian model selection criterion that in combination with the propagation model genetic algorithm and other computational aspects forms the cor

specification testing in markov switching time series models this paper develops a series of specification tests of markov switching time series models tests for omitted autocorrelation omitted arch misspecification of the markovian dynamics and omitted explanatory variables are proposed all of the tests can be constructed as a natural byproduct of the routine used to calculate the smoothed probability that a given observation came from a particular regime and do not require estimation of additional parameters the paper performs monte carlo analysis of the tests and briefly illustrates their use with an empirical application

quasi maximum likelihood multiuser detection using semi definite relaxation with application to synchronous cdma the maximum likelihood ml multiuser detector is well known to exhibit better bit error rate ber performance than many other multiuser detectors unfortunately ml detection mld is a nondeterministic polynomial time hard np hard problem for which there is no known algorithm that can find the optimal solution with polynomial time complexity in the number of users in this paper a polynomial time approximation method called semi definite sd relaxation is applied to the mld problem with antipodal data transmission sd relaxation is an accurate approximation method for certain np hard problems the sd relaxation ml sdr ml detector is efficient in that its complexity is of the order of k3 5 where k is the number of users we illustrate the potential of the sdr ml detector by showing that some existing detectors such as the decorrelator and the linear minimum mean square error detector can be interpreted as degenerate forms of the sdr ml detector simulation results indicate that the ber performance of the sdr ml detector is better than that of these existing detectors and is close to that of the true ml detector even when the cross correlations between users are strong or the near far effect is significant

complex variable and regularization methods of inversion of the laplace transform in this paper three methods are derived for approximating f given its laplace transform g on 0 oo i e fo f t exp st dt g s assuming that g e l 0 oo the first method is based on a sinc like rational approximation of 9 the second on a sinc solution of the integral equation fo0 f t exp st dt g s via standard regularization and the third method is based on first converting fo0 f t exp st dt g s to a convolution integral over r and then finding a sinc approximation to f via the application of a special regularization procedure to solve the fourier transform problem we also obtain bounds on the error of approximation which depend on both the method of approximation and the regularization parameter

optimal feedback controls for a class of nonlinear distributed parameter systems it is shown that the optimal controls for a distributed control problem governed by semilinear parabolic equations can be expressed in feedback form the corresponding hamilton jacobi equation is studied and applications to infinite time horizon and time optimal control problems are given some extensions to control problems governed by parabolic variational inequalities are indicated also

quickest change detection and kullback leibler divergence for two state hidden markov models in this paper the quickest change detection problem is studied in two state hidden markov models hmm where the vector parameter theta of the hmm changes from theta 0 to theta 1 at some unknown time and one wants to detect the true change as quickly as possible while controlling the false alarm rate it turns out that the generalized likelihood ratio glr scheme while theoretically straightforward is generally computationally infeasible for the hmm to develop efficient but computationally simple schemes for the hmm we first discuss a subtlety in the recursive form of the generalized likelihood ratio glr scheme for the hmm then we show that the recursive cusum scheme proposed in fuh ann statist 2003 can be regarded as a quasi glr scheme for pseudo post change hypotheses with certain dependence structure between pre and postchange observations next we extend the quasi glr idea to propose recursive score schemes in the scenario when the postchange parameter theta 1 of the hmm involves a real valued nuisance parameter finally the kullback leibler kl divergence plays an essential role in the quickest change detection problem and many other fields however it is rather challenging to numerically compute it in hmms here we develop a non monte carlo method that computes the kl divergence of two state hmms via the underlying invariant probability measure which is characterized by the fredholm integral equation numerical study demonstrates an unusual property of the kl divergence for hmm that implies the severe effects of misspecifying the postchange parameter for the hmm

an algorithm for detecting a change in a stochastic process the problem of detecting a change from one given stationary and ergodic stochastic process to another such process is considered it is assumed that both stochastic processes are processes with memory and that they are mutually independent a sequential test is proposed and analyzed it is proved that the proposed test is asymptotically optimal in a mathematically precise sense

multihypothesis sequential probability ratio tests ii accurate asymptotic expansions for the expected sample size for pt i see ibid vol 45 p 2448 61 1999 we proved in pt i that two specific constructions of multihypothesis sequential tests which we refer to as multihypothesis sequential probability ratio tests msprts are asymptotically optimal as the decision risks or error probabilities go to zero the msprts asymptotically minimize not only the expected sample size but also any positive moment of the stopping time distribution under very general statistical models for the observations in this paper based on nonlinear renewal theory we find accurate asymptotic approximations up to a vanishing term for the expected sample size that take into account the overshoot over the boundaries of decision statistics the approximations are derived for the scenario where the hypotheses are simple the observations are independent and identically distributed i i d according to one of the underlying distributions and the decision risks go to zero simulation results for practical examples show that these approximations are fairly accurate not only for large but also for moderate sample sizes the asymptotic results given here complete the analysis initiated by baum and veeravalli 1994 where first order asymptotics were obtained for the expected sample size under a specific restriction on the kullback leibler distances between the hypotheses

on certain large random hermitian jacobi matrices with applications to wireless communications in this paper we study the spectrum of certain large random hermitian jacobi matrices these matrices are known to describe certain communication setups in particular we are interested in an uplink cellular channel which models mobile users experiencing a soft handoff situation under joint multicell decoding considering rather general fading statistics we provide a closed form expression for the per cell sum rate of this channel in high signal to noise ratio snr when an intra cell time division multiple access tdma protocol is employed since the matrices of interest are tridiagonal their eigenvectors can be considered as sequences with second order linear recurrence therefore the problem is reduced to the study of the exponential growth of products of two by two matrices for the case where k users are simultaneously active in each cell we obtain a series of lower and upper bound on the high snr power offset of the per cell sum rate which are considerably tighter than previously known bounds

random walks on three strand braids and on related hyperbolic groups we investigate the statistical properties of random walks on the simplest nontrivial braid group b3 and on related hyperbolic groups we provide a method using cayley graphs of groups allowing us to compute explicitly the probability distribution of the basic statistical characteristics of random trajectories the drift and the return probability the action of the groups under consideration in the hyperbolic plane is investigated and the distribution of a geometric invariant the hyperbolic distance is analysed it is shown that a random walk on b3 can be viewed as a magnetic random walk on the group psl 2

sprt and cusum in hidden markov models in this paper we study the problems of sequential probability ratio tests for parameterized hidden markov models we investigate in some detail the performance of the tests and derive corrected brownian approximations for error probabilities and expected sample sizes asymptotic optimality of the sequential probability ratio test for testing simple hypotheses based on hidden markov chain data is established next we consider the cumulative sum cusum procedure for change point detection in this model based on the renewal property of the stopping rule cusum can be regarded as a repeated one sided sequential probability ratio test asymptotic optimality of the cusum procedure is proved in the sense of lorden 1971 motivated by the sequential analysis in hidden markov models wald s likelihood ratio identity and wald s equation for products of markov random matrices are also given we apply these results to several types of hidden markov models i i d hidden markov models switch gaussian regression and switch gaussian autoregression which are commonly used in digital communications speech recognition bioinformatics and economics

identifiability of hidden markov information sources and their minimum degrees of freedom if only a function of the state in a finite state markov chain is observed then the stochastic process is no longer markovian in general this type of information source is found widely and the basic problem of its identifiability remains open that is the problem of showing when two different markov chains generate the same stochastic process the identifiability problem is completely solved by linear algebra where a block structure of a markov transition matrix plays a fundamental role and from which the minimum degree of freedom for a source is revealed

quickest change detection of a markov process across a sensor array recent attention in quickest change detection in the multisensor setting has been on the case where the densities of the observations change at the same instant at all the sensors due to the disruption in this work a more general scenario is considered where the change propagates across the sensors and its propagation can be modeled as a markov process a centralized bayesian version of this problem is considered with a fusion center that has perfect information about the observations and a priori knowledge of the statistics of the change process the problem of minimizing the average detection delay subject to false alarm constraints is formulated in a dynamic programming framework insights into the structure of the optimal stopping rule are presented in the limiting case of rare disruptions it is shown that the structure of the optimal test reduces to thresholding the a posteriori probability of the hypothesis that no change has happened under a certain condition on the kullback leibler k l divergence between the post and the pre change densities it is established that the threshold test is asymptotically optimal in the vanishing false alarm probability regime it is shown via numerical studies that this low complexity threshold test results in a substantial improvement in performance over naive tests such as a single sensor test or a test that incorrectly assumes that the change propagates instantaneously

optimality of cusum rule approximations in change point detection problems application to nonlinear state space systems the well known cumulative sum cusum sequential rule for abrupt model change detection in stochastic dynamic systems relies on the knowledge of the probability density functions of the system output variables conditional on their past values and on the system functioning mode at each time step this paper shows how to build an asymptotically optimal detection rule under the common average run length arl constraint when these densities are not available but can be consistently estimated this is the case for nonlinear state space systems observed through output variables for such systems a new class of particle filters based on convolution kernels allows to get consistent estimates of the conditional densities leading to an optimal cusum like filter detection rule fdr

constrained state estimation for nonlinear discrete time systems stability and moving horizon approximations state estimator design for a nonlinear discrete time system is a challenging problem further complicated when additional physical insight is available in the form of inequality constraints on the state variables and disturbances one strategy for constrained state estimation is to employ online optimization using a moving horizon approximation we propose a general theory for constrained moving horizon estimation sufficient conditions for asymptotic and bounded stability are established we apply these results to develop a practical algorithm for constrained linear and nonlinear state estimation examples are used to illustrate the benefits of constrained state estimation our framework is deterministic

order estimation and sequential universal data compression of a hidden markov source by the method of mixtures we consider first the estimation of the order i e the number of states of a discrete time finite alphabet stationary ergodic hidden markov source hms our estimator uses a description of the observed data in terms of a uniquely decodable code with respect to a mixture distribution obtained by suitably mixing a parametric family of distributions on the observation space this procedure avoids maximum likelihood calculations the order estimator is shown to be strongly consistent with the probability of underestimation decaying exponentially fast in the number n of observations while the probability of overestimation does not exceed cn sup 3 where c is a constant next we present a sequential algorithm for the uniquely decodable universal data compression of the hms which performs an on line estimation of source order followed by arithmetic coding this code asymptotically attains optimum average redundancy

a sequential procedure for multihypothesis testing the sequential testing of more than two hypotheses has important applications in direct sequence spread spectrum signal acquisition multiple resolution element radar and other areas a useful sequential test which we term the msprt is studied in this paper the test is shown to be a generalization of the sequential probability ratio test under bayesian assumptions it is argued that the msprt approximates the much more complicated optimal test when error probabilities are small and expected stopping times are large bounds on error probabilities are derived and asymptotic expressions for the stopping time and error probabilities are given a design procedure is presented for determining the parameters of the msprt two examples involving gaussian densities are included and comparisons are made between simulation results and asymptotic expressions comparisons with bayesian fixed sample size tests are also made and it is found that the msprt requires two to three times fewer samples on average

asymptotic operating characteristics of an optimal change point detection in hidden markov models let 0 1 1 be observations from the hidden markov model with probability distribution p 0 and let 1 be observations from the hidden markov model with probability distribution p 1 the parameters 0 and 1 are given while the change point is unknown the problem is to raise an alarm as soon as possible after the distribution changes from p 0 to p 1 but to avoid false alarms specifically we seek a stopping rule n which allows us to observe the s sequentially such that e n is large and subject to this constraint sup k e k n k n k is as small as possible here e k denotes expectation under the change point k and e denotes expectation under the hypothesis of no change whatever in this paper we investigate the performance of the shiryayev roberts pollak srp rule for change point detection in the dynamic system of hidden markov models by making use of markov chain representation for the likelihood function the structure of asymptotically minimax policy and of the bayes rule and sequential hypothesis testing theory for markov random walks we show that the srp procedure is asymptotically minimax in the sense of pollak ann statist 13 1985 206 227 next we present a second order asymptotic approximation for the expected stopping time of such a stopping scheme when w 1 motivated by the sequential analysis in hidden markov models a nonlinear renewal theory for markov random walks is also given

sequential detection of targets in multichannel systems it is supposed that there is a multichannel sensor system which performs sequential detection of a target sequential detection is done by implementing a generalized wald s sequential probability ratio test which is based on the maximum likelihood ratio statistic and allows one to fix the false alarm rate and the rate of missed detections at specified levels we present the asymptotic performance of this sequential detection procedure and show that it is asymptotically optimal in the sense of minimizing the expected sample size when the probabilities of erroneous decisions are small we do not assume that the observations are independent and identically distributed i i d the first order asymptotic optimality result holds for general statistical models that are not confined to the restrictive i i d assumption however for i i d and quasi i i d cases where the log likelihood ratios can be represented in the form of sums of random walks and slowly changing sequences we obtain much stronger results specifically using the nonlinear renewal theory we are able to obtain both tight expressions for the error probabilities and higher order approximations for the average sample size up to a vanishing term the performance of the multichannel sequential detection algorithm is illustrated by an example of detection of a deterministic signal in correlated colored gaussian noise in this example we provide both the results of theoretical analysis and the results of a monte carlo experiment these results allow us to conclude that the use of the sequential detection algorithm substantially reduces the required resources of the system compared to the best nonsequential algorithm

sequential detection of almost harmonic signals to detect a purely harmonic signal it is difficult to beat a fast fourier transform fft however when the signal is very long and weak parker and white 1999 have shown that a sequential probability ratio test sprt operating on magnitude square fft data is far more efficient indeed both from a numerical error perspective and in terms of robustness against a deviation from a precisely tonal signal the block fft sprt idea is very appealing here the approach is extended to the case that the frequency is unknown and expressions are developed for performance both in terms of detection and of sample number the approach is applicable to a large number of practical problems but particular attention is paid to the continuous gravitational wave gw example the computational savings as compared with a fixed test vary as a function of signal strength block length bandwidth and operating point however gains of a factor of two are easy that these gains are not more exciting relates mostly to the underlying fft structure although many sprts end early it is difficult to take advantage of that with an efficient fft algorithm however the progressive reduction of the number of working sprts implies a substantial reduction of the ensemble of the candidate frequencies with time which is an appealing feature particularly in the gw case

multihypothesis sequential probability ratio tests i asymptotic optimality the problem of sequential testing of multiple hypotheses is considered and two candidate sequential test procedures are studied both tests are multihypothesis versions of the binary sequential probability ratio test sprt and are referred to as msprts the first test is motivated by bayesian optimality arguments while the second corresponds to a generalized likelihood ratio test it is shown that both msprts are asymptotically optimal relative not only to the expected sample size but also to any positive moment of the stopping time distribution when the error probabilities or more generally risks associated with incorrect decisions are small the results are first derived for the discrete time case of independent and identically distributed i i d observations and simple hypotheses they are then extended to general possibly continuous time statistical models that may include correlated and nonhomogeneous observation processes it also demonstrated that the results can be extended to hypothesis testing problems with nuisance parameters where the composite hypotheses due to nuisance parameters can be reduced to simple ones by using the principle of invariance these results provide a complete generalization of the results given by veeravalli and baum see ibid vol 41 p 1994 97 1995 where it was shown that the quasi bayesian msprt is asymptotically efficient with respect to the expected sample size for i i d observations

on the multifractal analysis of bernoulli convolutions ii dimensions we show how the formalism developed in a previous paper allows us to exhibit the multifractal nature of the infinitely convolved bernoulli measures for the golden mean in this second part we show how the hausdorff dimension of the set where the measure has a power law singularity of strength is related to the large deviation function given in part i

efficient likelihood estimation in state space models motivated by studying asymptotic properties of the maximum likelihood estimator mle in stochastic volatility sv models in this paper we investigate likelihood estimation in state space models we first prove under some regularity conditions there is a consistent sequence of roots of the likelihood equation that is asymptotically normal with the inverse of the fisher information as its variance with an extra assumption that the likelihood equation has a unique root for each n then there is a consistent sequence of estimators of the unknown parameters if in addition the supremum of the log likelihood function is integrable the mle exists and is strongly consistent edge worth expansion of the approximate solution of likelihood equation is also established several examples including markov switching models arma models g arch models and stochastic volatility sv models are given for illustration

sequential detection and identification of a change in the distribution of a markov modulated random sequence the problem of detection and identification of an unobservable change in the distribution of a random sequence is studied via a hidden markov model hmm approach the formulation is bayesian on line discrete time allowing both single and multiple disorder cases dealing with both independent and identically distributed i i d and dependent observations scenarios allowing for statistical dependencies between the change time and change type in both the observation sequence and the risk structure and allowing for general discrete time disorder distributions several of these factors provide useful new generalizations of the sequential analysis theory for change detection and or hypothesis testing taken individually in this paper a unifying framework is provided that handles each of these considerations not only individually but also concurrently optimality results and optimal decision characterizations are given as well as detailed examples that illustrate the myriad of sequential change detection and identification problems that fall within this new framework

the bad truth about laplace s transform inverting the laplace transform is a paradigm for exponentially ill posed problems for a class of operators including the laplace transform we give forward and inverse formulae that have fast implementations using the fast fourier transform these formulae lead easily to regularized inverses whose effects on noise and filtered data can be precisely described our results give cogent reasons for the general sense of dread most mathematicians feel about inverting the laplace transform

failure detection and identification using the geometric concept of an unobservability subspace a solution is given to the problem of detecting and identifying control system component failures in linear time invariant systems conditions are developed for the existence of a causal linear time invariant processor that can detect and uniquely identify a component failure first for the case where components can fail simultaneously and then for the case where they fail only one at a time explicit design algorithms are provided when these conditions are satisfied in addition to time domain solvability conditions frequency domain interpretations of the results are given and connections are drawn with results already available in the literature