some majorization inequalities in multivariate statistical analysis this paper contains a review of certain majorization inequalities in multivariate statistical analysis the results apply to a large class of distributions including the multivariate normal distribution and have implications in estimation hypothesis testing and other related problems in statistics

asynchronous deterministic rendezvous in graphs two mobile agents robots having distinct labels and located in nodes of an unknown anonymous connected graph have to meet we consider the asynchronous version of this well studied rendezvous problem and we seek fast deterministic algorithms for it since in the asynchronous setting meeting at a node which is normally required in rendezvous is in general impossible we relax the demand by allowing meeting of the agents inside an edge as well the measure of performance of a rendezvous algorithm is its cost for a given initial location of agents in a graph this is the number of edge traversals of both agents until rendezvous is achieved if agents are initially situated at a distance d in an infinite line we show a rendezvous algorithm with cost o d lmin 2 when d is known and o d lmax 3 if d is unknown where lmin and lmax are the lengths of the shorter and longer label of the agents respectively these results still hold for the case of the ring of unknown size but then we also give an optimal algorithm of cost o n lmin if the size n of the ring is known and of cost o n lmax if it is unknown for arbitrary graphs we show that rendezvous is feasible if an upper bound on the size of the graph is known and we give an optimal algorithm of cost o d lmin if the topology of the graph and the initial positions are known to agents

unimodality convexity and applications properties of univariate unimodal distributions concepts of multivariate unimodality some more notions of unimodality unimodality for discrete distributions unimodality of infinitely divisible distributions unimodality and notions of dependence ordering of distributions by peakedness applications of unimodality in statistical inference convexity in reliability theory appendix

on the nonlinear and nonnormal filter using rejection sampling a nonlinear and or nonnormal filter is proposed using rejection sampling generating random draws of the state vector directly from the filtering density the filtering estimate is simply obtained as the arithmetic average of the random draws in the proposed filter the random draws are recursively generated at each time monte carlo experiments indicate that the proposed nonlinear and nonnormal filter shows a good performance

non gaussian state space modeling of nonstationary time series abstract a non gaussian state space approach to the modeling of nonstationary time series is shown the model is expressed in state space form where the system noise and the observational noise are not necessarily gaussian recursive formulas of prediction filtering and smoothing for the state estimation and identification of the non gaussian state space model are given also given is a numerical method based on piecewise linear approximation to the density functions for realizing these formulas significant merits of non gaussian modeling and the wide range of applicability of the method are illustrated by some numerical examples a typical application of this non gaussian modeling is the smoothing of a time series that has mean value function with both abrupt and gradual changes simple gaussian state space modeling is not adequate for this situation here the model with small system noise variance cannot detect jump whereas the one with large system noise variance yields unfavorable wiggle to work

the kantorovich inequality for error analysis of the kalman filter with unknown noise distributions the state space model with random noises is widely used to model dynamic systems in many practical problems the noise terms will have unknown probability distributions which is contrary to the usual assumption of gaussian noises made in state vector estimation this assumption is usually made for reasons of mathematical tractability a problem arises however in that inference based on the incorrect gaussian assumption can lead to misleading or erroneous conclusions this note shows how the kantorovich inequality from probability theory has potential for characterizing the estimation error of a kalman filter in such a non gaussian distribution free setting

the generalized moment based filter can we solve the filtering problem from the only knowledge of few moments of the noise terms in this technical note by exploiting set of distributions based filtering we solve this problem without introducing additional assumptions on the distributions of the noises e g gaussianity or on the final form of the estimator e g linear estimator given the moments e g mean and variance of random variable x it is possible to define the set of all distributions that are compatible with the moments information this set can be equivalently characterized by its extreme distributions a family of mixtures of dirac s deltas the lower and upper expectation of any function g of x are obtained in correspondence of these extremes and can be computed by solving a linear programming problem the filtering problem can then be solved by running iteratively this linear programming problem in this technical note we discuss theoretical properties of this filter we show the connection with set membership estimation and its practical applications

rendezvous search on the line with more than two players suppose n blind speed one players are placed by a random permutation onto the integers 1 to n and each is pointed randomly to the right or left what is the least expected time required form m n of them to meet together at a single point if they must all use the same strategy we call this time the symmetric rendezvous value rn ms otherwise the asymmetric value rn ma we show that r3 2a 47 48 and that rn ns is asymptotic to n 2 these results respectively extend those for two players given by alpern and gal alpern s s gal 1995 rendezvous search on the line with distinguishable players siam j control optim 33 1270 1276 and anderson and essegaier anderson e j s essegaier 1995 rendezvous search on the line with indistinguishable players siam j control optim 33 1637 1642

rendezvous search on the line with bounded resources expected time minimization abstract two players are placed on the line and want to meet neither knows the direction of the other but they know the distance between them or perhaps the distribution of this distance they can move with speed at most one and each has a resource constraint on the total distance he can travel we first consider the question of whether the two players can ensure that they meet when they can then we seek the least expected meeting time otherwise we maximize the probability of a meeting this generalizes the similar problem studied by the first author and s gal siam j control and optimization 33 4 1995 1270 without any resource constraint and indeed for sufficiently large resources gives the same rendezvous time i e least expected time to meet the paper may also be considered a generalization of the linear search problem studied by the second author and others which corresponds to the case when one of the resource constraints is zero so that player cannot move more specifically it generlaizes the bounded resource version of the linear search problem presented by foley hill and spruill naval research logistics 38 1991 555 565

static search games played over graphs and general metric spaces we define a general game which forms a basis for modelling situations of static search and concealment over regions with spatial structure the game involves two players the searching player and the concealing player and is played over a metric space each player simultaneously chooses to deploy at a point in the space the searching player receiving a payoff of 1 if his opponent lies within a predetermined radius r of his position the concealing player receiving a payoff of 1 otherwise the concepts of dominance and equivalence of strategies are examined in the context of this game before focusing on the more specific case of the game played over a graph methods are presented to simplify the analysis of such games both by means of the iterated elimination of dominated strategies and through consideration of automorphisms of the graph lower and upper bounds on the value of the game are presented and optimal mixed strategies are calculated for games played over a particular family of graphs

closed geodesies on compact lorentzian manifolds of splitting type in this paper we consider the problem of the existence of a spacelike closed geodesic on compact lorentzian manifolds tipler and galloway proved that under suitable topological properties of the manifold there exists a closed timelike geodesic in their proofs they use the hypothesis that the time coordinate of one timelike geodesic has derivative always different from zero this clearly fails for spacelike geodesies using variational methods and applying the relative category theory we prove the existence of a closed spacelike geodesic on a compact manifold of splitting type observe that thanks to the previous results the existence of at least two geometrically distinct closed geodesies on follows

slide window bounded error time varying systems identification this technical note presents a new identification method for discrete linear systems with time varying parameters based on bounded error approach it is assumed a bounded additive error on measurements and a bound on parameter variation each time instant the identification method uses a slide window along historical data and an explicit expression to provide an outer bound of the set of model parameters that is consistent with measurements model structure and bounds on error and variations considered furthermore the center estimation of the obtained outer solution set is a suitable nominal estimation of the parameters this center is the optimal solution of a least squares problem that penalizes the prediction error of the model and the parameter variations the explicit expression that provides the outer bound of the parameters and the optimal property of the center estimation are the main contributions of the technical note in order to clarify the proposed method an application example and a comparison with a previous recursive bounded error method are included

input output modeling of nonlinear systems with time varying linear models time varying arma autoregressive moving average and armax autoregressive moving average with exogenous inputs models are proposed for input output modeling of nonlinear deterministic and stochastic systems the coefficients of these models are estimated by a random walk kalman filter rwkf this method requires no prior assumption on the nature of the model coefficients and is suitable for real time implementation since no off line training is needed a simulation example illustrates the method goodness of performance is judged by the quality of the residuals histograms autocorrelation functions and the kolmogorov smirnoff test

prediction mean squared error for state space models with estimated parameters summary we obtain a conditional prediction mean squared error for a state space model with estimated parameters an important application of our results is the derivation of conditional forecast and interpolation mean squared errors for autoregressive moving average models with estimated parameters we also obtain the conditional mean squared error for filtered and smoothed estimates of the state vector of z t and s t i n 0 var z t z t i n 0 is the mean squared error both z t i n 0 and s t i n 0 can be computed efficiently using the kalman filter and the smoothing algorithms in chapters 3 and 7 respectively of anderson moore 1979 if 0 is unknown then we can estimate it by maximum likelihood to obtain 0 say and a a then use z t i n 0 and s t in 0 in place of z t i n 0 and s t i n 0 however the estimate s t i n 0 of mean squared error underestimates the true mean squared error because it does not take into account the extra variation due to estimating the parameters one important example of this problem is the adjustment of the forecast mean squared error of an arma process to allow for the variability due to parameter estimation with the exception of phillips 1979 and fuller hasza 1981 whose work is confined to the autoregressive model results in the literature on arma and armax models with estimated parameters have dealt with the unconditional forecast mean squared error see for example bloomfield 1972 yamamoto 1976 1981 and baillie 1979 1980 their unconditional forecast mean squared error is an average over all realizations of y whereas the actual prediction error depends on the particular realization y we observe this will be made clear for the first order autoregressive model in 2 therefore the

minimax rendezvous on the line suppose that n players are placed randomly on the real line at consecutive integers and faced in random directions each player has maximum speed one cannot see the others and doesn t know his relative position what is the minimum time m n required to ensure that all the players can meet together at a single point regardless of their initial placement we prove that m 2 3 m 3 4 and m n is asymptotic to n 2 we also consider a variant of the problem which requires players who meet to stick together and find in this case that three players require 5 time units to ensure a meeting this paper is thus a minimax version of the rendezvous search problem which has hitherto been studied only in terms of minimizing the expected meeting time

event based state estimation with time synchronous updates to reduce the amount of data transfer in networked systems measurements are usually taken only when an event occurs rather than at each synchronous sample instant however this complicates estimation problems considerably especially in the situation when no measurement is received anymore the goal of this paper is therefore to develop a state estimator that can successfully cope with event based measurements and attains an asymptotically bounded error covariance matrix to that extent a general mathematical description of event sampling is proposed this description is used to set up a state estimator with a hybrid update i e when an event occurs the estimated state is updated using the measurement while at synchronous instants the update is based on knowledge that the sensor value lies within a bounded subset of the measurement space furthermore to minimize computational complexity of the estimator the algorithm is implemented using a sum of gaussians approach the benefits of this implementation are demonstrated by an illustrative example of state estimation with event sampling

error constrained filtering for a class of nonlinear time varying delay systems with non gaussian noises in this technical note the quadratic error constrained filtering problem is formulated and investigated for discrete time varying nonlinear systems with state delays and non gaussian noises both the lipschitz like and ellipsoid bounded nonlinearities are considered the non gaussian noises are assumed to be unknown bounded and confined to specified ellipsoidal sets the aim of the addressed filtering problem is to develop a recursive algorithm based on the semi definite programme method such that for the admissible time delays nonlinear parameters and external bounded noise disturbances the quadratic estimation error is not more than a certain optimized upper bound at every time step the filter parameters are characterized in terms of the solution to a convex optimization problem that can be easily solved by using the semi definite programme method a simulation example is exploited to illustrate the effectiveness of the proposed design procedures

a method for discovering kantorovich type inequalities and a probabilistic interpretation abstract the kantorovich inequality z az z a 1z 1 n 2 4 1 n can be proved using convexity and elementary geometry this method of proof is generalized so that it can often yield the upper and lower bounds for f z az z bz z cz where a b c are commuting n x n hermitian matrices and z is a unit vector in c n among the examples is the mulholland smith inequality it is also shown that our method finds probability distributions giving f e e e its upper and lower bounds

new inequalities of the kantorovich type for bounded linear operators in hilbert spaces some new inequalities of the kantorovich type are established they hold for larger classes of operators and subsets of complex numbers than considered before in the literature and provide refinements of the classical results in the case when the involved operator satisfies the usual conditions several new reverse inequalities for the numerical radius of a bounded linear operator are obtained as well

rendezvous search on the line we present two new results for the asymmetric rendezvous problem on the line we first show that it is never optimal for one player to be stationary during the entire search period in the two player rendezvous then we consider the meeting time of n players in the worst case and show that it has an asymptotic behavior of n 2 o log n

asymmetric rendezvous on the line is a double linear search problem we apply a new method of analysis to the asymmetric rendezvous search problem on the line arspl this problem previously studied in a paper of alpern and gal 1995 asks how two blind speed one players placed a distance d apart on the line can find each other in minimum expected time the distance d is drawn from a known cumulative probability distribution g and the players are faced in random directions we show that the arspl is strategically equivalent to a new problem we call the double linear search problem dlsp where an object is placed equiprobably on one of two lines and equiprobably at positions d a searcher is placed at the origin of each of these lines the two searchers move with a combined speed of one to minimize the expected time before one of them finds the object using results from a concurrent paper of the first author and j v howard 1998 we solve the dlsp and hence the arspl for the case where g is convex on its support and show that the solution is that conjectured in a paper of baston and gal 1998

robust filtering through coherent lower previsions the classical filtering problem is re examined to take into account imprecision in the knowledge about the probabilistic relationships involved imprecision is modeled in this paper by closed convex sets of probabilities we derive a solution of the state estimation problem under such a framework that is very general it can deal with any closed convex set of probability distributions used to characterize uncertainty in the prior likelihood and state transition models this is made possible by formulating the theory directly in terms of coherent lower previsions that is of the lower envelopes of the expectations obtained from the set of distributions the general solution is specialized to two particular classes of coherent lower previsions the first consists of a family of gaussian distributions whose means are only known to belong to an interval the second is the so called linear vacuous mixture model which is a family made of convex combinations of a known nominal distribution e g a gaussian with arbitrary distributions for the latter case we empirically compare the proposed estimator with the kalman filter this shows that our solution is more robust to the presence of modelling errors in the system and that hence appears to be a more realistic approach than the kalman filter in such a case