information theoretic lengths of jacobi polynomials the information theoretic lengths of the jacobi polynomials p n x which are information theoretic measures renyi shannon and fisher of their associated rakhmanov probability density are investigated they quantify the spreading of the polynomials along the orthogonality interval 1 1 in a complementary but different way as the root mean square or standard deviation because contrary to this measure they do not refer to any specific point of the interval the explicit expressions of the fisher length are given the renyi lengths are found by the use of the combinatorial multivariable bell polynomials in terms of the polynomial degree n and the parameters the shannon length which cannot be exactly calculated because of its logarithmic functional form is bounded from below by using sharp upper bounds to general densities on 1 1 given in terms of various expectation values moreover its asymptotics is also pointed out finally several computational issues relative to these three quantities are carefully analyzed

criteria for the assessment of random number generators abstract for any finite sequence of pseudo random numbers special simulation problems can be constructed such that this sequence yields useless results in order to study the quality of the result of a simulation the assessment of a pseudorandom number generator based on theoretical considerations is a great advantage several criteria for the assessment of random number generators are known the most important criteria are discussed

on the convergence of pseudo linear regression algorithms the convergence properties of a general iterative off line pseudo linear regression plr algorithm are examined it is found that the basic form of the algorithm converges under quite restrictive conditions a simple modification of the algorithm is then proposed which relaxes the convergence conditions considerably the refined plr algorithm converges under precisely the same conditions as the well known recursive on line plr procedure the problem of monitoring the algorithm to guarantee the stability of estimated models is also discussed a modified plr algorithm which does not require monitoring is proposed and its convergence properties analysed finally some numerical examples illustrating the main theoretical results are included

direct spreading measures of laguerre polynomials the direct spreading measures of the laguerre polynomials l n a x which quantify the distribution of its rakhmanov probability density r n a x 1d n 2x ae x l n a x 2 along the positive real line in various complementary and qualitatively different ways are investigated these measures include the familiar root mean square or standard deviation and the information theoretic lengths of fisher renyi and shannon types the fisher length is explicitly given the renyi length of order q such that email protected n is also found in terms of n a by means of two error free computing approaches one makes use of the lauricella function f a 2 q 1 1q 1q 1 which is based on the srivastava niukkanen linearization relation of laguerre polynomials and another one utilizes the multivariate bell polynomials of combinatorics the shannon length cannot be exactly calculated because of its logarithmic functional form but its asymptotics is provided and sharp bounds are obtained by the use of an information theoretic optimization procedure finally all these spreading measures are mutually compared and computationally analyzed in particular it is found that the apparent quasilinear relation between the shannon length and the standard deviation becomes rigorously linear only asymptotically i e for email protected 1

renyi entropies lq norms and linearization of powers of hypergeometric orthogonal polynomials by means of multivariate special functions the quantification of the spreading of the orthogonal polynomials p n x can be investigated by means of the renyi entropies r q r q being a positive integer number of the associated rakhmanov probability densities r x email protected x p n 2 x where w x is the corresponding weight function the renyi entropies are closely related to the l q norms of the polynomials in this manuscript the l q norms and the associated renyi entropies of the real hypergeometric orthogonal polynomials i e hermite laguerre and jacobi polynomials and the generalized hermite polynomials are expressed in an explicit way in terms of some generalized multivariate special functions of lauricella and srivastava daoust types which are evaluated at some specific values of 2q variables these functions depend on 4q 1 and 6q 2 parameters respectively which are determined by the order q the degree n of the polynomial and the parameters of the orthogonality weight function w x the key idea is based on some extended linearization formulas for these polynomials these results open the way to determine the renyi information entropies of the quantum systems whose wavefunctions are controlled by hypergeometric orthogonal polynomials

a measure of the tracking capability of recursive stochastic algorithms with constant gains a criterion is given for measuring the tracking capability of recursive algorithms when applied to slowly time varying systems the optimal gain for a given disturbance is also calculated this criterion is seen to have some connection with the fisher information matrix and allows us to select a priori the best algorithm for identifying a given unknown parameter which may be subject to smooth unknown disturbances examples of applications are given in the areas of identification theory and data communication theory

complexity analysis of hypergeometric orthogonal polynomials the complexity measures of the cramer rao fisher shannon and lmc lopez ruiz mancini and calvet types of the rakhmanov probability density n x x p n 2 x of the polynomials p n x orthogonal with respect to the weight function x x a b are used to quantify various two fold facets of the spreading of the hermite laguerre and jacobi systems all over their corresponding orthogonality intervals in both analytical and computational ways their explicit cramer rao and asymptotical fisher shannon lmc values are given for the three systems of orthogonal polynomials then these complexity type mathematical quantities are numerically examined in terms of the polynomial s degree n and the parameters which characterize the weight function finally several open problems about the generalized hypergeometric functions of lauricella and srivastava daoust types as well as on the asymptotics of weighted l q norms of laguerre and jacobi polynomials are pointed out

local shannon entropy measure with statistical tests for image randomness in this paper we propose a new image randomness measure using shannon entropy over local image blocks the proposed local shannon entropy measure overcomes several weaknesses of the conventional global shannon entropy measure including unfair randomness comparisons between images of different sizes failure to discern image randomness before and after image shuffling and possible inaccurate scores for synthesized images statistical tests pertinent to this new measure are also derived this new measure is therefore both quantitative and qualitative the parameters in the local shannon entropy measure are further optimized for a better capture of local image randomness the estimated statistics and observed distribution from 50 000 experiments match the theoretical ones finally two examples are given applying the proposed measure to image randomness among shuffled images and encrypted images both examples show that the proposed method is more effective and more accurate than the global shannon entropy measure

optimal instrumental variable estimation and approximate implementations the accuracy properties of instrumental variables iv methods are investigated extensions such as prefiltering of data and use of additional instruments are included in the analysis the parameter estimates are shown to be asymptotically gaussian distributed an explicit expression is given for the covariance matrix of their distribution the covariance matrix is then taken as a multivariable measure of accuracy it is shown how it can be optimized by an appropriate selection of instruments and prefilter the so obtained optimal instrumental variable estimates cannot be used directly since the true system and the statistical properties of the disturbance must be known in order to compute the optimal instruments and prefilters a multistep procedure consisting of three or four simple steps is then proposed as a way to overcome this difficulty this procedure includes modeling of the disturbance as an arma process using a statistically efficient method such as a prediction error method the statistical properties of the estimates obtained with the multistep procedure are also analyzed those estimates are shown to be asymptotically gaussian distributed as well the covariance matrix of the estimation errors is compared to that corresponding to a prediction error method for some model structures these two approaches give the same asymptotic accuracy the conclusion is that the multistep procedure which is quite easy to implement and also has nice uniqueness properties can be viewed as an interesting alternative to prediction error methods

a statistical complexity measure with nonextensive entropy and quasi multiplicativity the properties of a statistical complexity measure that are characterized by nonextensivity in entropy have been investigated which is of so called disequilibrium type considering the composition law for two systems with different nonextensivities quasi multiplicativity a nontrivial relation between the nonextensive parameters and the fluctuating bit number in information theory has been mentioned to see the time evolution of the nonextensive complexity measure we examine systems having a lognormal distribution the underlying dynamics for which is known to obey a random multiplicative process in the presence of a boundary constraint

multiplicative congruential random number generators with modulus 2 an exhaustive analysis for 32 and a partial analysis for 48 this paper presents the results of a search to find optimal maximal period multipliers for multiplicative congruential random number generators with moduli 2 32 and 2 48 here a multiplier is said to be optimal if the distance between adjacent parallel hyperplanes on which k tuples lie does not exceed the minimal achievable distance by more than 25 percent for k 2 6

on the asymptotic accuracy of pseudo linear regression algorithms the accuracy properties of a general pseudo linear regression plr method are examined both off line and on line algorithms are considered assuming the parameter estimates converge they will under weak conditions be asymptotically gaussian distributed expressions for the corresponding covariance matrices are given it is shown that the asymptotic covariance matrix of the off line plr algorithm is bounded from above by the matrix corresponding to the on line plr algorithm and from below by that corresponding to the prediction error method some simple numerical illustrations of the theoretical results are also included

the hierarchy of correlations in random binary sequences the meaning of randomness is studied for the simple case of binary sequences ensemble theory is used together with correlation coefficients of any order conservation laws for the total amount of correlation are obtained they imply that true randomness is an ensemble property and can never be achieved in a single sequence the relation with entropy is discussed for different ensembles well tempered pseudorandom sequences turn out to be suitable sources of random numbers and practical recipes to generate them for use in large scale monte carlo simulations are found

technical communique eigenvalue location of certain matrices arising in convergence analysis problems we give simple formulas for the eigenvalues of a certain type of matrices occurring in some convergence analysis problems application of the result to a pseudo linear regression algorithm gives explicit necessary and sufficient conditions for local convergence

the exact determination of rectangle discrepancy for linear congruential pseudorandom numbers l algorithme presente peut etre utilise dans les cas de grands generateurs avec un temps de calcul tres raisonnable

rapid parameter estimation algorithms using the els principle abstract two computationally efficient algorithms for estimating the parameters of linear discrete time systems are proposed the algorithms are based on the extended least squares els principle they are essentially a correlation version of the off line els method that eliminate all the redundant computations do not require construction and operations of large matrices and bypass the explicit evaluation of residuals examples are given to illustrate their feasibility and performance

comparison of some instrumental variable methods consistency and accuracy aspects a general analysis of various instrumental variable methods is given the investigation includes methods for open loop operation as well as methods designed for closed loop operation the instrumental variables are formed as different combinations of inputs delayed inputs delayed outputs filtered inputs and external setpoint variations necessary conditions as well as sufficient conditions for consistency are explicitly derived and discussed the paper also includes some simple counter examples to general consistency for some common instrumental variable methods also asymptotic accuracy properties are reviewed it is e g studied how the use of filters will influence the covariance matrix of the parameter estimates

regression procedures for arma estimation abstract regression procedures for parameter estimation in autoregression moving average arma models are discussed mainly for providing initial estimates for iterative maximization of a gaussian likelihood an iterative procedure of spliid 1983 is compared to a procedure of hannan and rissanen 1982 and a global convergence result is established for an iterative modification of spliid s procedure spliid s iteration does not always converge when it does it has the same asymptotic distribution as the second stage of the hannan rissanen procedure this second stage iteration gives the same problems as spliid s procedure so it is preferable to go immediately to the third stage which is asymptotically efficient in the gaussian case an example is provided by a first order arma model y t y t 1 e t e t 1 under the usual regularity conditions of stationarity and invertibility spliid s procedure fits an autoregression of predetermined order i e 2 to obtain estimates of the e t a

uniqueness of estimated k step prediction models of arma processes the direct estimation of the k step prediction models of arma processes is discussed the emphasis is on the uniqueness properties of the parameter estimates of such models obtained by using either a prediction error method pem or a pseudo linear regression plr algorithm the main result is that both pem and plr when applied to such models have a certain uniqueness property more specifically it is shown that all the limit models corresponding to either method behave precisely as the true optimal predictor furthermore if a minimal parameterization is used then the true predictor is the unique limit model both for pem and for plr

convergence and rate of convergence of a recursive identification and adaptive control method which uses truncated estimators a stochastic approximation like method is used for the recursive identification of the coefficients in y n sum min 1 max l 1 a i y n i sum min 0 max l 2 b i u n i sum min 1 max l 3 c i w n i w n where w n is a sequence of mutually independent and bounded random variables and is independent of the bounded u n such methods normally require the recursive estimation of the residuals or the w n and the algorithms for doing this can be unstable if the parameter estimates are not close enough to their true values the problem is solved here by use of a simple truncated estimator which is probably what would be used in implementation in any case then under a stability and strict positive real type condition with probability 1 w p 1 convergence is proved and the rate of convergence is obtained an associated adaptive control problem is also treated

features of the extension of a statistical measure of complexity to continuous systems we discuss some aspects of the extension to continuous systems of a statistical measure of complexity introduced by l opez ruiz mancini and calbet phys lett a 209 321 1995 in general the extension of a magnitude from the discrete to the continuous case is not a trivial process and requires some kind of choice in the present study several possibilities appear available one of them is examined in detail some interesting properties desirable for any magnitude of complexity are discovered on this particular extension

the lattice structure of pseudo random vectors generated by matrix generators abstract in many areas of science the problems treated by monte carlo simulations become more and more complex and more extensive because of that generators like linear congruential matrix generators are needed which produce enormously many pseudo random numbers in order to assess stochastical properties of the generated pseudo random vectors the lattice structure of these matrix generators is studied here