Alberto Lekuona

Sharp Bounds on the Entropy of the Poisson Law and Related Quantities

One of the difficulties in calculating the capacity of certain Poisson channels is that H(¿), the entropy of the Poisson distribution with mean ¿, is not available in a simple form. In this paper, we derive upper and lower bounds for H(¿) that are asymptotically tight and easy to compute. The derivation of such bounds involves only simple probabilistic and analytic tools. This complements the asymptotic expansions of Knessl (1998), Jacquet and Szpankowski (1999), and Flajolet (1999). The same method yields tight bounds on the relative entropy D(n, p) between a binomial and a Poisson, thus refining the work of Harremoe¿s and Ruzankin (2004). Bounds on the entropy of the binomial also follow easily.

RATES OF CONVERGENCE FOR THE ITERATES OF CESÀRO OPERATORS

We obtain sharp rates of convergence in the usual sup-norm for the nth iterates D n f and C n f of continuous and discrete Cesaro operators, respectively. In both cases the best possible rate of convergence is n -1/2 , and such a rate is attained under appropriate integrability conditions on f. Otherwise, the rates of convergence could be extremely poor, depending on the behavior of f near the boundary. We introduce probabilistic representations of D n f and C n f involving standardized sums of independent identically distributed random variables and binomial mixtures, respectively, which allow us to use the classical Berry-Esseen theorem.

Modeling and Monitoring Dynamic Systems by Chain Graphs

It is widely recognized that probabilistic graphical models provide a good framework for both knowledge representation and probabilistic inference (e.g., see [Cheeseman94], [Whittaker90]). The dynamic behaviour of a system which changes over time requires an implicit or explicit time representation. In this paper, an implicit time representation using dynamic graphical models is proposed. Our goal is to model the state of a system and its evolution over time in a richer and more natural way than other approaches together with a more suitable treatment of the inference on variables of interest.

EXACT VALUES AND SHARP ESTIMATES FOR THE TOTAL VARIATION DISTANCE BETWEEN BINOMIAL AND POISSON DISTRIBUTIONS

We present a method to obtain both exact values and sharp estimates for the total variation distance between binomial and Poisson distributions with the same mean λ. We give a simple efficient algorithm, whose complexity order is to compute exact values. Such an algorithm can be further simplified for moderate sample sizes n, provided that λ is neither close to from the left nor close to from the right. Sharp estimates, better than other known estimates in the literature, are also provided. The 0s of the second Krawtchouk and Charlier polynomials play a fundamental role.

Dynamic graphical models and nonhomogeneous hidden Markov models

We propose a dynamic graphical model which generalizes nonhomogeneous hidden Markov models. Inference and forecast procedures are developed. A comparison with an exact propagation algorithm is established and equivalence is stated.

Quantitative Estimates for Positive Linear Operators in terms of the Usual Second Modulus

We give accurate estimates of the constants appearing in direct inequalities of the form   , , ,  and   where is a positive linear operator reproducing linear functions and acting on real functions defined on the interval , is a certain subset of such functions, is the usual second modulus of , and is an appropriate weight function. We show that the size of the constants mainly depends on the degree of smoothness of the functions in the set and on the distance from the point to the boundary of . We give a closed form expression for the best constant when is a certain set of continuous piecewise linear functions. As illustrative examples, the Szasz-Mirakyan operators and the Bernstein polynomials are discussed.

Estimating transition-probabilities in a dynamic graphic model with unobservable variables

This paper emphasizes the utility of graphic models in describing partially observed dynamic systems, and establishes a method for estimating the parameters of the model. A dynamic graphic model with an associated graphic structure, which consists of a sequence of chain graphs with two consecutive graphs in the sequence connected by directed links, is described. The chain graphs describe relationships among the contemporaneous variables; the directed links describe the relations between noncontemporary variables. The paper assumes that some of the variables are unobservable when the model is in use, but partial observation of these variables is allowed in an estimation phase, by performing autopsies of the system: stopping the system and observing its state, including destructive observation. A recursive estimation method for the parameters is given and a simulation study evaluates its performance. In conclusion, the parameters of the model can be estimated, the RMS errors of the estimates are larger for those parameters associated with longer time (because of their dependence on previous estimates), and the larger the dependence between unobservable and observable variables, the better the parameter estimates.

A Graphical Model for Equipment Maintenance and Replacement Problems

From a statistical point of view, modelling stochastic temporal processes by graphical models turns out to be a suitable choice, specially when certain standard assumptions in classical modelling cannot be assumed.

Closed form expressions for Appell polynomials

We show that any Appell sequence can be written in closed form as a forward difference transformation of the identity. Such transformations are actually multipliers in the abelian group of the Appell polynomials endowed with the operation of binomial convolution. As a consequence, we obtain explicit expressions for higher order convolution identities referring to various kinds of Appell polynomials in terms of the Stirling numbers. Applications of the preceding results to generalized Bernoulli and Apostol–Euler polynomials of real order are discussed in detail.

Binomial Identities and Moments of Random Variables

Abstract We give unified simple proofs of some binomial identities, by using an elementary identity on moments of random variables.