Abstract: I would like to congratulate Professor Antoniadis for successfully outlining the current state-of-art of wavelet applications in statistics. Since wavelet techniques were introduced to statistics in the early 90’s, the applications of wavelet techniques have mushroomed. There is a vast forest of wavelet theory and techniques in statistics and one can find himself easily lost in the jungle. The review by Antoniadis, ranging from linear wavelets to nonlinear wavelets, addressing both theoretical issues and practical relevance, gives in-depth coverage of the wavelet applications in statistics and guides one entering easily into the realm of wavelets.

Abstract: In this paper the notion of coherent prevision given by de Finetti for bounded random quantities is extended to arbitrary ones It is shown that the main properties of de Finetti’s prevision are preserved by the extended notion and some of his conjectures on previsions of unbounded random quantities are proved Finally, a representation theorem for finite previsions in terms of Riemann-Stieltjes integral is given for random quantities defined on a common partition of the certain event and an ensuing possible interpretation for modelling real situations is discussed

Abstract: Consider the differential equation dX(t) = A(X(t))dt, X(t 0 ) = given, where the function A depends on some unknown (vector) parameter θ to be estimated from a realization X(t 0 ) = x0, X(t1) = x1,…, X(tn) = x n of a process X related to X but which incorporates measurement errors. As an example, consider the logistic equation for population growth in a finite environment and its solution $$ A(x) = rx\left( {1 - x/k} \right);\,\,X(t) = k/\left( {1 + \left( {k/X\left( {{t_0}} \right) - 1} \right)\exp \left\{ { - r\left( {t - {t_0}} \right)} \right\}} \right) $$ Two of the most common approaches to the estimation of the unknown parameters in the Presence of errors are Regression and Diffusion. We will follow an approach which models the randomness as pertaining inherently to the time lag between apparent time X−1(x j ) and chronological time t j . The observed process X(t) is viewed as X(t)=X(T(t)).

Abstract: In this paper a simple characterization of the geometric distribution, in the class of discrete distributions with monotone hazard ratio, is provided. This result is used to construct a test for the hypothesis that the anival process of a discrete queueing model is a geometric process. The properties of the test, as well as those of its «bootstrapped version », are studied both theoretically and by Monte Carlo simulation.

Abstract: In this paper, starting from some well known results for the structural identifiability of VARMA models of given maximum time lagsp andq, we derive parametric conditions which guarantee the identification of Generalized STARMA models

Abstract: A necessary condition for identification of latent class models is that the number of unknown independent parameters must not be greater than the number of observed cells in the contingency table. Such condition is not sufficient at all. Verifying Goodman’s sufficient condition for local identifiability may be, for complex models, a cumbersome procedure. In any case, local identifiability does not guarantee global identifability. The paper provides rules to ascertain global identifiability of some specifications of latent class Markov models, expressing the unknown parameters as a function of the observed frequencies. In the case that not all parameters of a model are identified, the outlined rules provide hints about the restrictions to impose in order to obtain fully identified models.

Abstract: This paper considers the asymptotic theory of estimating odds ratios ink tables when there is uncertain prior information about homogeneity constraint on them. Using a preliminary test approach, we propose seven estimators and study their properties of asymptotic dominance under local alternatives. In the process, we propose a Wald-type test statistic for testing homogeneity and obtain its asymptotic distribution under local alternatives.

Abstract: This paper develops a method of adaptive modeling that may be applied to forecast nonstationary time series. The starting point are time-varying coefficients models introduced in statistics, econometrics and engineering. The basic step of modeling is represented by the implementation of adaptive recursive estimators for tracking parameters. This is achieved by unifying basic algorithms — such as recursive least squares (RLS) and extended Kalman filter (EKF) — into a general scheme and next by selecting its coefficients with the minimization of the sum of squared prediction errors. This defines a non-linear estimation problem that may be analyzed in the context of the conditional least squares (CLS) theory. Anumerical application on the IBM stock price series of Box-Jenkins illustrates the method and shows its good forecasting ability.

Abstract: We consider the semiparametric problem of estimating a functional of the unknown drift coefficient of a dynamical system by the observation of a diffusion process with small diffusion coefficient. Under smoothness assumptions, we define an asymptotic minimax lower bound for the risk of any estimator and we propose an asymptotically efficient estimator according to this bound. The estimated integral-type functional is defined by means of the (unknown) solution of the deterministic limit dynamical system.

Abstract: We consider a model involvingk competing risks when the random variables of interest (risks) are censored from the left with the unobservable random variable. The nonparametrical estimators for survival functions of risks are presented and the estimators are strongly approximated with the best rates by appropriate Gaussian processes.

Abstract: An intriguing economic phenomenon in consumption studies is the law of diminishing return. In spite of its economic importance, this phenomenon has not received attention by stochastic modelers. In this article, an attempt is made to explain this phenomenon via a new probability model. Both the estimation and hypothesis testing procedures for the model parameters are derived, and illustrated using data on the number of viewers of a weekly television show in Denver, USA.

Abstract: This paper gives characterizations of normal and gamma distributions .via conditional structure.

Abstract: In the paper we propose a way to test the null hypothesis h(θ)=0 when the information matrix is singular. The approach is based on the introduction of a quadratic penalty log-likelihood function with the penalty parameter close to zero (but not zero). This slight change of the log-likelihood allows to obtain a consistent estimator close to the maximum likelihood estimator and whose limiting distribution is Normal with variance-covariance matrix given by the Moore-Penrose pseudoinverse of the information matrix. This result allows to obtain an asymptoticx 2-distribution that may be used in hypothesis testing when the infonnation matrix is singular.

Abstract: Professor Antoniadis deserves our warmest compliments for his complete and very clear review of this topic. In our discussion we want to point out several connections between wavelets (and their use) and Bayesian nonparametric statistics.

Abstract: This article deals with the linear Bayes estimator (LBE) for incorrect prior assumptions under the misspecified linear regression model. In order to obtain the mean square error (MSE)-matrix properties of theLBE, we compare theMSE-matrix of theLBE averaging over the correct prior assumptions with that averaging over incorrect prior assumptions. We also compare theMSE-matrices of the ordinary least square estimator (OLSE) and theLBE under correct prior assumptions, given that the original prior assumptions are incorrect. Finally, we examine theMSE-matrix properties of theLBE in prediction problems.

Abstract: We derive confidence interval procedures for the intraclass correlation coefficient σ based on the likelihood score, the bias corrected score and the bias and skewness corrected score These procedures are then compared, through simulation, with six other procedures, compared by Donner and Wells (1986), in terms of coverage probabilities and coverage lengths A methods due to Thomas and Hultquist (1978, the BAL method) and the method based on the maximum likelihood estimate of σ (the ML method) provide, on the average, the shortest lengths Both these methods are overly liberal Another method of Thomas and Hultquist (the TH method) and the method based on Fisher’s Z-transform (the F method) provide largest lengths, on the average The TH method is liberal and the F method is overly conservative The remaining methods, on the average, clump together in terms of average coverage lengths However, the bias corrected score procedure (the BAB Method) performs best in terms of coverage probabilities, in the sense that the coverage probabilities are closest, on the average, to the nominal confidence coefficient A closer look at the simulation results indicate that when σ is very small (ie σ ≤ 1) the BAL method is best, otherwise the BAB method is best

Abstract: This paper addresses the problem of choosing the optimal number of basis functions in constructing wavelet series density estimators. It is well known thatprojection estimators tend to overfit the density if the number of basis functions in the orthogonal expansion is too large. In extreme cases the estimator is close to the Dirac function concentrated at the observations. We propose a roughness measure of wavelet estimators and establish a data driven method for determining the number of levels to be included in the estimate. Our method exploits the idea of using the Fisher information functional as a roughness measure. The method is demonstrated on simulated data.