Sidney Yakowitz

Nonparametric Density Estimation, Prediction, and Regression for Markov Sequences

Abstract Let {Xi } be a stationary Markov sequence having a transition probability density function f(y | x) giving the pdf of X i +1 | (Xi = x). In this study, nonparametric density and regression techniques are employed to infer f(y | x) and m(x) = E[X i + 1 | Xi = x]. It is seen that under certain regularity and Markovian assumptions, the asymptotic convergence rate of the nonparametric estimator mn (x) to the predictor m(x) is the same as it would have been had the Xi s been independently and identically distributed, and this rate is optimal in a certain sense. Consistency can be maintained after differentiability and even the Markovian assumptions are abandoned. Computational and modeling ramifications are explored. I claim that my methodology offers an interesting alternative to the popular ARMA approach.

Contributions to Cournot oligopoly theory

Abstract This work contributes to a number of questions concerning oligopoly models. In particular, uniqueness of the Cournot equilibrium point is demonstrated under the assumption that either the unit price function is differentiable and the derivative is strictly negative or the cost functions are strictly convex. Also, under the assumption of either strictly decreasing unit price function or strictly convex cost functions, it is shown that (a) the total production level at equilibrium increases with entry of additional players, (b) that cooperation between some of the players necessarily entails profit for the others, and (c) cooperative grouping causes decrease in production levels.

Weakly convergent nonparametric forecasting of stationary time series

The conditional distribution of the next outcome given the infinite past of a stationary process can be inferred from finite but growing segments of the past. Several schemes are known for constructing pointwise consistent estimates, but they all demand prohibitive amounts of input data. We consider real-valued time series and construct conditional distribution estimates that make much more efficient use of the input data. The estimates are consistent in a weak sense, and the question whether they are pointwise-consistent is still open. For finite-alphabet processes one may rely on a universal data compression scheme like the Lempel-Ziv (1978) algorithm to construct conditional probability mass function estimates that are consistent in expected information divergence. Consistency in this strong sense cannot be attained in a universal sense for all stationary processes with values in an infinite alphabet, but weak consistency can. Some applications of the estimates to on-line forecasting, regression, and classification are discussed.

Limits to consistent on-line forecasting for ergodic time series

This article concerns problems of time-series forecasting under the weakest of assumptions. Related results are surveyed and are points of departure for the developments here, some of which are new and others are new derivations of previous findings. The contributions in this study are all negative, showing that various plausible prediction problems are unsolvable, or in other cases, are not solvable by predictors which are known to be consistent when mixing conditions hold.

Nonparametric density and regression estimation for Markov sequences without mixing assumptions

The nonparametric estimation results for time series described in the literature to date stem fairly directly from a seminal work of M. Rosenblatt. The gist of the current picture is that under either strong or G2 mixing, many properties of nonparametric estimation in the i.i.d. case carry over to Markov sequences as well. The present work shows that many of the above results remain valid even when mixing assumptions are removed altogether. It is seen here that if the Markov process has a stationary density function, then under standard smoothness conditions, the kernel estimators of the stationary density and the auto-regression functions are asymptotically normal, with the same limiting parameters as in the i.i.d. case. Even when no stationary law exists, there are circumstances lenient enough to include ARMA processes and random walks, for which a kernel auto-regression estimator with sample-driven bandwidths is asymptotically normal. The foundation for this study is developments by Orey and Harris.

Plane steady shear flow of a cohesionless granular material down an inclined plane. A model for flow avalanches Part II: numerical results

SummaryThe granular flow model proposed by Jenkins and Savage., [2], and extended by us [1] is used here to construct numerical solutions of steady chute flows thought to be typical of such flows.We briefly state the equations and boundary conditions and present numerical solutions when the following model parameters of the Senkins and Savage model are varied: (a) the coefficient of restitution of the particles under binary collision, (b) the number of particles per layer, (c) the inclination angle of the chute, and (d) the basal and free surface boundary conditions. We demonstrate that the Jenkins and Savage model may yield physically questionable results, that those of its extension differ markedly from them and are physically more reasonable in certain cases, but yield equally questionable results in others. The results are apt to redefine research directions which granular flow modellers might want to pursue in the near future.

Plane steady shear flow of a cohesionless granular material down an inclined plane: A model for flow avalanches Part I: Theory

SummaryA continuum mechanical model describing rapid shear flow of granular materials as deduced by Jenkins and Savage (1983) [11] from considerations of statistical mechanics is applied to steady plane shear flows down an inclined chute. Depending on the type and form of the physically suggested boundary conditions that are imposed at the base and the free surface, respectively, the emerging boundary value problems permit or prohibit existence of mathematical solutions. For instance, the model does not permit incorporation of an aerodynamic drag and requires special sliding boundary conditions at the base. Cause for the singular behavior is the fact that granular pressure and fluctuation energy vanish simultaneously. Rectification is e.g. possible by including particle density gradients in the constitutive relation of granular stress, but this requires postulation of additional boundary conditions.We present the differential equations and boundary conditions and suggest a procedure of non-dimensionalization which yields the dimensionless parameters governing the problem. Construction of local solutions close to the boundaries by means of Frobenius expansions discloses the singular behavior and yields the basis for the non-existence proof under limiting conditions. Adding to the particle stress a Newtonian viscous contribution is not sufficient to regularize the problem and neither is the form of the stress tensor resulting from Lun et al.s statistical model that incorporates kinetic terms. The stress tensor must have a term proportional to the dyadic product of the particle concentration gradient with itself. Numerical solution techniques and computational results are given in a companion paper (Hutter, Szidarovszky, Yakowitz, 1986 [9]).

Weighted Monte Carlo Integration

This work considers Monte Carlo methods for approximating the integral of any twice-differentiable function f over a hypercube. Whereas earlier Monte Carlo schemes have yilded on

The application of optimal control methodology to nonlinear programming problems

O({1 / n})

Small-Sample Hypothesis Tests of Markov Order, with Application to Simulated and Hydrologic Chains

convergence rate for the expected square error, we show that by allowing nonlinear operations on the random samples