Testing Parametric versus Semiparametric Modeling in Generalized Linear Models

Adaptive signal processing algorithms are often used in order to track an unknown time-varying parameter vector. This work develops an upper bound on the mean of the norm-squared error between the unknown parameter vector being tracked and the value obtained by the algorithm. The results require very mild covariance decay rate conditions on the training data and a bounded algorithm. The upper bound illustrates the relationship between the algorithm step size and the maximum rate of variation in the parameter vector being tracked.

We consider estimation in a semiparametric generalized linear model for clustered data using estimating equations. Our results apply to the case where the number of observations per cluster is finite, whereas the number of clusters is large. The mean of the outcome variable μ is of the form g(μ) = XTβ + θ(T), where g(·) is a link function, X and T are covariates, β is an unknown parameter vector, and θ(t) is an unknown smooth function. Kernel estimating equations proposed previously in the literature are used to estimate the infinite-dimensional nonparametric function θ(t), and a profile-based estimating equation is used to estimate the finite-dimensional parameter vector β. We show that for clustered data, this conventional profile-kernel method often fails to yield a √n-consistent estimator of β along with appropriate inference unless working independence is assumed or θ(t) is artificially undersmoothed, in which case asymptotic inference is possible. To gain insight into these results, we derive the semiparametric efficient score of β, which is found to have a complicated form, and show that, unlike for independent data, the profile-kernel method does not yield a score function asymptotically equivalent to the semiparametric efficient score of β, even when the true correlation is assumed and θ(t) is undersmoothed. We illustrate the methods with an application to infectious disease data and evaluate their finite-sample performance through a simulation study.

The present chapter deals with the study of an important but least known case in the theory of parametric estimation of controlled MS, the case when the vector of unknown system parameters, or that of unknown perturbations affecting the control object, is an unknown (uncontrollable, nonmeasurable) vector function of time. Certain models of solving the problems of adaptive stabilization and optimization under parametric drift conditions, based on utilizing the formalism of Lyapunov functions, were given in earlier researches under a variety of restrictions, for example on the information about the drift model and the rate of change of the parameters. The dominant difficulties that occur in these problems of forming convergent estimation algorithms, concern the proof of the fact that the Lyapunov function monotonically decreases on the trajectories of the controlled process.

In this note, we consider two of the major issues that have arisen in implementing a sequential quadratic programming (SQP) method for nonlinearly constrained optimization problems (the code NPSOL; Gill et al. , 1983). The problem of concern is assumed to be of the form[EQUATION]where F(x) is a smooth nonlinear function, A L is a constant matrix, and c(x) is a vector of smooth nonlinear constraint functions. The matrix A L and the vector c(x) may be empty. Note that upper and lower bounds are specified for all the variables and for all the constraints. This from allows full generality in constraint specification. In particular, the i -th constraint may be defined as an equality by setting l i = u i . If certain bounds are not present, the associated elements of l or u can be set to special values that will be treated as - ∞ or +∞.

Stability analysis of high-order Hopfield type neural networks with uncertainty

The iterative method, which can be applied to estimation of the unknown vector of discrete parameters in combination with filtering of the unknown vector of continuous signal parameters, is proposed. It is demonstrated that this method makes it possible to construct highly efficient algorithms for joint demodulation and filtering of communications channel parameters in code-division multiple-access and orthogonal frequency-division multiplexing systems.

This paper considers the estimation problem for linear continuous-time systems with the interrupted observation mechanism which is characterized in terms of the jump Markov process taking on the values of 0 or 1. The minimum variance estimator algorithm is derived. The approach adopted here is that we express the jump process in terms of the initial value and the jump times instead of its instantaneous values and that in order to apply Lainiotis' formula we regard the initial value and jump times as the unknown system parameter vector. The parameter vector is, however, infinite dimensional so that Lainiotis' formula cannot be applied directly. This is because the formula is applicable only to linear systems with a finite dimensional parameter vector. Therefore, the minimum variance estimator algorithm is derived by using some limiting procedure. The resultant optimal algorithm is infinite dimensional, so that feasible approximate algorithms should be developed for practical implementation. A simple illustrating example is also worked out.

Adaptive signal processing algorithms derived from LS (least squares) cost functions are known to converge extremely fast and have excellent capabilities to "track" an unknown parameter vector. This paper treats analytically and experimentally the steady-state operation of RLS (recursive least squares) adaptive filters with exponential windows for stationary and nonstationary inputs. A new formula for the "estimation-noise" has been derived involving second- and fourth-order statistics of the filter input as well as the exponential windowing factor and filter length. Furthermore, it is shown that the adaptation process associated with "lag effects" depends solely on the exponential weighting parameter λ. In addition, the calculation of the excess mean square error due to the lag for an assumed Markov channel provides the necessary information about tradeoffs between speed of adaptation and steady-state error. It is also the basis for comparison to the simple LMS algorithm, in a simple case of channel identification, it is shown that the LMS and RLS adaptive filters have the same tracking behavior. Finally, in the last part, we present new RLS restart procedures applied to transversal structures for mitigating the disastrous results of the third source of noise, namely, finite precision arithmetic.

We consider a semi-parametric model for recurrent events. The model consists of an unknown hazard rate function, the infinite-dimensional parameter of the model, and a parametrically specified effective age function. We will present a condition on the family of effective age functions under which the profile likelihood function evaluated at the parameter vector theta, say, exceeds the profile likelihood function evaluated at the parameter vector (theta) over bar, say, with probability p. From this we derive a condition under which profile likelihood inference for the finite-dimensional parameter of the model leads to inconsistent estimates. Examples will be presented. In particular, we will provide an example where the profile likelihood function is monotone with probability one regardless of the true data generating process. We also discuss the relation of our results to other semi-parametric models like the accelerated failure time model and Cox's proportional hazards model.

A robust output feedback control which globally stabilizes a class of SISO minimum-phase nonlinear systems with known and constant relative degree containing a vector of unknown parameters is developed. The constant parameter vector is not restricted to enter linearly in the state equations but it is assumed to belong to a known compact set and an imprecise knowledge of the nonlinearities (e.g., lookup tables) is allowed. The class of nonlinear systems is determined by geometric conditions; an additional assumption, which generalizes the knowledge of the sign of high-frequency gain for linear systems, is also required. The nonlinearities are restricted to depend, in suitable coordinates, on the output only: no growth conditions, such as sector or Lipschitz, are required. The robust output feedback control stabilizes the system for every value of the parameter vector in a known compact set. The order of the compensator is equal to the relative degree minus one. >

The identification problem of systems with logarithmic quantized data is analyzed in this paper. A discrete-time system with an unknown parameter vector and Gaussian noise is considered. The output of the system is quantized by a logarithmic quantizer. The estimator of the unknown parameter vector is obtained using the quantile-based estimation. The convergence almost everywhere of the estimator and the upper bound of the estimation error are also analyzed, respectively.

This paper presents a solution to a robust optimal regulation problem for a nonlinear polynomial system affected by parametric and matched uncertainties, which is based only on partial state information. The parameters describing the dynamics of the nonlinear polynomial plant depend on a vector of unknown parameters, which belongs to a finite parametric set, and the application of a certain control input is associated with the worst or least favorable value of the unknown parameter. A high-order sliding mode state reconstructor is designed for the nonlinear plant in such a way that the previously designed control can be applied for a system with incomplete information. Additionally, the matched uncertainty is also compensated by means of the same output-based regulator. The obtained algorithm is applied to control a mechanical pendulum of the third order, successfully verifying the effectiveness of the developed approach.

In this paper, we present a solution to the problem of the quadratic Mini-Max regulator for polynomial uncertain systems. The main characteristic of this type of problems is that the parameters describing the dynamic of the nonlinear plant depend on a vector of unknown parameters, which belongs to a finite parametric set, and the solution is given in terms of the worst case scenario. That is to say, the result of the application of a certain control input (in terms of the cost function value) is associated with the worst or least favorable value of the unknown parameter. Based on the general necessary conditions for mini-max optimality, a closed form for the control is provided, which makes use of a p-linear form tensor representation of the polynomial system. In its turn, this allows one to present the solution in a way similar to the so called Riccati technique. The final control is shown to be a convex combination (with some weights) of the optimal controls for each fixed parameter of the polynomial system. A numerical example for the robust regulation of the well-known Duffing equation which represents a typical (and challenging to control) nonlinear polynomial system is presented, to show the effectiveness of our approach.

Let \( \{ ({x'_i},{y_i})\} _{i = 1}^N \) be an iid sample drawn from a known distribution F(x i,y i, s), where s is a k × 1 vector of unknown parameters. Let f y|x (y, β) denote the likelihood function of y | x, which is the density function of y | x if y |x is continuous or the probability of y | x if y | x is discrete. Define f x (x) analogously, which is not a function of s.

The paper considers the problem of estimating the parameters in a continuous time regression model with a non-Gaussian noise of pulse type. The vector of unknown parameters is assumed to belong to a compact set. The noise is specified by the Ornstein–Uhlenbeck process driven by the mixture of a Brownian motion and a compound Poisson process. Improved estimates for the unknown regression parameters, based on a special modification of the James–Stein procedure with smaller quadratic risk than the usual least squares estimates, are proposed. The developed estimation scheme is applied for the improved parameter estimation in the discrete time regression with the autoregressive noise depending on unknown nuisance parameters.