Real-valued Parameters Research Articles

The optimal projection equations with Petersen-Hollot bounds: robust stability and performance via fixed-order dynamic compensation for systems with structured real-valued parameter uncertainty

A feedback control design problem involving structured real-valued plant parameter uncertainties is considered. A quadratic Lyapunov bound suggested by recent work of I.R. Petersen and C.V. Hollot (1986) is utilized in conjunction with the guaranteed cost approach of S.S.L. Chang and T.K.C. Peng (1972) to guarantee robust stability with robust performance bound. Necessary conditions that generalize the optimal projection equations for fixed-order dynamic compensation are used to characterize the controller that minimizes the performance bound. The design equations thus effectively serve as sufficient conditions for synthesizing dynamic output-feedback controllers that provide robust stability and performance.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the attainment of a lower bound for the bates risk in estimating a parametric function

A variation of a BAYESian analogue to the RAO-CRAMER Inequality Is examined under the assumption of a quadratic loss function for extimating some function τ(θ) of a real-valued parameter θ A necessary and sufficient condition under which the BAYES estimator attains the lower bound for the BAYES risk provided by the inequality is given when the posterior density is of the exponential type. As application an inequality involving gamma functions Is then derived

A NOTE ON EMBEDDING A DISCRETE PARAMETER ARMA MODEL IN A CONTINUOUS PARAMETER ARMA MODEL

Abstract. We have shown that it is not always possible to embed a real‐valued discrete parameter Gaussian AR(1) model in a real‐valued continuous parameter Gaussian AR(1). The problem with general ARMA models is also discussed.

Pseudo-Stokes parameter representation of light propagation in layered inhomogeneous uniaxial media in the geometric optics approximation

We study the propagation of normally incident, elliptically polarized light in layered inhomogeneous uniaxial structures by using the geometric optics approximation (GOA). We show that, in the GOA, light propagation can be described by the evolution of four real-valued pseudo-Stokes parameters. These parameters are linearly related to the familiar Stokes parameters and obey a simple first-order differential equation. Conservation of energy and angular momentum is preserved under this approximation.

Multiple Comparisons With the Best Treatment

Abstract Let π1, π2, …, π k be k ≥ 2 sources of observations (treatments, populations) and suppose the “goodness” of treatment π i is characterized by the size of an unknown real-valued parameter θ i . Let θ[k] = max1≤i≤k θ i . If π i is preferred to π j when θ i > θ j , the parameters δ i = θ[k] — θ i , i = 1, 2, …, k reflect in an inverse sense the “goodness” of each treatment relative to the “best” treatment. A general technique for obtaining simultaneous confidence intervals on the δ i is demonstrated with several examples. This technique can be applied in any setting where comparison-with-control intervals can be computed regarding any π j as the control. These results have special importance in ranking and selection problems in that the process of generating upper bounds on the δ i generates traditional confidence statements of both the indifference zone and the subset selection schools, simultaneously, as established by Hsu (1981).

Multiple Comparisons with the Best Treatment

Abstract Let π1, π2, …, π k be k ≥ 2 sources of observations (treatments, populations) and suppose the “goodness” of treatment π i is characterized by the size of an unknown real-valued parameter θ i . Let θ[k] = max1≤i≤k θ i . If π i is preferred to π j when θ i > θ j , the parameters δ i = θ[k] — θ i , i = 1, 2, …, k reflect in an inverse sense the “goodness” of each treatment relative to the “best” treatment. A general technique for obtaining simultaneous confidence intervals on the δ i is demonstrated with several examples. This technique can be applied in any setting where comparison-with-control intervals can be computed regarding any π j as the control. These results have special importance in ranking and selection problems in that the process of generating upper bounds on the δ i generates traditional confidence statements of both the indifference zone and the subset selection schools, simultaneously, as established by Hsu (1981).

A Universal Prior for Integers and Estimation by Minimum Description Length

An earlier introduced estimation principle, which calls for minimization of the number of bits required to write down the observed data, has been reformulated to extend the classical maximum likelihood principle. The principle permits estimation of the number of the parameters in statistical models in addition to their values and even of the way the parameters appear in the models; i.e., of the model structures. The principle rests on a new way to interpret and construct a universal prior distribution for the integers, which makes sense even when the parameter is an individual object. Truncated real-valued parameters are converted to integers by dividing them by their precision, and their prior is determined from the universal prior for the integers by optimizing the precision.

Model robust confidence intervals

Abstract Confidence intervals are constructed for real-valued parameter estimation in a general regression model with normal errors. When the error variance is known these intervals are optimal (in the sense of minimizing length subject to guaranteed probability of coverage) among all intervals estimates which are centered at a linear estimate of the parameter. When the error variance is unknown and the regression model is an approximately linear model (a class of models which permits bounded systematic departures from an underlying ideal model) then an independent estimate of variance is found and the intervals can then be appropriately scaled.

Nuisance parameters, mixture models, and the efficiency of partial likelihood estimators

This paper establishes lower bounds for estimation in parametric statistical models in which one wishes to estimate a real-valued parameter of interest in the presence of nuisance parameters which are accruing in number in direct proportion to the number of independent observations. The formal setting requires that the nuisance parameters be independent observations from an unknown distribution. In this setting an information measure analogous to the Fisher information is derived. It is then used to generate lower bounds for the variance of unbiased estimators and also for the asymptotic variance of consistent asymptotically normal estimators. Under certain conditions, consistent asymptotically normal estimators can be generated by maximizing factors of the complete likelihood, even though the maximum likelihood estimator is inconsistent. These estimators can be fully efficient in the sense of meeting the lower bounds despite their apparent wasteful use of the likelihood, as is demonstrated, in several important examples, by the use of a natural sufficient condition.

The use of Markov Random Fields as models of texture

We propose Markov Random Fields (MRFs) as probabilistic models of digital image texture where a textured region is viewed as a finite sample of a two-dimensional random process describable by its statistical parameters. MRFs are multidimensional generalizations of Markov chains defined in terms of conditional probabilities associated with spatial neighborhoods. We present an algorithm that generates an MRF on a finite toroidal square lattice from an independent identically distributed (i.i.d.) array of random variables and a given set of independent real-valued statistical parameters. The parametric specification of a consistent collection of MRF conditional probabilities is a general result known as the MRF-Gibbs Random Field (GRF) equivalence. The MRF statistical parameters control the size and directionality of the clusters of adjacent similar pixels which are basic to texture discrimination and thus seem to constitute an efficient model of texture. In the last part of this paper we outline an MRF parameter estimation method and goodness of fit statistical tests applicable to MRF models for a given unknown digital image texture on a finite toroidal square lattice. The estimated parameters may be used as basic features in texture classification. Alternatively these parameters may be used in conjunction with the MRF generation algorithm as a powerful data compression scheme.

An essentially complete class of multiple decision procedures

Abstract Let π 1 ,…, π k represent k (⩾2) independent populations. The quality of the i th population π i is characterized by a real-valued parameter θ i , usually unknown. We define the best population in terms of a measure of separation between θ i 's. A selection of a subset containing the best population is called a correct selection (CS). We restrict attention to rules for which the size of the selected subset is controlled at a given point and the infimum of the probability of correct selection over the parameter space is maximized. The main theorem deals with construction of an essentially complete class of selection rules of the above type. Some classical subset selection rules are shown to belong to this class.

Inference in contingency tables with ordered categories using Plackett's coefficient of association for bivariate distributions

SUMMARY By using a one-parameter class of bivariate distributions proposed by Plackett (1965), methods are derived for analysing the pattern of association in contingency tables with ordered categories. In an r x c table (r - 1) (c - 1) estimates of this parameter can be con- sidered and all of them, or subsets of them, can be tested for equality. This can answer the question whether the pattern of association is consistently the same over the whole table or not. Two well-known examples are considered using this method. The problems arising in the analysis of contingency tables with ordered categories usually go further than mere tests for independence. Generally, a specific analysis of the pattern of association between the two categories is required. For this purpose, contingency tables with ordered categories can be considered as samples from bivariate distributions. Models for symmetry, quasisymmetry, marginal homogeneity or palindromic symmetry can be applied (Bishop, Fienberg & Holland, 1975, p. 281 ff.; McCullagh, 1978) in order to analyse such tables. It is the major aim of these analyses to evaluate differences between the marginal variables by taking into account the association between them. In the present paper the central emphasis is put on the association itself. We use a class of bivariate distributions proposed by Plackett (1965) which seems to be very useful in this context. This class of bivariate distributions, introduced in detail in ? 2, is characterized by one real-valued parameter representing a measure of association. This parameter is closely related to the cross-product ratio in 2 x 2 tables. In an r x c table this parameter can be estimated in (r - 1) (c - 1) different ways. We derive the asymptotic distribution for this set of estimates. This enables us to construct tests for the equality of all parameters estimated or certain subsets of them. By this means we can determine whether the structure of association between the two marginal random variables is consistently the same over the whole table. This would mean that the bivariate distribu- tion, incorporating the interaction of the two marginal variables, can be explained by a simple one-parameter model. The interpretation of the model depends on the numerical value of the parameter and ranges from negative association over independence to positive association. Regardless of the numerical values of the estimates, a rejection of the hypothesis of equality of the para- meters would indicate that there is an interaction between the marginal variables which

Modeling by shortest data description

The number of digits it takes to write down an observed sequence x 1, …, x N of a time series depends on the model with its parameters that one assumes to have generated the observed data. Accordingly, by finding the model which minimizes the description length one obtains estimates of both the integer-valued structure parameters and the real-valued system parameters.

Minimax codes for finite alphabets (Corresp.)

Minimax codes are designed for applications where the symbol probabilities are not known except for their order. A minimax code is defined by real-valued parameters/, (resembling the lengths of codewords) which minimize the maximum ratio <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(\Sigma_{i}p_{i}l_{i})/(--\Sigma_{i}p_{i}\logp_{i})</tex> subject to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Sigma_{i}2^{-l_{i}}\leq1</tex> over the set of monotonically nonincreasing distributions such that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p_{1}\leq1/m</tex> for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> an integer greater than one.

Estimation of ordered parameters from k stochastically increasing distributions

AbstractThere are given k (⩾ 2) univariate cumulative distribution functions (c.d.f.'s) G(x; θi) indexed by a real‐valued parameter θi, i=1,…, k. Assume that G(x; θi) is stochastically increasing in θi. In this paper interval estimation on the ith smallest of the θ's and related topics are studied. Applications are considered for location parameter, normal variance, binomial parameter, and Poisson parameter.

Monotonicity of a Power Function: An Elementary Probabilistic Proof

Abstract It is pointed out that in many one-sided testing situations for a real-valued parameter θ, the monotonicity of the power function hinges on the stochastic order of the underlying family of distributions [Fθ] rather than on the stronger property of monotone likelihood ratio of the family. An elementary proof, accessible to students of introductory probability and statistics, is presented.

Moment space upper and lower error bounds for digital systems with intersymbol interference

A method for the evaluation of upper and lower bounds to the error probability of a linear pulse-amplitude modulation (PAM) system with bounded intersymbol interference and additive Gaussian noise is obtained via an isomorphism theorem from the theory of moment spaces. These upper and lower bounds are seen to be equivalent to upper and lower envelopes of some compact convex body generated from a set of kernel functions. Depending on the selection of these kernels and their corresponding moments, different classes of bounds are obtained. In this paper, upper and lower bounds that depend on the absolute moment of the intersymbol interference random variable, the second moment, the fourth moment, and an "exponential moment" are found by analytical, graphical, or iterative approaches. We study in detail the exponential moment case and obtain a family of new upper and a family of new lower bounds. Within each family, expressions for these bounds are given explicitly as a function of an arbitrary real-valued parameter. For two channels of interest, upper and lower bounds are evaluated and compared. Results indicate these bounds to be tight and useful.

On the Solution of an Integral Equation of Convolution Type

The solution $Q_\lambda ^\mu $ of the integral equation $xQ_\lambda ^\mu (x) = [\{ (\lambda - \mu ) + \mu f\} ^ * Q_\lambda ^\mu ](x)$ is considered as a transform of the function f; f is a function of the real variable x, and $\lambda $ and $\mu $ are real-valued parameters with $\lambda $ positive. The functions $Q_\lambda ^\mu [f]$ with f fixed form a group under the convolution operator. The functions $F_\lambda ^\mu (x) = \Gamma (\lambda )x^{1 - \lambda } Q_\lambda ^\mu [f](x)$, occurring in elementary particle physics, exhibit properties, such as their analyticity and behavior at the origin and at infinity, similar to those of the input function $f(x)$. Besides several applications concerning hypergeometric functions, a table is presented giving various input functions f and their transforms $F_\lambda ^\mu $.

Bidirectional Unbiased Procedures

Abstract When the hypothesis H 0: θ = θ0 is tested against the alternative Ha: θ ≠ θ0, θ a real-valued parameter, it would often be more appropriate to make simultaneous tests of the two hypotheses 1 H 0: θ ≤ θ0 (with alternative 1 Ha: θ > θ0) and 2 H 0: θ ≥ θ0 (with alternative 2 Ha: θ < θ0). The criterion of unbiasedness in choosing an optimal test of H 0 is inapplicable for choosing an optimal simultaneous procedure for testing 1 H 0 and 2 H 0. A bidirectional unbiased procedure is defined and its properties enumerated. In many important applications such a procedure for testing 1 H 0 and 2 H 0 is essentially equivalent to an “equal tails” test of H 0.

Bidirectional Unbiased Procedures

Abstract When the hypothesis H 0: θ = θ0 is tested against the alternative Ha: θ ≠ θ0, θ a real-valued parameter, it would often be more appropriate to make simultaneous tests of the two hypotheses 1 H 0: θ ≤ θ0 (with alternative 1 Ha: θ > θ0) and 2 H 0: θ ≥ θ0 (with alternative 2 Ha: θ < θ0). The criterion of unbiasedness in choosing an optimal test of H 0 is inapplicable for choosing an optimal simultaneous procedure for testing 1 H 0 and 2 H 0. A bidirectional unbiased procedure is defined and its properties enumerated. In many important applications such a procedure for testing 1 H 0 and 2 H 0 is essentially equivalent to an “equal tails” test of H 0.