Stability of the roots of random algebraic polynomials

In this paper we consider the behavior of the roots of random algebraic polynomials. A code was developed which generates a sample of random algebraic polynomials, calculates the roots of each sample polynomial, and then calculates the averages of the roots. Finally, the roots of the deterministic algebraic polynomial whose coefficients are the averages of the sample coefficients are calculated. These data are then tabulated and graphically displayed. The relationship between the averages of the roots of the sample polynomials and the roots of the average polynomial is discussed.     

Estimating Mixing Densities in Exponential Family Models for Discrete Variables

This paper is concerned with estimating a mixing density g using a random sample from the mixture distribution f(x)=∫f x | θ)g(θ)dθ where f(· | θ) is a known discrete exponen tial family of density functions. Recently two techniques for estimating g have been proposed. The first uses Fourier analysis and the method of kernels and the second uses orthogonal polynomials. It is known that the first technique is capable of yielding estimators that achieve (or almost achieve) the minimax convergence rate. We show that this is true for the technique based on orthogonal polynomials as well. The practical implementation of these estimators is also addressed. Computer experiments indicate that the kernel estimators give somewhat disappoint ing finite sample results. However, the orthogonal polynomial estimators appear to do much better. To improve on the finite sample performance of the orthogonal polynomial estimators, a way of estimating the optimal truncation parameter is proposed. The resultant estimators retain the convergence rates of the previous estimators and a Monte Carlo finite sample study reveals that they perform well relative to the ones based on the optimal truncation parameter.     

Bayesian curve estimation by polynomial of random order

A Bayesian method of estimating an unknown regression curve by a polynomial of random order is proposed. A joint distribution is assigned over both the set of possible orders of the polynomial and the polynomial coefficients. Reversible jumps Markov chain Monte Carlo (MCMC) (Green, Biometrika 82 (1995) 711-32), are used to compute required posteriors. The methodology is extended to polynomials of random order with discontinuities and to piecewise polynomials of random order to handle wiggly curves. The effectiveness of the methodology is illustrated with a number of examples.     

Least squares adaptive polynomials

A concept of adaptive least squares polynomials is introduced for modelling time series data. A recursion algorithm for updating coefficients of the adaptive polynomial (of a fixed degree) is derived. This concept assumes that the weights w are such that i) the importance of the data values, in terms of their weights, relative to each other stays fixed, and that ii) they satisfy the update property, i.e., the polynomial does not change if a new data value is a polynomial extrapolate. Closed form results are provided for exponential weights as a special case as they are shown to possess the update property when used with polynomials.

The concept of adaptive polynomials is similar to the linear adaptive prediction provided by the Kalman filter or the Least Mean Square algorithm of Widrow and Hoff. They can be useful in interpolating, tracking and analyzing nonstationary data.     

Addendum to fitting polynomials to main effects in multi factor experiments: A note concerning polynomial coefficients when all factors are quantitative

When all factors are quantitative, cell means may be expressed as a polynomial function of products of powers of the associated quantitative classification variables. Existence and uniqueness of this polynomial is established for unbalanced data with unequal spacing for no missing cells. The relationship between the coefficients of this polynomial and the main effects polynomials are derived for main effects which are arbitrary weighted averages of the cell means.     

Local orthogonal polynomial expansion for density estimation

A local orthogonal polynomial expansion (LOrPE) of the empirical density function is proposed as a novel method to estimate the underlying density. The estimate is constructed by matching localised expectation values of orthogonal polynomials to the values observed in the sample. LOrPE is related to several existing methods, and generalises straightforwardly to multivariate settings. By manner of construction, it is similar to local likelihood density estimation (LLDE). In the limit of small bandwidths, LOrPE functions as kernel density estimation (KDE) with high-order (effective) kernels inherently free of boundary bias, a natural consequence of kernel reshaping to accommodate endpoints. Consistency and faster asymptotic convergence rates follow. In the limit of large bandwidths LOrPE is equivalent to orthogonal series density estimation (OSDE) with Legendre polynomials, thereby inheriting its consistency. We compare the performance of LOrPE to KDE, LLDE, and OSDE, in a number of simulation studies. In terms of mean integrated squared error, the results suggest that with a proper balance of the two tuning parameters, bandwidth and degree, LOrPE generally outperforms these competitors when estimating densities with sharply truncated supports.     

Orthogonal comparisons for Fourier analysis within mixed level factorial experiments

Tables of orthogonal comparisons for Fourier analysis have been derived from general expressions applicable to periodic data over one cycle, for equal spacing of the levels of the independent variable. The orthogonal comparisons can be used in conjunction with orthogonal polynomials to analyse factorial experiments with any number of levels and factors of periodic or polynomial type. The appropriate grouping of mean squares for significance testing is considered. A numerical example is given.     

Improved orthogonal polynomial density estimates

Density estimates that are expressible as the product of a base density function and a linear combination of orthogonal polynomials are considered in this paper. More specifically, two criteria are proposed for determining the number of terms to be included in the polynomial adjustment component and guidelines are suggested for the selection of a suitable base density function. A simulation study reveals that these stopping rules produce density estimates that are generally more accurate than kernel density estimates or those resulting from the application of the Kronmal–Tarter criterion. Additionally, it is explained that the same approach can be utilized to obtain multivariate density estimates. The proposed orthogonal polynomial density estimation methodology is applied to several univariate and bivariate data sets, some of which have served as benchmarks in the statistical literature on density estimation.     

An algebraic and Bayesian technology for object recognition

This paper presents a new robust, low computational cost technology for recognizing free-form objects in three-dimensional (3D) range data, or, in two dimensional (2D) curve data in the image plane. Objects are represented by implicit polynomials (i.e. 3D algebraic surfaces or 2D algebraic curves) of degree greater than two, and are recognized by computing and matching vectors of their algebraic invariants (which are functions of their coefficients that are invariant to translations, rotations and general linear transformations). Such polynomials of the fourth degree can represent objects considerably more complicated than quadrics and super-quadrics, and can realize object recognition at significantly lower computational cost. Unfortunately, the coefficients of high degree implicit polynomials are highly sensitive to small changes in the data to which the polynomials are fit, thus often making recognition based on these polynomial coefficients or their invariants unreliable. We take two approaches to the problem: one involves restricting the polynomials to those which represent bounded curves and surfaces, and the other approach is to use Bayesian recognizers. The Bayesian recognizers are amazingly stable and reliable, even when the polynomials have unbounded zero sets and very large coefficient variability. The Bayesian recognizers are a unique interplay of algebraic functions and statistical methods. In this paper, we present these recognizers and show that they work effectively, even when data are missing along a large portion of an object boundary due, for example, to partial occlusion.     

An algebraic and Bayesian technology for object recognition

This paper presents a new robust, low computational cost technology for recognizing free-form objects in three-dimensional (3D) range data, or, in two dimensional (2D) curve data in the image plane. Objects are represented by implicit polynomials (i.e. 3D algebraic surfaces or 2D algebraic curves) of degree greater than two, and are recognized by computing and matching vectors of their algebraic invariants (which are functions of their coefficients that are invariant to translations, rotations and general linear transformations). Such polynomials of the fourth degree can represent objects considerably more complicated than quadrics and super-quadrics, and can realize object recognition at significantly lower computational cost. Unfortunately, the coefficients of high degree implicit polynomials are highly sensitive to small changes in the data to which the polynomials are fit, thus often making recognition based on these polynomial coefficients or their invariants unreliable. We take two approaches to the problem: one involves restricting the polynomials to those which represent bounded curves and surfaces, and the other approach is to use Bayesian recognizers. The Bayesian recognizers are amazingly stable and reliable, even when the polynomials have unbounded zero sets and very large coefficient variability. The Bayesian recognizers are a unique interplay of algebraic functions and statistical methods. In this paper, we present these recognizers and show that they work effectively, even when data are missing along a large portion of an object boundary due, for example, to partial occlusion.     

E-optimal designs for polynomial regression without intercept

We give all E-optimal designs for the mean parameter vector in polynomial regression of degree d without intercept in one real variable. The derivation is based on interplays between E-optimal design problems in the present and in certain heteroscedastic polynomial setups with intercept. Thereby the optimal supports are found to be related to the alternation points of the Chebyshev polynomials of the first kind, but the structure of optimal designs essentially depends on the regression degree being odd or even. In the former case the E-optimal designs are precisely the (infinitely many) scalar optimal designs, where the scalar parameter system refers to the Chebyshev coefficients, while for even d there is exactly one optimal design. In both cases explicit formulae for the corresponding optimal weights are obtained. Remarks on extending the results to E-optimality for subparameters of the mean vector (in heteroscdastic setups) are also given.     

A further remark on testing linear inequality constraints in the standard linear model

In this paper we will show that the expressions obtained by Hillier (1986) for the null distribution of the likelihood ratio test of zero restrictions on nonnegative regression coefficients in the standard linear model are the same as those reported by Farebrother (1986).     

Distribution approximation and modelling via orthogonal polynomial sequences

A general methodology is developed for approximating the distribution of a random variable on the basis of its exact moments. More specifically, a probability density function is approximated by the product of a suitable weight function and a linear combination of its associated orthogonal polynomials. A technique for generating a sequence of orthogonal polynomials from a given weight function is provided and the coefficients of the linear combination are explicitly expressed in terms of the moments of the target distribution. On applying this approach to several test statistics, we observed that the resulting percentiles are consistently in excellent agreement with the tabulated values. As well, it is explained that the same moment-matching technique can be utilized to produce density estimates on the basis of the sample moments obtained from a given set of observations. An example involving a well-known data set illustrates the density estimation methodology advocated herein.     

On Stein Identity,Chernoff Inequality,and Orthogonal Polynomials

In this article, we present a general method for deriving Stein-like identity and Chernoff-like inequality based on orthogonal polynomials. In order to illustrate our method, some applications are given with respect to normal, Gamma, Beta, Poisson, binomial, and negative binomial distribution, not only for random variables but also for random vectors, resulting corresponding Stein-like identity and Chernoff-like inequality are obtained consequently. Within our best knowledge, some of our matrix version results are new in the literature. In addition, forward difference formulae of Charlier polynomials, Krawtchouk polynomials and Meixner polynomials, Stein-like identity, and Chernoff-like inequality with respect to Beta distribution, as well as Rodrigues formula of Meixner polynomials are also prepared in the first time within our limited information. Interestingly, as far as normal, Gamma, Beta, Poisson, binomial, and negative binomial distribution are concerned, we found that their Stein-like identity and corresponding Chernoff-like inequality are related closely, by examining their Rodrigues formula.     

Local linear regression with adaptive orthogonal fitting for the wind power application

Short-term forecasting of wind generation requires a model of the function for the conversion of meteorological variables (mainly wind speed) to power production. Such a power curve is nonlinear and bounded, in addition to being nonstationary. Local linear regression is an appealing nonparametric approach for power curve estimation, for which the model coefficients can be tracked with recursive Least Squares (LS) methods. This may lead to an inaccurate estimate of the true power curve, owing to the assumption that a noise component is present on the response variable axis only. Therefore, this assumption is relaxed here, by describing a local linear regression with orthogonal fit. Local linear coefficients are defined as those which minimize a weighted Total Least Squares (TLS) criterion. An adaptive estimation method is introduced in order to accommodate nonstationarity. This has the additional benefit of lowering the computational costs of updating local coefficients every time new observations become available. The estimation method is based on tracking the left-most eigenvector of the augmented covariance matrix. A robustification of the estimation method is also proposed. Simulations on semi-artificial datasets (for which the true power curve is available) underline the properties of the proposed regression and related estimation methods. An important result is the significantly higher ability of local polynomial regression with orthogonal fit to accurately approximate the target regression, even though it may hardly be visible when calculating error criteria against corrupted data.     

On approximating the central and noncentral multivariate gamma distributions

In this paper a finite series approximation involving Laguerre polynomials is derived for central and noncentral multivariate gamma distributions. It is shown that if one approximates the density of any k nonnegative continuous random variables by a finite series of Laguerre polynomials up to the (n₁, …, n_k)th degree, then all the mixed moments up to the order (n₁, …, n_k) of the approximated distribution equal to the mixed moments up to the same order of the random variables. Some numerical results are given for the bivariate central and noncentral multivariate gamma distributions to indicate the usefulness of the approximations.     

EXPECTATIONS OF RATIOS OF QUADRATIC FORMS IN NORMAL VARIABLES: EVALUATING SOME TOP-ORDER INVARIANT POLYNOMIALS

Chikuse's (1987) algorithm constructs top-order invariant polynomials with multiple matrix arguments. Underlying it is a set of simultaneous equations for which all integer solutions must be found. Each solution represents a component of the sum of terms which comprise the polynomial. The system of equations has a specialised structure which may be exploited to obtain a polynomial with r matrix arguments in terms of a polynomial with r-1 matrix arguments. This is demonstrated for two particular polynomials that have two matrix arguments. These results are applied to problems involving expectations of ratios of quadratic forme in normal variables; analytic as well as computable formulae are derived.     

Inference in affine shape theory under elliptical models

This paper studies the elliptical statistical affine shape theory under certain particular conditions on the evenness or oddness of the number of landmarks. In such a case, the related distributions are polynomials, and the inference is easily performed; as an example, a landmark data is studied, and the performance of the polynomial density versus the usual series density is compared.     

Jacobi and Laguerre polynomial approximations for the distributions of statistics useful in testing for outliers in exponential and gamma samples

Recently, Sanjel and Balakrishnan [A Laguerre Polynomial Approximation for a goodness-of-fit test for exponential distribution based on progressively censored data, J. Stat. Comput. Simul. 78 (2008), pp. 503–513] proposed the use of Laguerre orthogonal polynomials for a goodness-of-fit test for the exponential distribution based on progressively censored data. In this paper, we use Jacobi and Laguerre orthogonal polynomials in order to obtain density approximants for some test statistics useful in testing for outliers in gamma and exponential samples. We first obtain the exact moments of the statistics and then the density approximants, based on these moments, are expressed in terms of Jacobi and Laguerre polynomials. A comparative study is carried out of the critical values obtained by using the proposed methods to the corresponding results given by Barnett and Lewis [Outliers in Statistical Data, 3rd ed., John Wiley & Sons, New York, 1993]. This reveals that the proposed techniques provide very accurate approximations to the distributions. Finally, we present some numerical examples to illustrate the proposed approximations. Monte Carlo simulations suggest that the proposed approximate densities are very accurate.     

Smooth Goodness-of-Fit Specification Tests Under the Lagrange Multiplier Principle

This article generalizes Neyman's smooth test for the goodness-of-fit hypothesis using orthogonal polynomials of the density function under the null hypothesis, and derives a Lagrange Multiplier (LM) statistic based on the generalized form of the smooth test. Under the null hypothesis, using the joint limiting normality of the orthogonal functions imbedded into the smooth alternative density function and the restricted parameter estimators, the covariance matrix of the LM statistic can be estimated. The procedure of constructing monic orthogonal polynomials from a given moment function is developed. This procedure is applied to examples of testing for normal, Poisson, and gamma distributions.