Invariant Estimators Research Articles

Formula omitted]-Consistent robust integration-based estimation

We propose a new robust estimator of the regression coefficients in a linear regression model. The proposed estimator is the only robust estimator based on integration rather than optimization. It allows for dependence between errors and regressors, is n -consistent, and asymptotically normal. Moreover, it has the best achievable breakdown point of regression invariant estimators, has bounded gross error sensitivity, is both affine invariant and regression invariant, and the number of operations required for its computation is linear in n . An extension would result in bounded local shift sensitivity, also.

Properties of the explicit estimators in the extended growth curve model

Let be the extended growth curve model with error matrix ℰ distributed as a normal distribution with mean 0 and covariance I⊗ Σ, subject to some specified conditions. A quadratic statistic Σˆ(Y), distributed as a Wishart distribution, is proposed and proved to be a uniformly minimum variance unbiased invariant estimator of the second-order parameter matrix Σ. In addition, unbiased and explicit estimators Θˆ i (Y) of the first-order parameter matrices Θ i are given. And explicit formulae to compute the traces of the covariance matrices of Θˆ i (Y) are derived.

Minimum Risk Invariant Estimators of a Continuous Cumulative Distribution Function

The problem of nonparametric minimum risk invariant estimation has engaged a good deal of attention in the literature and minimum risk invariant estimators (MRIE's) have been constructed for some special statistical models. We present a new and simple method of obtaining the MRIE's of a continuous cumulative distribution function (cdf) under a general invariant loss function. All the MRIE's, which are known from the literature, can be constructed by the method presented in the article, in particular, under the weighted quadratic, LINEX and entropy loss functions. This method enables also to construct the MRIE's in nonparametric statistical models which have not been considered until now. In particular, considering a family of nonparametric precautionary loss functions, a new class of MRIE's of the cdf has been found. We also give some general remarks on obtaining the MRIE's and a review concerning minimaxity and admissibility of MRIE's.

Estimation of Parameters of Uniform Distribution Based on K-Record Values

Abtsrcat There are many situations where experimental outcomes are a sequence of record-breaking observations. In this paper, an extension of record models, well known as k-records, is considered. The best linear unbiased estimators (BLUEs) for the location and scale parameters of the uniform distribution are derived using k-record values. In addition, best linear invariant estimators of two parameter uniform distribution are obtained on the basis of k-record values. We have also determined the e�ciency of BLUEs relative to maximum likelihood estimators (adjusted for bias). The best linear unbiased predictor of future k-record values is also determined.

Use of Spacings in the Estimation of Common Scale Parameter of Several Distributions

In this article, we consider the problem of best linear unbiased estimation and best linear invariant estimation of the common scale parameter of several distributions using spacing of the pooled sample of all observations of individual samples. We derived conditions for the non negativity of the scale estimator obtained by the above methods. Further, we obtained necessary and sufficient conditions for the derived estimators to be constant multiples of the pooled sample range.

Complex algebraic plane curves via Poincaré-Hopf formula. II. Annuli

We give a complete classification of algebraic curves in ℂ2 which are homeomorphic with ℂ* and which satisfy a certain natural condition about codimensions of its singularities. In the proof we use the method developed in [BZI]. It relies on estimation of certain invariants of the curve, the so-called numbers of double points hidden at singularities and at infinity. The sum of these invariants is given by the Poincare-Hopf formula applied to a suitable vector field.

On-line Rotation Invariant Estimation and Recognition

Rotation invariant estimation is an important and computationally difficult process in the real-time human computer interaction. Our new methodologies propose here for on-line image rotation angle estimation, correction and feature extractions based on line integrals. We reveal that a set of projection data of line integrals from single (fan-arc and fan-beam) or multi point sources (Radon transform) are employed for orientation estimation. After estimating orientation, image angle variations are altered to its principal direction. We further combine Boltzmann machine and k-mean clustering to obtain parameter optimized Gabor filters, which are used to extract non-redundant compact set of features for classification. The proposed method of fan-line, fan-arc and Radon transform are compared for real-time image orientation detection. Accuracy of classification is evaluated with the classifiers viz., back propagation, Hamming neural network, Euclidean-norm distance, and k-nearest neighbors. Experiment on a database of 535 images consisting of license plate and iris images. The viability of suggested algorithms has been tested with different classifiers. Thus, this paper proposes an efficient rotation invariant recognition for on-line images recognition.

Asymptotic expansions for the location invariant moment-type estimator

In this paper, based on moment-type and location invariant Hill estimators, a new kind of location invariant moment-type extreme value index estimator is proposed. The weak and strong consistency of the estimator are discussed. The asymptotic expansion of the estimator and its distribution are also considered under second order regularly varying conditions. The asymptotic normality is employed to construct the confidence interval. Monte Carlo simulations are performed to compare the Hill estimator and the moment estimator in terms of mean squared error.

AN EFFICIENT AND ACCURATE METHOD FOR THE ESTIMATION OF ENTROPY AND OTHER DYNAMICAL INVARIANTS FOR PIECEWISE AFFINE CHAOTIC MAPS

In this paper, we discuss an efficient iterative method for the estimation of the chief dynamical invariants of chaotic systems based on stochastically stable piecewise affine maps (e.g. the invariant measure, the Lyapunov exponent as well as the Kolmogorov–Sinai entropy). The proposed method represents an alternative to the Monte-Carlo methods and to other methods based on the discretization of the Frobenius–Perron operator, such as the well known Ulam's method. The proposed estimation method converges not slower than exponentially and it requires a computation complexity that grows linearly with the iterations. Referring to the theory developed by C. Liverani, we discuss a theoretical tool for calculating a conservative estimation of the convergence rate of the proposed method. The proposed approach can be used to efficiently estimate any order statistics of a symbolic source based on a piecewise affine mixing map.

A note on classical Stein-type estimators in elliptically contoured models

In this paper the conditions under which a broad class of Stein-type estimators dominates the best invariant unbiased estimator of the mean of an elliptically contoured population have been established. The superiority conditions are derived for both known and unknown scale structures. Also an example is given when the general scale matrix is assumed to be known in linear regression.

Automated quality assessment of autonomously acquired microscopic images of fluorescently stained bacteria

Quality assessment of autonomously acquired microscopic images is an important issue in high-throughput imaging systems. For example, the presence of low quality images (>or=10%) in a dataset significantly influences the counting precision of fluorescently stained bacterial cells. We present an approach based on an artificial neural network (ANN) to assess the quality of such images. Spatially invariant estimators were extracted as ANN input data from subdivided images by low level image processing. Different ANN designs were compared and >400 ANNs were trained and tested on a set of 25,000 manually classified images. The optimal ANN featured a correct identification rate of 94% (3% false positives, 3% false negatives) and could process about 10 images per second. We compared its performance with the image quality assessment by different humans and discuss the difficulties in assigning images to the correct quality class. The computer program and the documented source code (VB.NET) are provided under General Public Licence.

Linear invariant estimation of clock phase and trend from noisy phase data

We show how to solve two problems of optimal linear estimation from a set of noisy measurements of clock phase. The phase is modelled as a process with stationary dth increments, e.g. a sum of processes with power-law spectra. The additive measurement noise can be any mean-zero process with a known autocovariance function. The estimation targets are the phase at a given time and the coefficient of the overall trend, e.g. the frequency drift. A feasibility condition called ‘invariance’ is imposed according to the target and the degree of non-stationarity of the phase model. The solution of a set of linear equations gives the regression coefficients and the mean squared error of the best linear invariant estimate. Proofs of these results are available from the online version of this journal.

Analysing Wage Differences between the USA and Germany Using Proportional Hazards Models

AbstractWe analyse differences between the wage distributions in the USA and Germany in 2001 for both women and men. The empirical analysis is based on the decomposition of differences using Cox's marginal (partial) likelihood. The approach based on rank invariant estimators such as Cox's is borrowed from the literature on failure time data. Donald et al. pioneered this approach in 2000. However, they did not use the full power of the semi‐parametric approach. Instead, they argued for using a piecewise constant hazard rate model. We improve on their work by showing that the semi‐parametric features of Cox's marginal likelihood are as appropriate for the analysis of wage decompositions and as easy to interpret. Moreover, we extend their approach by allowing for non‐linear regression effects. We show empirically that this formulation both increases the flexibility of their approach and improves the discriminatory power between wage regimes.

Minimax invariant estimator of a continuous distribution function under a general loss function

The problem of invariant estimation of a continuous distribution function is considered under a general loss function. Minimaxity of the minimum risk invariant estimator of a continuous distribution function is proved for any sample size n ≥ 2.

What determines wage differentials across the EU?

We analyse determinants of the wage differentials observed between thirteen European countries using the European Community Household Panel (ECHP). The empirical decomposition is based on a proportional hazards model. The approach based on rank invariant estimators is borrowed from the literature on failure time data and has been adapted by Donald et al. (Rev Econ Stud 67:609-633, 2000) to analyse differences in wage distributions. Extending this approach and using natural splines to allow for nonlinear regression effects we isolate three different underlying forces for wage differentials: differences in characteristics, differences in weights of characteristics and differences in the returns-to-skill functions. The analysis reveals that country differences in returns-to-skill functions contribute most to observed wage differences while differences in observed characteristics contribute surprisingly little.

Spectral Peak-Shift Estimation With Wavelength Dependent Sources and Detectors

This correspondence extends Karl and Pien's [ldquoHigh-Resolution Biosensor Spectral Peak Shift Estimation,rdquo IEEE Transactions on Signal Processing, vol 53, no. 12, pp. 4631-4639, December 2005] approach to spectral-shift estimation for optical sensors to account for wavelength dependent illumination sources and detectors. Relaxing the assumption of spectrally uniform source power and detector efficiency alters the previously derived Cramer-Rao bound and maximum-likelihood estimator. The new estimator offers significantly reduced error compared to the wavelength invariant estimator and common ad hoc estimators.

Elliptic estimates independent of domain expansion

In this paper, we consider elliptic estimates for a system with smooth variable coefficients on a domain \({\Omega \subset \mathbb{R}^n,\, n \ge 2}\) containing the origin. We first show the invariance of the estimates under a domain expansion defined by the scale that \({y = Rx,\, x,\,y \in \mathbb{R}^n}\) with parameter R > 1, provided that the coefficients are in a homogeneous Sobolev space. Then we apply these invariant estimates to the global existence of unique strong solutions to a parabolic system defined on an unbounded domain.

Maintaining parameter invariance in seemingly unrelated regressions estimation

Seemingly unrelated regressions (SUR) is an important estimation methodology in demand analysis. While possessing qualities that make it attractive to applied demand analysts, the SUR estimation technique poses several difficulties, including the lack of parameter invariance in some instances e.g. estimation of consumer demand systems with micro-level data. When data includes zero consumption, corrections used to account for censoring may result in different parameter estimates when different share equations are dropped for estimation. This article proposes an alternative estimation objective which results in invariant parameter estimates when imposing adding-up by dropping equations.

A two-step estimator of the extreme value index

In this paper, we build a two-step estimator $\hat{\gamma}_{\rm STEP}$ , which satisfies $\sqrt{k}(\hat{\gamma}_{\rm STEP}-\hat{\gamma}_{ML})\stackrel{P}{\rightarrow} 0$ , where $\hat{\gamma}_{ML}$ is the well-known maximum likelihood estimator of the extreme value index. Since the two-step estimator $\hat{\gamma}_{\rm STEP}$ can be calculated easily as a function of the observations, it is much simpler to use in practice. By properly choosing the first step estimator, such as the Pickands estimator, we can even get a shift and scale invariant estimator with the above property.

Admissibility and minimaxity of Bayes estimators for a normal mean matrix

In some invariant estimation problems under a group, the Bayes estimator against an invariant prior has equivariance as well. This is useful notably for evaluating the frequentist risk of the Bayes estimator. This paper addresses the problem of estimating a matrix of means in normal distributions relative to quadratic loss. It is shown that a matricial shrinkage Bayes estimator against an orthogonally invariant hierarchical prior is admissible and minimax by means of equivariance. The analytical improvement upon every over-shrinkage equivariant estimator is also considered and this paper justifies the corresponding positive-part estimator preserving the order of the sample singular values.