Statistical Estimation Theory Research Articles

Quantum Fisher metric and estimation for pure state models

A statistical parameter estimation theory for quantum pure state models is presented. We first investigate the basic framework of pure state estimation theory and derive quantum counterparts of the Fisher metric. We then formulate a one-parameter estimation theory, based on symmetric logarithmic derivatives, and clarify the differences between pure state models and strictly positive models.

Memorial: Nikolai Nikolaevitch Chentsov

Memorial: Nikolai Nikolaevitch Chentsov

On greatest accuracy credibility with limited fluctuation.

Abstract In greatest accuracy credibility theory the credibility estimator is constructed so as to be as close as possible to the estimand whereas in limited fluctuation credibility theory one tries to limit the fluctuations from year to year. It has been argued that in practice the greatest accuracy credibility estimators often give too great year-to-year fluctuations. In the present paper we modify the greatest accuracy credibility theory to limit these fluctuations, thus trying to combine the advantages of both theories. We also discuss other aspects of the relation between rate-making and statistical estimation theory.

Quantum Statistical Inference

The three main points of this article are: 1. Quantum mechanical data differ from conventional data: for example, joint distributions usually cannot be defined conventionally; 2. rigorous methods have been developed for analyzing such data; the methods often use quantum-consistent analogs of classical statistical procedures; 3. with these procedures, statisticians, both data-analytic and more theoretically oriented, can become active participants in many new and emerging areas of science and biotechnology. In the physical realm described by quantum mechanics, many conven- tional statistical and probabilistic assumptions no longer hold. Probabi- listic ideas are central to quantum theory but the standard Kolmogorov axioms are not uniformly applicable. Studying such phenomena requires an altered model for sample spaces, for random variables and for inference and decision making. The appropriate decision theory has been in devel- opment since the mid-1960s. It is both mathematically and statistically rigorous and conforms to the requirements of the known physical results. This article provides a tour of the structure and current applications of quantum-consistent statistical inference and decision theory. It presents examples, outlines the theory and considers applications and open prob- lems. Certain central concepts of quantum theory are more clearly appre- hended in terms of the quantum-consistent statistical decision theory. For example, the Heisenberg uncertainty principle can be obtained as a consequence of the quantum version of the Cramer-Rao inequality. This places concepts of statistical estimation and decision theory, and thus the statistician, at the center of the quantum measurement process. Quantum statistical inference offers considerable scope for participa- tion by the statistical community, in both applications and foundational questions.

On the time a diffusion process spends along a line

For an arbitrary diffusion process X with time-homogeneous drift and variance parameters μ( x) and σ 2( x), let V ε be 1 ε times the total time X( t) spends in the strip [a + bt − 1 2 ε, a + bt + 1 2 ε] . The limit V as ε → 0 is the full halfline version of the local time of X( t) − a − bt at zero, and can be thought of as the time X spends along the straight line x = a + bt. We prove that V is either infinite with probability 1 or distributed as a mixture of an exponential and a unit point mass at zero, and we give formulae for the parameters of this distribution in terms of μ(·), σ(·), a, b, and the starting point X(0). The special case of a Brownian motion is studied in more detail, leading in particular to a full process V( b) with continuous sample paths and exponentially distributed marginals. This construction leads to new families of bivariate and multivariate exponential distributions. Truncated versions of such ‘total relative time’ variables are also studied. A relation is pointed out to a second order asymptotics problem in statistical estimation theory, recently investigated in Hjort and Fenstad (1992a, 1992b).

Análise rotineira de dados de vigilância em saúde pública: que procedimentos estatísticos utilizar?

In recent years an increasing interest in epidemiological surveillance (that we prefer to label public health surveillance) has emerged. The viewpoint that the ease of access to computers and statistical software may permit the use of more sophisticated statistical methodologies in the analysis of surveillance data has been profounded in many studies. It is a cause of concern that this attitude, when used indiscriminated by, may lead to analysis lacking in theoretical support. Thus, first, a viewpoint about surveillance in public health is presented and the basic propositions of the theory of statistical hypothesis tests and interval estimation are described briefly and without technicalities. The nature of surveillance data, their nonsampling character and non-random selection are also commented on. Some descriptive procedures that may be used without loss of quality in analysis are described than some procedures that are proposed in the relevant international literature but that need fuller investigation before being introduced in routine data analysis are given.

Estimation and detection issues in matched-field processing

A conceptual framework in which the model-based, space-time acoustic signal processing procedure known as matched field processing (MFP) can be handled in a consistent manner is established. A framework for strong-signal MFP based on standard statistical estimation theory, in which MFP is regarded as essentially an estimation problem in the strong-signal regime, is developed. In the weak-signal case, the necessary requirement of detection dictates that MFP then be considered a joint detection-estimation task. It is demonstrated that, generally, MFP is essentially a space-time processing problem rather than simply an array processing (spatial processing only) procedure. An overview of the processing schemes used to date in MFP is given, showing how these methods relate to the optimal space-time structure. Weak-signal detection and estimation scenarios relevant to MFP are then noted. Present methods for dealing with the inherent instability of MFP algorithms (mismatch) are discussed. The current status of MFP is summarized, and recommendations for future research are made.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Reinterpreting network measures for models of disease transmission.

In the wake of AIDS and HIV, a better framework is needed to model the pattern of contacts between infectious and susceptible individuals. Several network measures have been defined elsewhere which quantify the distance between 2 nodes, and the centrality of a node, in a network. Stephenson and Zelen (S-Z), however, have recently presented a new measure based on statistical estimation theory and applied it to a network of AIDS cases. This paper shows that the closeness measure proposed by S-Z is equivalent to the effective conductance in an electrical network, fits the measure into the existing theory of percolation, and provides a more efficient algorithm for computing S-Z closeness. The S-Z methodology is compared with the closeness measures of maximal flow, first passage time, and random hitting time. Computational problems associated with the measure are discussed in a closing section.

The cue for contour-curvature discrimination

Seven potential geometric cues for contour-curvature discrimination were tested: curvature, turningangle, arc-length, arc-length-divided-by-chord-length, maximum-deviation (sag), mean-deviation and area. Three experiments were performed, each requiring the discrimination of two simultaneously presented, 1-sec-duration, curved-line stimuli, whose chord-lengths ranged from 12 to 48 arcmin visual angle and whose curvatures ranged from 0 to 0.13 arcmin −1. Experiments 1 and 2 determined for each cue the smallest detectable increment (the increment threshold) as a function of cue value, for a set of spatial transformations of the stimulus (one- and two-dimensional scalings) equivalent to changes in viewing distance and direction. In accordance with statistical estimation theory, the “best” cue was defined as the most efficient one, that is, the one which best accounted for the variance in the data. As a control, Expt 3 compared increment-threshold functions for circular and elliptical arcs of constant chord-length and circular arcs of constant arc-length. Over all three experiments, only sag and its linear approximation, mean-deviation, accounted well for the variance in the data; sag provided the best predictor, and its increment-threshold function satisfied Weber's law over almost all of the stimulus range. Additionally, sag has a special theoretical property (shared only with mean-deviation and area): the relationship it defines (in proportional terms) between curved contours in an image of an object or a scene is constant and independent of viewing distance and direction.

Estimation Theory for Dynamic Systems with Unknown but Bounded Uncertainty: An Overview

In many problems such as linear and nonlinear regressions, parameter and state estimation of dynamic systems, state space and time series prediction, interpolation, smoothing, functions approximation, one has to evaluate some unknown variable using available data. The data are always associated with some uncertainty and it is necessary to evaluate how this uncertainty affects the estimated variables. Typically the problem is approached assuming a probabilistic description of uncertainty and applying statistical estimation theory. An interesting alternative approach, referred to as set membership or Unknown But Bounded (UBB) error description, has been investigated since the late 60's. In this approach, uncertainty is described by an additive noise which is known only to have given integral (typically l 2 or l 1 ) or componentwise ( l ∞ ) bounds. In this paper we review the main results of this theory, with special attention to the most recent advances obtained in the case of componentwise bounds.

Characterizing the microstructure of alumina using image processing

Quantitative structural and chemical information can be obtained from high angle annular dark field scanning transmission electron microscopy (HAADF STEM) images when using statistical parameter estimation theory. In this approach, we assume an empirical parameterized imaging model for which the total scattered intensities of the atomic columns are estimated. These intensities can be related to the material structure or composition. Since the experimental probe profile is assumed to be known in the description of the imaging model, we will explore how the uncertainties in the probe profile affect the estimation of the total scattered intensities. Using multislice image simulations, we analyze this effect for Cs corrected and non-Cs corrected microscopes as a function of inaccuracies in cylindrically symmetric aberrations, such as defocus and spherical aberration of third and fifth order, and non-cylindrically symmetric aberrations, such as 2-fold and 3-fold astigmatism and coma.

On invariance principles for distributed parameter identification algorithms

We consider a class of identification algorithms for distributed parameter systems. Utilizing stochastic optimization techniques, sequences of estimat.ors are constructed by minimizing appropriate functionals. The main effort is to devel<?p weak and strong invariance principles for the underlying algorithms. By means of weak convergence methods, a functional central limit theorem is established. Using the Skorohod imbedding, a strong invariance principle is obtained. These invariance principles provide very precise rates of convergence results for parameter estimates, yielding important information for experimental design. / Key wor6s: identification, distributed parameter system, stochastic optimization, invari~nce principle. 1. Introd l1ction. In a wide range of applications, various problems have been formulated by using partial differential equations appropriate boundary and initial conditions. Quite frequently, the underlying systems involve some unknown parameters, typically in the form of coefficients in the equation. As a consequence, distributed parameter identification, in which parameters are estimated from observed data, has witnessed rapid progress in recent years. To illustrate, we consider the following examples. , 1 Research of this author was supported in part by the National Science Foundation under grant DMS-9022139. 2 Research of this author was supported in part by the Air Force Office of Scientific Research under grant AFOSR-91-0021. G. Yin and B. G. Fitzpatrick 99 EXAMPLE 1. The following differential equation models fluid tlansport in cat brain tissue: Here u represents' fluid concentration, and the parameters V and 1) are c')nvectio,n and diffusion coefficients; respectively. Banks and Kareiva (1983) (sce also Banks and Fitzpatrick, 1989) used least squares techniques to fit this model to observed data, and went on to apply ANOVA-type' hypothesis tests for V = 0, in order to verify conjectures concerning the role of convection in grey and white matter. EXAMPLE 2. The determination of damping terms in flexible st.ructures is crucial to modeling and control objectives. Banks et. al. (1987) applied an Euler-Bernoulli model viscous and Kelvin-Voigt damping terms: where El is the stiffness, f is the forcing function, and cDl and 'Y are the Kelvin-Voigt and viscous damping coefficients, respectively. The functionu represents dipl Banks and Fitzpatrick (1989), Fitzpatrick (1988), Banks .and Fitzpatrick (1990), the effects of noisy observations on the class of stochastic optimization and parameter estimation procedures w~reanalyzed. In particular, consistency and asymptotic normality were established, a primary objective of developing appropriate statistics for hypothesis tests. This work complements the papers of Fitzpatrick (1988), Banks and Fitzpatrick (1990) by developing weak and strong functional invariance principles of the least squares algorithms for distributed parameter identification. Our main concerns are to investigate further the asymptotic properties and to develop rate of convergence result~. The importance of these results for applications is obvious: the a;nount of data required to achieve s~me specified estimation accu~acy would be very helpful information for designing experiments. i Functional central limit theorems and functional laws of iterated logarithms 'have played important roles in statistical estimation theory involving large samples. In (Heyde, 1981), Heyde gives an extensive survey on the usefulness and recent pmgress in these invariance theorems, which both and extend the interplay between statistical estimation and stochastic processes. The results to be presented in the sequel deal the convergence of functions constructed out of the sequence of least squares estimators (suitably scaled), and provide portmanteau forms from which other limit theorems may be obtained. A wide range oflimit distribution results involving functionals of the sequence of estimators can be inferred by employing the weak invariance principle. and the with probability one convergence rate of the algorithm G. Yin and B. G. Fitzpatrick 101 can be derived by virtue of the strong invariance theorem. These results provide us further insight on the behavior of the nonlinear least squares type of stochastic optimization and identification algorithms. The rest of. the paper is organized as follows. In the next section, we set up the notations, and summarize some previous results. Section 3 is devoted to the weak convergence issue. Under suitable conditions, we show an appropriately scaled sequence converges weakly to a Brownian motion. Exploiting this function space setting further, we derive an almost sure estimate on the error bound in Section 4. As a consequence, the functional law of iterated logarithm holds. To proceed, a brief explanation about the notations is in order. We shall use I to denote the transpose of a matrix and J{ to denote a generic positive constant; its value may change from time to time. The short hand notion w.p.P is meant to be with probability one. 2. The general least squares problem. We begin this section by setting up the least squares identifica.tion problem. Let X be a compa.ct subset of Rm , 9 : X -+ R be an unknown continuous function. Wc make a sequence of observations {li}

Optimal mean-square N-observation digital morphological filters : II. Optimal gray-scale filters

The second part of this two-part study interprets digital gray-scale morphological operations as numerical functionals on integer-valued vectors. The result is a gray-scale morphological statistical estimation theory based on N observation random variables and a consequent theory of mean-square optimization. Whereas in the binary setting the search for optimal structuring elements is clearly restricted by the requirement of binary N-vector structuring elements, gray-scale structuring elements are N-vectors of integers (not confined to the range of the images). Thus, it is necessary to find a set of structuring elements from which both singleand multiple-erosion filters can be optimized, the latter being characterized within the framework of the gray-scale Matheron representation. Consequently, a significant part of the paper is concerned with the derivation of the fundamental set, this set being a minimal family of structuring elements from which it is always possible to select the erosions comprising an optimal filter. As in the case of binary optimization, we treat constrained optimality; however, here we apply it to construction of optimal linear filters. Finally, we demonstrate that erosion optimization is equivalent to dilation optimization.

Optimal estimation theory for dynamic systems with set membership uncertainty: An overview

In many problems such as linear and nonlinear regressions, parameter and state estimation of dynamic systems, state-space and time series prediction, interpolation, smoothing and functions approximation, one has to evaluate some unknown variable using available data. The data are always associated with some uncertainty and it is necessary to evaluate how this uncertainty affects the estimated variables. Typically, the problem is approached assuming a probabilistic description of uncertainty and applying statistical estimation theory. An interesting alternative approach, referred to as set membership or unknown but bounded (UBB) error description, has been investigated since the late 1960s. In this approach, uncertainty is described by an additive noise which is known only to have given integral (typically l 2 of l 1) or componentwise ( l ∞) bounds. In this paper we review the main results of this theory, with special attention to the most recent advances obtained in the case of componentwise bounds.

Mathematical statistics and metastatistical analysis

This paper deals with meta-statistical questions concerning frequentist statistics. In Sections 2 to 4 I analyse the dispute between Fisher and Neyman on the so called logic of statistical inference, a polemic that has been concomitant of the development of mathematical statistics. My conclusion is that, whenever mathematical statistics makes it possible to draw inferences, it only uses deductive reasoning. Therefore I reject Fisher's inductive approach to the statistical estimation theory and adhere to Neyman's deductive one. On the other hand, I assert that Neyman-Pearson's testing theory, as well as Fisher's tests of significance, properly belong to decision theory, not to logic, neither deductive nor inductive. I then also disagree with Costantini's view of Fisher's testing model as a theory of hypothetico-deductive inferences.

空間相関関数とその統計的検定の実用的計算手法と視覚化 : 実用的メッシュデータ解析システム構築のための空間相関分析法の体系化 その1

The space correlation analysis (SCA) was proposed with its possibility of application on urban analysis in our previous papers. The calculating program of SCA, however, is not easy to develop for urban planners because the program has a relatively large size and complicated. The calculating theory is reconstructed to make SCA fit for practical use. The following possibilities of calculating method of SCA are shown ; 1) the space influence function (SIF) can be calculated by using ordinary library program, 2) the space correlation function (SCF) can be calculated very rapidly by using FFT (fast Fourier transform). Also presented in this paper are visual representation methods of SCA model with reliable range of estimated parameters. It is very important in actual planning to know the structure of spatial influence relation of urban activities accurately and intuitionally. Using the statistical range estimation and testing theory, 1) the method for getting the confidence interval of SCF is shown, and 2) the method for grouping the value of SCF is shown. The visual presentation of the results of the above methods are very effective to assist urban planners to understand the characteristics of spatial relation of urban activities.

At-sea Experiment of a Floating Offshore Structure

The at-sea experiment of the moored floating structure, which is called POSEIDON, has been done at the Japan sea since september in 1986.This paper deals with the characteristics of wind speeds measured at the test field. Especially, we concentrate upon varying wind speed spectrum, 1/nth highest expected values, and gust factors.As an anemometer, a sonic anemometer was used and it was installed at the top of mast of POSEIDON, at which is the position of 19.5 m height above sea surface.The main results are as follows.i) A new formula of varying wind speed spectrum is developed under physical considerations. The presented spectrum can be determined by two parameters, i. e. the peak frequency and the power, those are prescribed by the friction coefficient between air and sea surface.The measured spectra do not agree with the well-known spectra (Davenport's and Hino's spectra) but accord with the presented spectra.ii) 1/nth highest expected values and gust factors can be estimated by using the well-known statistical estimation theory (i. e. Longuet-Higgins theory) and the presented spectrum.

Il miglioramento di una previsione in presenza di sigma-algebre sufficienti in senso predittivo

The paper deals with the problem of finding a better predictor starting from a given one. After introducing some basic notions of sufficiency in terms of a 'sigma-field', a theorem due to K. Tacheuchi and M. Akahira is used to show criterion leading to a better predictor similar to Rao-Blackwell in statistical estimation theory. Then the wide sense sufficiency is introduced showing a criterion to get a better predictor in presence of a sufficient 'sigma-field ' in a wide sense.

A theory for the use of visual orientation information which exploits the columnar structure of striate cortex.

A neural model is constructed based on the structure of a visual orientation hypercolumn in mammalian striate cortex. It is then assumed that the perceived orientation of visual contours is determined by the pattern of neuronal activity across orientation columns. Using statistical estimation theory, limits on the precision of orientation estimation and discrimination are calculated. These limits are functions of single unit response properties such as orientation tuning width, response amplitude and response variability, as well as the degree of organization in the neural network. It is shown that a network of modest size, consisting of broadly orientation selective units, can reliably discriminate orientation with a precision equivalent to human performance. Of the various network parameters, the discrimination threshold depends most critically on the number of cells in the hypercolumn. The form of the dependence on cell number correctly predicts the results of psychophysical studies of orientation discrimination. The model system's performance is also consistent with psychophysical data in two situations in which human performance is not optimal. First, interference with orientation discrimination occurs when multiple stimuli activate cells in the same hypercolumn. Second, systematic errors in the estimation of orientation can occur when a stimulus is composed of intersecting lines. The results demonstrate that it is possible to relate neural activity to visual performance by an examination of the pattern of activity across orientation columns. This provides support for the hypothesis that perceived orientation is determined by the distributed pattern of neural activity. The results also encourage the view that limits on visual discrimination are determined by the responses of many neurons rather than the sensitivity of individual cells.

Concepts, Theory, and Techniques ROBUST METHODS: AN ALTERNATIVE APPROACH FOR ANALYZING DATA SETS CONTAINING INFLUENTIAL DATA POINTS

ABSTRACTRecent advances in statistical estimation theory have resulted in the development of new procedures, called robust methods, that can be used to estimate the coefficients of a regression model. Because such methods take into account the impact of discrepant data points during the initial estimation process, they offer a number of advantages over ordinary least squares and other analytical procedures (such as the analysis of outliers or regression diagnostics).This paper describes the robust method of analysis and illustrates its potential usefulness by applying the technique to two data sets. The first application uses artificial data; the second uses a data set analyzed previously by Tufte [15] and, more recently, by Chatterjee and Wiseman [6].