Point Estimation Theory Research Articles

Generalized measurement error: Intrinsic and incidental measurement error.

In this paper, we generalize the notion of measurement error on deterministic sample datasets to accommodate sample data that are random-variable-valued. This leads to the formulation of two distinct kinds of measurement error: intrinsic measurement error, and incidental measurement error. Incidental measurement error will be recognized as the traditional kind that arises from a set of deterministic sample measurements, and upon which the traditional measurement error modelling literature is based, while intrinsic measurement error reflects some subjective quality of either the measurement tool or the measurand itself. We define calibrating conditions that generalize common and classical types of measurement error models to this broader measurement domain, and explain how the notion of generalized Berkson error in particular mathematicizes what it means to be an expert assessor or rater for a measurement process. We then explore how classical point estimation, inference, and likelihood theory can be generalized to accommodate sample data composed of generic random-variable-valued measurements.

Probabilistic real-time natural gas jet fire consequence modeling of offshore platforms by hybrid deep learning approach

Natural gas jet fire induced by igniting blowouts has the potential to cause critical structure damage and great casualties of offshore platforms. Real-time natural gas jet fire plume prediction is essential to support the emergency planning to mitigate subsequent damage consequence and ocean pollution. Deep learning based on a large amount of Computational fluid dynamics (CFD) simulations has recently been applied to real-time fire modeling. However, existing approaches based on point-estimation theory are ‘over-confident’ when prediction deficiency exists, which reduce robustness and accuracy for emergency planning support. This study proposes probabilistic deep learning approach for real-time natural gas jet fire consequence modeling by integrating variational Bayesian inference with deep learning. Numerical model of natural gas jet fire from offshore platform is built and the natural gas jet fire scenarios are simulated to construct the benchmark dataset. Sensitivity analysis of pre-defined parameters such as MC (Monte Carlo) sampling number m and dropout probability p is conducted to determine the trade-off between model's accuracy and efficiency. The results demonstrated our model exhibits competitive accuracy with R2 = 0.965 and real-time capacity with an inference time of 12 ms. In addition, the predicted spatial uncertainty corresponding to spatial jet fire flame plume provides more comprehensive and reliable support for the following mitigation decision-makings compared to the state-of-the-art point-estimation based deep learning model. This study provides a robust alternative for constructing a digital twin of fire and explosion associated emergency management on offshore platforms.

On minimax optimality of sparse Bayes predictive density estimates

We study predictive density estimation under Kullback–Leibler loss in ℓ0-sparse Gaussian sequence models. We propose proper Bayes predictive density estimates and establish asymptotic minimaxity in sparse models. Fundamental for this is a new risk decomposition for sparse, or spike-and-slab priors. A surprise is the existence of a phase transition in the future-to-past variance ratio r. For r<r0=(√5−1)/4, the natural discrete prior ceases to be asymptotically optimal. Instead, for subcritical r, a ‘bi-grid’ prior with a central region of reduced grid spacing recovers asymptotic minimaxity. This phenomenon seems to have no analog in the otherwise parallel theory of point estimation of a multivariate normal mean under quadratic loss. For spike-and-uniform slab priors to have any prospect of minimaxity, we show that the sparse parameter space needs also to be magnitude constrained. Within a substantial range of magnitudes, such spike-and-slab priors can attain asymptotic minimaxity.

Asymptotic theory for time series with changing mean and variance

The paper develops point estimation and asymptotic theory with respect to a semiparametric model for time series with moving mean and unconditional heteroscedasticity. These two features are modelled nonparametrically, whereas autocorrelations are described by a short memory stationary parametric time series model. We first study the usual least squares estimate of the coefficient of the first-order autoregressive model based on constant but unknown mean and variance. Allowing for both the latter to vary over time in a general way we establish its probability limit and a central limit theorem for a suitably normed and centred statistic, giving explicit bias and variance formulae. As expected mean variation is the main source of inconsistency and heteroscedasticity the main source of inefficiency, though we discuss circumstances in which the estimate is consistent for, and asymptotically normal about, the autoregressive coefficient, albeit inefficient. We then consider standard implicitly-defined Whittle estimates of a more general class of short memory parametric time series model, under otherwise more restrictive conditions. When the mean is correctly assumed to be constant, estimates that ignore the heteroscedasticity are again found to be asymptotically normal but inefficient. Allowing a slowly time-varying mean we resort to trimming out of low frequencies to achieve the same outcome. Returning to finite order autoregression, nonparametric estimates of the varying mean and variance are given asymptotic justification, and forecasting formulae developed. Finite sample properties are studied by a small Monte Carlo simulation, and an empirical example is also included.

Uncertainty estimation with a small number of measurements, part II: a redefinition of uncertainty and an estimator method

This paper is the second (Part II) in a series of two papers (Part I and Part II). Part I has quantitatively discussed the fundamental limitations of the t-interval method for uncertainty estimation with a small number of measurements. This paper (Part II) reveals that the t-interval is an ‘exact’ answer to a wrong question; it is actually misused in uncertainty estimation. This paper proposes a redefinition of uncertainty, based on the classical theory of errors and the theory of point estimation, and a modification of the conventional approach to estimating measurement uncertainty. It also presents an asymptotic procedure for estimating the z-interval. The proposed modification is to replace the t-based uncertainty with an uncertainty estimator (mean- or median-unbiased). The uncertainty estimator method is an approximate answer to the right question to uncertainty estimation. The modified approach provides realistic estimates of uncertainty, regardless of whether the population standard deviation is known or unknown, or if the sample size is small or large. As an application example of the modified approach, this paper presents a resolution to the Du–Yang paradox (i.e. Paradox 2), one of the three paradoxes caused by the misuse of the t-interval in uncertainty estimation.

A Genetic Algorithm based Hybrid Approach to Solve Multi-objectiveInterval Assignment Problem by Estimation Theory

Objectives: In industrial organization and management science, Assignment Problem (AP) is most studied problem with multiple uncertain objectives; normally this uncertainty is represented by an interval. The main aim of this paper is to find a solution of such Multi-objective Interval Assignment Problem (MOIAP). Methods/Statistical analysis: This paper proposes a Genetic Algorithm (GA) based hybrid approach to solve MOIAP by using point and interval estimation theory. Estimation theory is used to manage such uncertainty with sampling. For that the samples are taken from the past data and find point estimation form the samples of each parameter to specified the range of interval with confidence level and this range is considered as an interval in AP. Findings: Traditional method like fuzzy, weighted min-max, goal programming is utilized by several researchers for find solution of MOIAP but here GA provided a proper analysis based solution. In this paper realistic example is provided to represents the solution effectiveness by estimation theory which shows that developed approach provide improved and analysis based solution compare to other approach.

On the guaranteed convergence of a cubically convergent Weierstrass-like root-finding method

Initial conditions that provide guaranteed and fast convergence of the Weierstrass-like cubically convergent iterative method for the simultaneous determination of all simple zeros of a polynomial are considered. It is proved that this method is convergent under suitable conditions stated in the spirit of Smale's point estimation theory. The proposed convergence conditions are computationally verifiable since they depend only on initial approximations and the degree of a given polynomial, which is of practical importance.

On the guaranteed convergence of new two-point root-finding methods for polynomial zeros

We presented new two-point methods for the simultaneous approximation of all n simple (real or complex) zeros of a polynomial of degree n. We proved that the R-order of convergence of the total-step version is three. Moreover, computationally verifiable initial conditions that guarantee the convergence of one of the proposed methods are stated. These conditions are stated in the spirit of Smale's point estimation theory; they depend only on available data, the polynomial coefficients, polynomial degree n and initial approximations x 1 ( 0 ) , Â? , x n ( 0 ) $x_{1}^{(0)},\ldots ,x_{n}^{(0)}$ , which is of practical importance. Using the Gauss-Seidel approach we state the corresponding single-step version and consequently its prove that the lower bound of its R-order of convergence is at least 2 + y n > 3, where y n Â? (1, 2) is the unique positive root of the equation y n Â? y Â? 2 = 0. Two numerical examples are given to demonstrate the convergence behavior of the considered methods, including global convergence.

New Theoretical Results on Channelized Hotelling Observer Performance Estimation With Known Difference of Class Means

Task-based assessments of image quality constitute a rigorous, principled approach to the evaluation of imaging system performance. To conduct such assessments, it has been recognized that mathematical model observers are very useful, particularly for purposes of imaging system development and optimization. One type of model observer that has been widely applied in the medical imaging community is the channelized Hotelling observer (CHO). Since estimates of CHO performance typically include statistical variability, it is important to control and limit this variability to maximize the statistical power of image-quality studies. In a previous paper, we demonstrated that by including prior knowledge of the image class means, a large decrease in the bias and variance of CHO performance estimates can be realized. The purpose of the present work is to present refinements and extensions of the estimation theory given in our previous paper, which was limited to point estimation with equal numbers of images from each class. Specifically, we present and characterize minimum-variance unbiased point estimators for observer signal-to-noise ratio (SNR) that allow for unequal numbers of lesion-absent and lesion-present images. Building on this SNR point estimation theory, we then show that confidence intervals with exactly-known coverage probabilities can be constructed for commonly-used CHO performance measures. Moreover, we propose simple, approximate confidence intervals for CHO performance, and we show that they are well-behaved in most scenarios of interest.

A note on the improved derivative free root-solvers

Using a fixed point relation of the square-root type and the basic fourth-order method, improved methods of fifth and sixth order for the simultaneous determination of simple zeros of a polynomial are obtained. An increase in convergence is achieved without additional numerical operations, which points to high computational efficiency of the accelerated methods. The main aim of this work is the convergence analysis of improved simultaneous methods given under computationally verifiable initial conditions in the spirit of Smale’s point estimation theory.

Point estimation of simultaneous methods for solving polynomial equations: A survey (II)

The construction of computationally verifiable initial conditions which provide both the guaranteed and fast convergence of the numerical root-finding algorithm is one of the most important problems in solving nonlinear equations. Smale's “point estimation theory” from 1981 was a great advance in this topic; it treats convergence conditions and the domain of convergence in solving an equation f ( z ) = 0 using only the information of f at the initial point z 0 . The study of a general problem of the construction of initial conditions of practical interest providing guaranteed convergence is very difficult, even in the case of algebraic polynomials. In the light of Smale's point estimation theory, an efficient approach based on some results concerning localization of polynomial zeros and convergent sequences is applied in this paper to iterative methods for the simultaneous determination of simple zeros of polynomials. We state new, improved initial conditions which provide the guaranteed convergence of frequently used simultaneous methods for solving algebraic equations: Ehrlich–Aberth's method, Ehrlich–Aberth's method with Newton's correction, Börsch-Supan's method with Weierstrass’ correction and Halley-like (or Wang–Zheng) method. The introduced concept offers not only a clear insight into the convergence analysis of sequences generated by the considered methods, but also explicitly gives their order of convergence. The stated initial conditions are of significant practical importance since they are computationally verifiable; they depend only on the coefficients of a given polynomial, its degree n and initial approximations to polynomial zeros.

The convergence of a family of parallelzero-finding methods

One of the crucial problems in solving polynomial equations is to state some kind of quantitative conditions that provide both safe and fast convergence. In the spirit of Smale's point estimation theory introduced in 1981, we construct in this paper computationally verifiable initial conditions which guarantee the convergence of the recently proposed one parameter family of methods of the fourth order for the simultaneous finding polynomial zeros. These conditions are computationally verifiable since they depend only on initial approximations, polynomial coefficients, and polynomial degree, which is of significant practical importance.

Approximation for the expectation of a function of the sample mean

Let X¯ n be the mean of a random sample of size n from a distribution with mean μ and variance σ2. Under some conditions it is shown that Ef(X¯ n ) = f(μ) + (σ2/2n) f″(μ) + O(n −2), and var(f(X¯ n )) = (σ2/n) (f′(μ))2 + O(n −2), where f is a continuous function with a suitable growth condition. This complements a result of Lehmann [(1991). Theory of Point Estimation. Wadsworth, California] and Cramér [(1946). Mathematical Methods of Statistics. Princeton University Press, Princeton, N.J.] for wider application. An illustrative example is given to show an application where the usual approximations do not apply.

On the guaranteed convergence of the square-root iteration method

The construction of initial conditions which guarantee the convergence of the applied iterative method, is one of the most important problems in solving nonlinear equations which attracted the great attention for many years. In this paper, we give a precise convergent analysis of the Ostrowski-like method of the fourth order for the simultaneous determination of polynomial zeros. Using a procedure based on Smale's point estimation theory and some recent results related to the localization of complex polynomial zeros, we state initial conditions which enable both the guaranteed and fast convergence of this method. These conditions are computationally verifiable since they depend only on polynomial coefficients, its degree and initial approximations, which is of practical importance.

On the guaranteed convergence of the fourth order simultaneous method for polynomial zeros

One of the most important and challenging problems in solving nonlinear equations is the construction of computationally verifiable initial conditions that provide both the guaranteed and fast convergence of the considered numerical method. A suitable convergence procedure, based partially on Smale's “point estimation theory” from 1981, is applied in this paper to the fourth order iterative method for the simultaneous approximation of simple zeros of polynomials, proposed by Zheng and Sun in 1999. We have stated initial conditions which ensure the guaranteed convergence of this method. These conditions are of significant practical importance since they depend only on available data: the coefficients of a given polynomial, its degree n and initial approximations to polynomial zeros.

The guaranteed convergence of Laguerre-like method

The construction of initial conditions that provide a guaranteed convergence of zero-finding methods has attracted a great deal of attention for many years. In this paper, we consider convergent properties of the Laguerre-like method of the fourth order for the simultaneous approximation of polynomial zeros. Using a procedure based on Smale's point estimation theory and some recent results concerned with localization of complex polynomial zeros, we state initial conditions which enable both the guaranteed and fast convergence of this method. These conditions are computationally verifiable since they depend only on initial approximations, polynomial coefficients, and polynomial degree, which is of practical importance.

Point estimation of simultaneous methods for solving polynomial equations: a survey

One of the most important problems in solving nonlinear equations is the construction of such initial conditions which provide both the guaranteed and fast convergence of the considered numerical algorithm. Smale's approach from 1981, known as “point estimation theory”, treats convergence conditions and the domain of convergence in solving an equation f( z)=0 using only the information of f at the initial point z 0. A procedure of this type is applied in this paper to iterative methods for the simultaneous approximation of simple zeros of polynomial equations. We have stated new, refined initial conditions which ensure the guaranteed convergence of the most frequently used simultaneous methods for solving algebraic equations: the Durand–Kerner method, Börsch-Supan method, the Ehrlich–Aberth method and the square-root family of one parameter methods. The stated initial conditions are of significant practical importance since they are computationally verifiable; they depend only on the coefficients of a given polynomial, its degree n and initial approximations to polynomial zeros.

Safe convergence of simultaneous methods for polynomial zeros

The theory of point estimation treating the initial conditions for the safe convergence of iterative processes for the simultaneous determination of polynomial zeros is considered. A general approach which makes use of corrections appearing in iterative formulas is given and demonstrated in the case of three well known methods without derivatives and based on Weierstrass’ corrections. The established convergence conditions are of practical importance since they depend only on available data: coefficients of a polynomial and initial approximations to the zeros.

Intrinsic Analysis of Statistical Estimation

The parametric statistical models with suitable regularity conditions have a natural Riemannian manifold structure, given by the information metric. Since the parameters are merely labels for the probability measures, an inferential statement should be formulated through intrinsic objects, invariant under reparametrizations. In this context the estimators will be random objects valued on the manifold corresponding to the statistical model. In spite of these considerations, classical measures of an estimator's performance, like the bias and the mean square error, are clearly dependent on the statistical model parametrizations. In this paper the authors work with extended notions of mean value and moments of random objects which take values on a Hausdorff and connected manifold, equipped with an affine connection. In particular, the Riemannian manifold case is considered. This extension is applied to the bias and the mean square error study in statistical point estimation theory. Under this approach an intrinsic version of the Cramer-Rao lower bound is obtained: a lower bound, which depends on the intrinsic bias and the curvature of the statistical model, for the mean square of the Rao distance, the invariant measure analogous to the mean square error. Further, the behavior of the mean square of the Rao distance of an estimator when conditioning with respect to a sufficient statistic is considered, obtaining intrinsic versions of the Rao-Blackwell and Lehmann-Scheffe theorems. Asymptotic properties complete the study.

The theory of Smale's point estimation and its applications

The main result of this paper is that we exact Smale's point estimation theory, i.e., without assuming γ k = ‖P′(z) −1P (k)(z) k! ‖ (k ⩾ 2) being bounded by γ, the point estimation convergence theorem of the Ne method is set up by making use of the majorizing method. The proof of the theorem is simple and precise, while the required point estimation conditions are weaker than all those of known point estimation convergence theorems. Another result of this paper is an application of the above new theory to the Durand-Kerner method. We compare the point estimation conditions for the Durand-Kerner method with other known point estimation conditions. Numerical results show that our results have evident advantages.