Confidence Regions For Parameters Research Articles

Confidence disc and square for Cauchy distributions

UDC 519.21 We construct a confidence region of parameters for a sample of size N from the Cauchy distributed random variables. Although Cauchy distribution has two parameters, a location parameter μ ∈ ℝ and a scale parameter σ > 0 , we infer them simultaneously by regarding them as a single complex parameter γ : = μ + i σ . The region should be a domain in the complex plane. We give a simple and concrete formula to give the region as a disc and as a square.

Confidence intervals of the Kling-Gupta efficiency

• The confidence intervals of the KG efficiency do not have a closed-form description. • The uncertainty of KG efficiency is attributed to measurement errors of training data. • Statistical inference with an informal goodness-of-fit metric is coined diagnostic regression. • Parameter confidence regions of KG efficiency in agreement with linear regression theory. • Marginal parameter distributions of KG efficiency are approximately Gaussian. The Kling-Gupta efficiency, hereafter referred to as KG efficiency rather than its common abbreviation KGE, proposed by Gupta et al. (2009) has become a widely used metric for evaluating the goodness-of-fit of n -vectors of observations, y ̃ = [ y ̃ 1 y ̃ 2 … y ̃ n ] ⊤ , and corresponding model simulations, y θ = y 1 θ y 2 θ … y n θ ⊤ . This metric rectifies some of the shortcomings of the coefficient of determination, R 2 , also known to hydrologists as the efficiency of Nash and Sutcliffe (1970) , by using a Euclidean-distance based weighting of the correlation, bias and temporal variability of the observed, y ̃ , and simulated, y θ , data. But as the KG efficiency is not borne out of assumptions with respect to the statistical distribution of the residuals, e θ = y ̃ - y θ , we cannot formally characterize its uncertainty. The NS efficiency suffers a similar problem, yet, statistical theory postulates that its confidence intervals should follow a beta distribution in certain special cases. Without a formal description of the confidence intervals of the KG efficiency, we cannot (amongst others) quantify parameter uncertainty, compute confidence and prediction limits on simulated model responses, inform decision makers about critical modeling uncertainties, evaluate model adequacy and assess the information content of calibration data. More fundamentally, without confidence intervals we cannot establish whether the KG efficiency is a consistent, efficient and unbiased estimator. In this paper we present an empirical description of the confidence intervals of the KG efficiency. We relate the unknown probability distribution of the KG efficiency to the measurement errors of the training data record, y ̃ , and use the bootstrap method to carry out statistical inference. We illustrate our method by application to a simple linear regression function for which the least squares parameter confidence regions are exactly known and two hydrologic models of contrasting complexity. The empirical parameter confidence regions and/or intervals of the KG efficiency are compared to those derived from generalized least squares, objective function contouring and Bayesian analysis using Markov chain Monte Carlo simulation. The marginal parameter distributions of the KG efficiency are generally well described by a normal distribution. Results further confirm that the distribution of the KG efficiency is a complex function of data length and the magnitude, distribution and structure of the measurement errors. This prohibits an analytic description of the empirical confidence regions and/or intervals of the KG efficiency and reiterates the need for the bootstrap method.

Nonparametric confidence regions via the analytic wild bootstrap

AbstractThe wild bootstrap is a nonparametric tool that can be used to estimate a sampling distribution in the presence of heteroscedastic errors. In particular, the wild bootstrap enables us to compute confidence regions for regression parameters under non‐i.i.d. models. While the wild bootstrap may perform well in these settings, its obvious drawback is a lack of computational efficiency. The wild bootstrap requires a large number of bootstrap replications, making the use of this tool impractical when dealing with big data. We introduce the analytic wild bootstrap (ANWB), which provides a nonparametric alternative way of constructing confidence regions for regression parameters. The ANWB is superior to the wild bootstrap from a computational standpoint while exhibiting similar finite‐sample performance. We report simulation results for both least squares and ridge regression. Additionally, we test the ANWB on a real dataset and compare its performance with that of other standard approaches.

Confidence Regions for Parameters in Stationary Time Series Models With Gaussian Noise

This article develops two new empirical likelihood methods for long-memory time series models based on adjusted empirical likelihood and mean empirical likelihood. By application of Whittle likelihood, one obtains a score function that can be viewed as the estimating equation of the parameters of the long-memory time series model. An empirical likelihood ratio is obtained which is shown to be asymptotically chi-square distributed. It can be used to construct confidence regions. By adding pseudo samples, we simultaneously eliminate the non-definition of the original empirical likelihood and enhance the coverage probability. Finite sample properties of the empirical likelihood confidence regions are explored through Monte Carlo simulation, and some real data applications are carried out.

A WILD BOOTSTRAP FOR DEPENDENT DATA

This paper introduces a novel wild bootstrap for dependent data (WBDD) as a means of calculating standard errors of estimators and constructing confidence regions for parameters based on dependent heterogeneous data. The consistency of the bootstrap variance estimator for smooth function of the sample mean is shown to be robust against heteroskedasticity and dependence of unknown form. The first-order asymptotic validity of the WBDD in distribution approximation is established when data are assumed to satisfy a near epoch dependent condition and under the framework of the smooth function model. The WBDD offers a viable alternative to the existing non parametric bootstrap methods for dependent data. It preserves the second-order correctness property of blockwise bootstrap (provided we choose the external random variables appropriately), for stationary time series and smooth functions of the mean. This desirable property of any bootstrap method is not known for extant wild-based bootstrap methods for dependent data. Simulation studies illustrate the finite-sample performance of the WBDD.

Determination of a Strain Energy Density Function for the Tricuspid Valve Leaflets Using Constant Invariant-Based Mechanical Characterizations.

The tricuspid valve (TV) regulates the blood flow within the right side of the heart. Despite recent improvements in understanding TV mechanical and microstructural properties, limited attention has been devoted to the development of TV-specific constitutive models. The objective of this work is to use the first-of-its-kind experimental data from constant invariant-based mechanical characterizations to determine a suitable invariant-based strain energy density function (SEDF). Six specimens for each TV leaflet are characterized using constant invariant mechanical testing. The data is then fit with three candidate SEDF forms: (i) a polynomial model-the transversely isotropic version of the Mooney-Rivlin model, (ii) an exponential model, and (iii) a combined polynomial-exponential model. Similar fitting capabilities were found for the exponential and the polynomial forms (R2=0.92-0.99 versus 0.91-0.97) compared to the combined polynomial-exponential SEDF (R2=0.65-0.95). Furthermore, the polynomial form had larger Pearson's correlation coefficients than the exponential form (0.51 versus 0.30), indicating a more well-defined search space. Finally, the exponential and the combined polynomial-exponential forms had notably smaller but more eccentric model parameter's confidence regions than the polynomial form. Further evaluations of invariant decoupling revealed that the decoupling of the invariant terms within the exponential form leads to a less satisfactory performance. From these results, we conclude that the exponential form is better suited for the TV leaflets owing to its superb fitting capabilities and smaller parameter's confidence regions.

Improved Statistical Pattern Analysis Monitoring for Complex Multivariate Processes Using Empirical Likelihood

This article developed an improved statistical pattern analysis (SPA) monitoring strategy for fault detection of complex multivariate processes using empirical likelihood. The technique based on statistical pattern analysis performs fault detection by inspecting change in the statistics of process variables (e.g., mean value, correlation coefficient, variance, kurtosis, etc.). It is capable of monitoring non-Gaussian or even nonlinear processes. However, the original SPA framework explicitly computes all the high-order statistics, which significantly increases the scale and dimensionality of the problem, especially in the case of complex multivariate processes. To alleviate this difficulty, we propose monitoring changes in the statistics with the same order using empirical likelihood, which is a widely used estimation method to construct confidence limits or regions for parameters with similar properties. As a result, changes in statistics of the same order can be translated into a single index; hence more information on the faulty conditions can be observed. Furthermore, by considering statistics of the same order, the scale of the problem is reduced significantly. The improved statistical pattern analysis monitoring strategy is suitable for monitoring complex multivariate processes. The performance of the improved method is illustrated by an application study to fault detection of the Tennessee Eastman (TE) process.

Hyrec-2: A highly accurate sub-millisecond recombination code

We present the new recombination code hyrec-2, which achieves the same accuracy as the state-of-the-art codes hyrec and cosmorec and, at the same time, surpasses the computational speed of the code recfast commonly used for cosmic microwave background (CMB) anisotropy data analyses. hyrec-2 is based on an effective four-level atom model, accounting exactly for the nonequilibrium of highly excited states of hydrogen, and very accurately for radiative transfer effects with a correction to the Lyman-$\ensuremath{\alpha}$ escape rate. The latter is computed with the original hyrec and tabulated as a function of temperature, along with its derivatives with respect to the relevant cosmological parameters. This enables the code to keep the same accuracy as the original hyrec over the full 99.7% confidence region of cosmological parameters currently allowed by Planck, while running in under one millisecond on a standard laptop. Our code leads to no noticeable bias in any cosmological parameters even in the case of an ideal cosmic-variance-limited experiment up to $\ensuremath{\ell}=5000$. Beyond CMB anisotropy calculations, hyrec-2 will be a useful tool to compute various observables that depend on the recombination and thermal history, such as the recombination spectrum or the 21-cm signal.

Automated label flows for excited states of correlation functions in lattice gauge theory.

Extracting excited states from lattice gauge theory correlation functions can be achieved through χ^{2} minimization fits or algebraic approaches such as the variational method and Prony's method. Performing any kind of error analysis often leads to overlapping confidence regions of model parameters, even when the spectrum is not particularly dense. To correctly estimate errors, one must beware of mislabeling the states. In this work, we provide an algorithm that we call automated label flows which consistently and systematically identifies a deterministic labeling of states. This is a black-box approach in the sense that it gives a sensible set of labels without user guidance. As an example, we pair one black box method with another, analyzing a lattice correlation function from real data using automated label flows in the context of Prony's method.

Empirical likelihood-based inferences for median medical cost regression models with censored data

ABSTRACT Recent studies show that medical cost data can be heavily censored and highly skewed, which leads to have more complex cost data analysis. In this paper, we propose influence function and empirical likelihood (EL)-based methods to construct confidence regions for regression parameters in median cost regression models with censored data. We further propose confidence intervals for the median cost with given covariates using the proposed EL-based confidence regions. Simulation studies are conducted to compare the proposed EL-based confidence regions with the existing normal approximation-based confidence regions in terms of coverage probabilities. The new EL-based methods are observed to have better finite sample performances than existing methods particularly when the censoring proportion is high. The new methods are also illustrated through a real data example.

Inference for Long-memory Time Series Models Based on Modified Empirical Likelihood

Empirical likelihood method has been applied to short-memory time series models by Monti (1997) through the Whittle's estimation method. Yau (2012) extended this idea to long-memory time series models. Asymptotic distributions of the empirical likelihood ratio statistic for short and long-memory time series have been derived to construct confidence regions for the corresponding model parameters. However, it experiences the undercoverage issue which causes the coverage probabilities of parameters lower than the given nominal levels, especially for small sample sizes. In this paper, we propose a modified empirical likelihood which combines the advantages of the adjusted empirical likelihood and the transformed empirical likelihood to modify the one proposed by Yau (2012) for autoregressive fractionally integrated moving average (ARFIMA) model for the purpose of improving coverage probabilities.Asymptotic null distribution of the test statistic has been established as the standard chi-square distribution with the degree of freedom one. Simulations have been conducted to investigate the performance of the proposed method as well as the comparisons of other existing methods to illustrate that the proposed method can provide better coverage probabilities especially for small sample sizes.

A general frequency domain method for assessing spatial covariance structures

When examining dependence in spatial data, it can be helpful to formally assess spatial covariance structures that may not be parametrically specified or fully model-based. That is, one may wish to test for general features regarding spatial covariance without presupposing any particular, or potentially restrictive, assumptions about the joint data distribution. Current methods for testing spatial covariance are often intended for specialized inference scenarios, usually with spatial lattice data. We propose instead a general method for estimation and testing of spatial covariance structure, which is valid for a variety of inference problems (including nonparametric hypotheses) and applies to a large class of spatial sampling designs with irregular data locations. In this setting, spatial statistics have limiting distributions with complex standard errors depending on the intensity of spatial sampling, the distribution of sampling locations, and the process dependence. The proposed method has the advantage of providing valid inference in the frequency domain without estimation of such standard errors, which are often intractable, and without particular distributional assumptions about the data (e.g., Gaussianity). To illustrate, we develop the method for formally testing isotropy and separability in spatial covariance and consider confidence regions for spatial parameters in variogram model fitting. A broad result is also presented to justify the method for application to other potential problems and general scenarios with testing spatial covariance. The approach uses spatial test statistics, based on an extended version of empirical likelihood, having simple chi-square limits for calibrating tests. We demonstrate the proposed method through several numerical studies.

Active Learning of Bayesian Linear Models with High-Dimensional Binary Features by Parameter Confidence-Region Estimation.

In this letter, we study an active learning problem for maximizing an unknown linear function with high-dimensional binary features. This problem is notoriously complex but arises in many important contexts. When the sampling budget, that is, the number of possible function evaluations, is smaller than the number of dimensions, it tends to be impossible to identify all of the optimal binary features. Therefore, in practice, only a small number of such features are considered, with the majority kept fixed at certain default values, which we call the working set heuristic. The main contribution of this letter is to formally study the working set heuristic and present a suite of theoretically robust algorithms for more efficient use of the sampling budget. Technically, we introduce a novel method for estimating the confidence regions of model parameters that is tailored to active learning with high-dimensional binary features. We provide a rigorous theoretical analysis of these algorithms and prove that a commonly used working set heuristic can identify optimal binary features with favorable sample complexity. We explore the performance of the proposed approach through numerical simulations and an application to a functional protein design problem.

Semiparametric efficient inferences for generalised partially linear models

In this paper, we consider semiparametric efficient inferences in the generalised partially linear models. A novel bias-corrected estimating procedure and a bias-corrected empirical log-likelihood ratio are developed, respectively, for point estimation and confidence regions for parameters of interest. Under mild conditions, the resulting likelihood ratio is shown to be standard chi-squared distributed asymptotically. Moreover, it is noteworthy that the range of bandwidth in this paper covers the optimal bandwidth due to the implementation of a new bias-corrected technique. Therefore, no undersmoothing is needed here for guaranteeing the asymptotically standard chi-squared distribution of the proposed statistic. Simulation study and real application are also provided in order to illustrate the performance of resulting procedure.

Effect of Electric Field on Pectinesterase Inactivation During Orange Juice Pasteurization by Ohmic Heating

This study evaluated effect of alternating electric field on pectinesterase activity in 5-mL Navel orange juice samples subject to ohmic pasteurization. An empirical Weibull kinetic model for enzyme inactivation was fitted to data from conventional and ohmic heating runs. Discretization of time-dependent inactivation rates allowed accounting for a nonisothermal process, with similar equivalent times between paired runs indicating comparable time-temperature profiles across heating methods. Conditions were 60 Hz, 32–36 V/cm, holding temperatures 60–90 °C, holding times 0–200 s in a custom cell with stirring, and two Ti-Pt electrodes. Residual pectinesterase activity was assayed with a traditional titration method at pH 7.5 and 30 °C. The stochastic Monte Carlo resampling method of smoothed bootstrap estimated the parameter populations to propagate experimental uncertainty on enzyme activity. Joint parameter populations and confidence regions agree with least squares estimates, and indicate that kinetic parameters depend on heating technology. This difference is evidence of electric field effects on pectinesterase activity, which is more evident at lower heat process intensities; however, the thermal denaturation seems to overcome other effects at higher processing times and temperatures. The finding that there are conditions where electric fields significantly affect enzyme inactivation highlights the importance of thoroughly assessing effects of alternative processing technologies on targets and indicators in food processing.

Monte Carlo two-stage indirect inference (2SIF) for autoregressive panels

Persistence and mean-nonstationarity often undermine reliability of asymptotically justified inference in dynamic panels. We combine the Monte Carlo test (MCT) and the indirect inference estimation (IIE) principles to construct confidence regions for autoregressive panel parameters with valid coverage whether mean-stationarity is imposed or relaxed and whether autoregressive roots are at or close to the unit boundary or far from unity. Procedures are based on the standard least squares dummy variables (LSDV) estimator and an augmented counterpart which we introduce to restore finite sample exactness in mean-nonstationary settings. We also put forth a ‘Durbin–Wu–Hausman-type’ test for mean-stationarity given a tested autoregressive parameter, based on the distance between the LSDV and its augmented counterpart. Location-scale invariance is shown analytically, and the MCT methods involve multiple stages that preserve exchangeability. The above are formally shown to control size exactly for finite N and T, and provide a new perspective to a literature that is primarily asymptotic. The advantages of the proposed approaches are also illustrated via comparative simulation studies. The identification problems arising from relaxing mean-stationarity are assessed and addressed; we also consider heteroskedastic directions. Results show concrete promise even with very small sample sizes and more broadly, underscore the usefulness of combining MCT and IIE principles.

Minimum Volume Confidence Regions for Parameters of Exponential Distributions from Different Samples

Independent samples from different exponential distributions are available in many statistical applications, for example, in queuing or reliability models, where the random variables describe waiting times and lifetimes, respectively. When two samples are observed, inference is then carried out for two location and two scale parameters. In multivariate setups including common location and common scale parameter assumptions, we provide confidence regions for the parameters of interest, which have minimum Lebesgue measure among all those based on the usual pivotal quantities and with the same or higher confidence level; in particular, they improve in terms of area (volume) upon the standard ‘trapezoidal’ confidence regions being constructed by combining independent univariate pivot statistics. The proposed confidence regions do not require any factorization of the overall confidence level, and their calculations need simple Monte Carlo simulations, only. Although focusing on two complete samples, generalizations of the results to more than two and doubly type-II censored samples are possible.

Empirical Likelihood Inference for Generalized Partially Linear Models with Longitudinal Data

In this article, we propose a generalized empirical likelihood inference for the parametric component in semiparametric generalized partially linear models with longitudinal data. Based on the extended score vector, a generalized empirical likelihood ratios function is defined, which integrates the within-cluster correlation meanwhile avoids direct estimating the nuisance parameters in the correlation matrix. We show that the proposed statistics are asymptotically Chi-squared under some suitable conditions, and hence it can be used to construct the confidence region of parameters. In addition, the maximum empirical likelihood estimates of parameters and the corresponding asymptotic normality are obtained. Simulation studies demonstrate the performance of the proposed method.

On an asymptotic relative efficiency concept based on expected volumes of confidence regions

ABSTRACTThe paper deals with an asymptotic relative efficiency concept for confidence regions of multidimensional parameters that is based on the expected volumes of the confidence regions. Under standard conditions the asymptotic relative efficiencies of confidence regions are seen to be certain powers of the ratio of the limits of the expected volumes. These limits are explicitly derived for confidence regions associated with certain plugin estimators, likelihood ratio tests and Wald tests. Under regularity conditions, the asymptotic relative efficiency of each of these procedures with respect to each one of its competitors is equal to 1. The results are applied to multivariate normal distributions and multinomial distributions in a fairly general setting.

Bartlett correction of frequency domain empirical likelihood for time series with unknown innovation variance

The Bartlett correction is a desirable feature of the likelihood inference, which yields the confidence region for parameters with improved coverage probability. This study examines the Bartlett correction for the frequency domain empirical likelihood (FDEL), based on the Whittle likelihood of linear time series models. Nordman and Lahiri (Ann Stat 34:3019–3050, 2006) showed that the FDEL does not have an ordinary Chi-squared limit when the innovation is non-Gaussian with unknown variance, which restricts the use of the FDEL inference in time series. We show that, by profiling the innovation variance out of the Whittle likelihood function, the FDEL is Chi-squared-distributed and Bartlett correctable. In particular, the order of the coverage error of the confidence region can be reduced from $$O(n^{-1})$$ to $$O(n^{-2})$$ .