Approximate Confidence Regions Research Articles

Neural likelihood surfaces for spatial processes with computationally intensive or intractable likelihoods

In spatial statistics, fast and accurate parameter estimation, coupled with a reliable means of uncertainty quantification, can be challenging when fitting a spatial process to real-world data because the likelihood function might be slow to evaluate or wholly intractable. In this work, we propose using convolutional neural networks to learn the likelihood function of a spatial process. Through a specifically designed classification task, our neural network implicitly learns the likelihood function, even in situations where the exact likelihood is not explicitly available. Once trained on the classification task, our neural network is calibrated using Platt scaling which improves the accuracy of the neural likelihood surfaces. To demonstrate our approach, we compare neural likelihood surfaces and the resulting maximum likelihood estimates and approximate confidence regions with the equivalent for exact or approximate likelihood for two different spatial processes—a Gaussian process and a Brown–Resnick process which have computationally intensive and intractable likelihoods, respectively. We conclude that our method provides fast and accurate parameter estimation with a reliable method of uncertainty quantification in situations where standard methods are either undesirably slow or inaccurate. The method is applicable to any spatial process on a grid from which fast simulations are available.

Likelihood-type confidence regions for optimal sensitivity and specificity of a diagnostic test

New methods are proposed that provide approximate joint confidence regions for the optimal sensitivity and specificity of a diagnostic test, i.e., sensitivity and specificity corresponding to the optimal cutpoint as defined by the Youden index criterion. Such methods are semi-parametric or non-parametric and attempt to overcome the limitations of alternative approaches. The proposed methods are based on empirical likelihood pivots, giving rise to likelihood-type regions with no predetermined constraints on the shape and automatically range-respecting. The proposal covers three situations: the binormal model, the binormal model after the use of Box-Cox transformations and the fully non-parametric model. In the second case, it is also shown how to use two different transformations, for the healthy and the diseased subjects. The finite sample behaviour of our methods is investigated using simulation experiments. The simulation results also show the advantages offered by our methods when compared with existing competitors. Illustrative examples, involving three real datasets, are also provided.

A Bivariate Index Vector to Measure Departure from Quasi-symmetry for Ordinal Square Contingency Tables

This study proposes a bivariate index vector to concurrently analyze both the degree and direction of departure from the quasi-symmetry (QS) model for ordinal square contingency tables. The QS model and extended QS (EQS) models identify the symmetry and asymmetry between the probabilities of normal circulation and reverse circulation when the order exists for arbitrary three categories. The asymmetry parameter of the EQS model implies the degree of departure from the QS model; the EQS model is equivalent to the QS model when the asymmetry parameter equals to one. The structure of the EQS model differs depending on whether the asymmetry parameter approaches zero or infinity. Thus, the asymmetry parameter of the EQS model also implies the direction of departure from the QS model. The proposed bivariate index vector is constructed by combining existing and original sub-indexes that represent the degree of departure from the QS model and its direction. These sub-indexes are expressed as functions of the asymmetry parameter under the EQS model. We construct an estimator of the proposed bivariate index vector and an approximate confidence region for the proposed bivariate index vector. Using real data, we show that the proposed bivariate index vector is important to compare degrees of departure from the QS model for plural data sets.

Algorithms of Confidence Intervals of WG Distribution Based on Progressive Type-II Censoring Samples

The purpose of this article offers different algorithms of Weibull Geometric (WG) distribution estimation depending on the progressive Type II censoring samples plan, spatially the joint confidence intervals for the parameters. The approximate joint confidence intervals for the parameters, the approximate confidence regions and percentile bootstrap intervals of confidence are discussed, and several Markov chain Monte Carlo (MCMC) techniques are also presented. The parts of mean square error (MSEs) and credible intervals lengths, the estimators of Bayes depend on non-informative implement more effective than the maximum likelihood estimates (MLEs) and bootstrap. Comparing the models, the MSEs, average confidence interval lengths of the MLEs, and Bayes estimators for parameters are less significant for censored models.

Statistical inference for two exponential populations under joint progressive type-I censored scheme

ABSTRACTIn this article, we introduce a new scheme called joint progressive type-I (JPC-I) censored and as a special case, joint type-I censored scheme. Bayesian and non Bayesian estimators have been obtained for two exponential populations under both JPC-I censored scheme and joint type-I censored. The maximum likelihood estimators of the parameters, the asymptotic variance covariance matrix, have been obtained. Bayes estimators have been developed under squared error loss function using independent gamma prior distributions. Moreover, approximate confidence region based on the asymptotic normality of the maximum likelihood estimators and credible confidence region from a Bayesian viewpoint are also discussed and compared with two Bootstrap confidence regions. A numerical illustration for these new results is given.

Interval Estimation for the Two-Parameter Bathtub-Shaped Lifetime Distribution Based on Records

In this paper, we study the estimation problems for the two-parameter bathtub-shaped lifetime distribution based on upper record values. Exact confidence intervals and exact joint confidence regions for the parameters are constructed. Approximate confidence intervals and regions are also discussed based on the asymptotic normality of the maximum likelihood estimators. A simulation study is done for the performance of all proposed confidence intervals and regions. Two numerical examples with real data set and simulated data, are presented to illustrate the methods proposed here.

Parameter estimation for multiple weibull populations under joint type-II censoring

In this paper, we introduce the maximum likelihood estimation for k Weibull populations under joint type II censored scheme and different special cases have been obtained. The asymptotic variance covariance matrix and approximate confidence region based on the asymptotic normality of the maximum likelihood estimators have been obtained. A numerical example is considered to illustrate the proposed estimators. Keywords: Approximate Inference; Coverage Probabilities; Joint Type II Censored Scheme; Maximum Likelihood Estimation; Weibull Distribution.

Bayesian and Non–Bayesian Estimation for Two Generalized Exponential Populations Under Joint Type II Censored Scheme

In this paper, Bayesian and non-Bayesian estimators have been obtained for two generalized exponential populations under joint type II censored scheme, which generalize results of Balakrishnan and Rasouli (2008) and Shafay et al. (2013) . The maximum likelihood estimators (MLEs) of the parameters and Bayes estimators have been developed under squared error loss function as well as under LINEX loss function. Moreover, approximate confidence region are also discussed and compared with two Bootstrap confidence regions. Also the MLE and three confidence intervals for the stress–strength parameter are explored. A numerical illustration for these new results is given.

Confidence intervals for parameters of IWD based on MLE and bootstrap

In this paper, we will study the joint confidence regions for the parameters of inverse Weibull distribution (IWD) in the point of view of record values. Based on this new censoring scheme, the approximate confidence intervals and percentile bootstrap confidence intervals as well as approximate joint confidence region for the parameters of IWD, are developed. One of the applications of the joint confidence regions of the parameters is to find confidence bounds for the functions of the parameters. Joint confidence regions for the parameters of extreme value distribution are also discussed. In this way we will discuss some numerical examples with real data set and simulated data, to illustrate the proposed method. A simulation study is performed to compare the proposed joint confidence regions.

Confidence regions for repeated measures ANOVA power curves based on estimated covariance

BackgroundUsing covariance or mean estimates from previous data introduces randomness into each power value in a power curve. Creating confidence intervals about the power estimates improves study planning by allowing scientists to account for the uncertainty in the power estimates. Driving examples arise in many imaging applications.MethodsWe use both analytical and Monte Carlo simulation methods. Our analytical derivations apply to power for tests with the univariate approach to repeated measures (UNIREP). Approximate confidence intervals and regions for power based on an estimated covariance matrix and fixed means are described. Extensive simulations are used to examine the properties of the approximations.ResultsClosed-form expressions are given for approximate power and confidence intervals and regions. Monte Carlo simulations support the accuracy of the approximations for practical ranges of sample size, rank of the design matrix, error degrees of freedom, and the amount of deviation from sphericity. The new methods provide accurate coverage probabilities for all four UNIREP tests, even for small sample sizes. Accuracy is higher for higher power values than for lower power values, making the methods especially useful in practical research conditions. The new techniques allow the plotting of power confidence regions around an estimated power curve, an approach that has been well received by researchers. Free software makes the new methods readily available.ConclusionsThe new techniques allow a convenient way to account for the uncertainty of using an estimated covariance matrix in choosing a sample size for a repeated measures ANOVA design. Medical imaging and many other types of healthcare research often use repeated measures ANOVA.

Uncertainty inherent in empirical fitting of distributions to experimental data

Treatment of experimental data often entails fitting frequency functions, in order to draw inferences on the population underlying the sample at hand, and/or identify plausible mechanistic models. Several families of functions are currently resorted to, providing a broad range of forms; an overview is given in the light of historical developments, and some issues in identification and fitting procedure are considered. But for the case of fairly large, well behaved data sets, empirical identification of underlying distribution among a number of plausible candidates may turn out to be somehow arbitrary, entailing a substantial uncertainty component. A pragmatic approach to estimation of an approximate confidence region is proposed, based upon identification of a representative subset of distributions marginally compatible at a given level with the data at hand. A comprehensive confidence region is defined by the envelope of the subset of distributions considered, and indications are given to allow first order estimation of uncertainty component inherent in empirical distribution fitting.

Two models for linear comparative calibration

We consider the comparative calibration problem in the case when linear relationship is assumed between two considered measuring devices with possibly different units and precisions. The first method for obtaining the approximate confidence region for unknown parameters of the calibration line applies the maximum likelihood estimators of the unknown parameters. The second method is based on estimation of the calibration line via replicated errors-in-variables model. Essential point in this approach is approximation of the small sample distribution of the Wald-type test statistic. This enables to construct the interval estimators for the multiple-use calibration case.

Fuzzy confidence regions for the Taguchi capability index

Most of the traditional methods for assessing the capability of manufacturing processes are dealing with crisp quality. In quality control, such as other statistical problems, we may confront imprecise concepts. One case is a situation in which specification limits (SLs) are imprecise. In this situation, the fuzzy process capability indices (PCIs) , and are necessary for measuring the fuzzy quality in an in-control process. These fuzzy capability indices are also helpful for comparing manufacturing processes with SLs. The fuzzy capability index is used to provide an assessment of the ability of the fuzzy process to be clustered around the target value. The emphasis on the use of over the other two fuzzy indices, and , is due to its definition that provides indications of both the process variability and deviation of process mean from a specified target. In this article, by using triangular fuzzy SLs, we present four approximate fuzzy confidence regions for the fuzzy PCI . A numerical example is given to show the performance of the method.

Conditional Maximum Likelihood and Interval Estimation for Two Weibull Populations under Joint Type-II Progressive Censoring

Comparative lifetime experiments are important when the object of a study is to determine the relative merits of two competing duration of life products. This study considers the interval estimation for two Weibull populations when joint Type-II progressive censoring is implemented. We obtain the conditional maximum likelihood estimators of the two Weibull parameters under this scheme. Moreover, simultaneous approximate confidence region based on the asymptotic normality of the maximum likelihood estimators are also discussed and compared with two Bootstrap confidence regions. We consider the behavior of probability of failure structure with different schemes. A simulation study is performed and an illustrative example is also given.

Empirical Simultaneous Confidence Regions for Path-Forecasts

Measuring and displaying uncertainty around path-forecasts, i.e. forecasts made in period T about the expected trajectory of a random variable in periods T+1 to T+H is a key ingredient for decision making under uncertainty. The probabilistic assessment about the set of possible trajectories that the variable may follow over time is summarized by the simultaneous confidence region generated from its forecast generating distribution. However, if the null model is only approximative or altogether unavailable, one cannot derive analytic expressions for this confidence region, and its non-parametric estimation is impractical given commonly available predictive sample sizes. Instead, this paper derives the approximate rectangular confidence regions that control false discovery rate error, which are a function of the predictive sample covariance matrix and the empirical distribution of the Mahalanobis distance of the path-forecast errors. These rectangular regions are simple to construct and appear to work well in a variety of cases explored empirically and by simulation. The proposed techniques are applied to provide confidence bands around the Fed and Bank of England real-time path-forecasts of growth and inflation.

On confidence regions of semiparametric nonlinear reproductive dispersion models

This article investigates the confidence regions for semiparametric nonlinear reproductive dispersion models (SNRDMs), which is an extension of nonlinear regression models. Based on local linear estimate of nonparametric component and generalized profile likelihood estimate of parameter in SNRDMs, a modified geometric framework of Bates and Wattes is proposed. Within this geometric framework, we present three kinds of improved approximate confidence regions for the parameters and parameter subsets in terms of curvatures. The work extends the previous results of Hamilton et al. [in Accounting for intrinsic nonlinearity in nonlinear regression parameter inference regions, Ann. Statist. 10, pp. 386–393, 1982], Hamilton [in Confidence regions for parameter subset in nonlinear regression, Biometrika, 73, pp. 57–64, 1986], Wei [in On confidence regions of embedded models in regular parameter families (a geometric approch), Austral. J. Statist. 36, pp. 327–338, 1994], Tang et al. [in Confidence regions in quasi-likelihood nonlinear models: a geometric approach, J. Biomath. 15, pp. 55–64, 2000b] and Zhu et al. [in On confidence regions of semiparametric nonlinear regression models, Acta. Math. Scient. 20, pp. 68–75, 2000].

TU‐C‐BRB‐03: Radiation Dose Response of Plasma Cell Neoplasms

Purpose: To review the available clinical dose response data for extramedullary plasmacytomas (EMP) and solitary plasmacytomas of the bones (SPB), including standard 12 Gy TBI treatments for multiple myeloma (MM), to compute the expected dose response for plasma cell neoplasms and evaluate differences between EMP and SPB dose response. Method and Materials: Articles from 27 published studies on plasmacytomas were analyzed. Local control (LC) was used as the end point. Clinical data are often reported as LC for the median of a dose range — only data from ranges of width ⩽10 Gy were used. The maximum likelihood method (ML) was used to estimate the parameters of a tumour control probability (TCP) model based on Poisson statistics, and approximate likelihood confidence regions (CR) were determined. A Monte Carlo experiment (MC) assessed the parameters' uncertainty due to the 10 Gy dose interval. A statistical test based on the ability of the MC distributions of the parameters to discriminate between different kinds of tumors was performed. Results: Radiation therapy was used as the sole treatment in more than 70% of the patients and in 8 of the 12 studies selected. Parameters characterizing TCP and 95% confidence intervals from MC are reported, along with graphical representations of the dose response, and 2D MC histograms and the CRs on the parameter space. Conclusion: An extensive review of plasmacytoma clinical data was performed. Although the data suffer from a lack of low dose data and are mostly reported within a dose range, this approach is a preliminary assessment of dose response relationship for plasma cell neoplasms. The parameters of the TCP model were determined. Significant difference was seen between EMP and SPB dose response. The models could be used to interpolate clinical data and estimate TCP when assessing new therapies and comparing different treatment planning approaches.

Exact likelihood inference for two exponential populations under joint Type-II censoring

Comparative lifetime experiments are of paramount importance when the object of a study is to ascertain the relative merits of two competing products in regard to the duration of their service life. In this paper, we discuss exact inference for two exponential populations when Type-II censoring is implemented on the two samples in a combined manner. We obtain the conditional maximum likelihood estimators (MLEs) of the two exponential mean parameters. We then derive the moment generating functions and the exact distributions of these MLEs along with exact confidence intervals and simultaneous confidence regions. Moreover, simultaneous approximate confidence regions based on the asymptotic normality of the MLEs and simultaneous credible confidence regions from a Bayesian viewpoint are also discussed. A comparison of the exact, approximate, bootstrap and Bayesian intervals is also made in terms of coverage probabilities. Finally, an example is presented in order to illustrate all the methods of inference discussed here.

Empirical likelihood for the difference of quantiles under censorship

In this paper, we use a smoothed empirical likelihood method to investigate the difference of quantiles under censorship. An empirical log-likelihood ratio is derived and its asymptotic distribution is shown to be chi-squared. Approximate confidence regions based on this method are constructed. Simulation studies are used to compare the empirical likelihood and the normal approximation method in terms of its coverage accuracy. It is found that the empirical likelihood method provides a much better performance.

PEAS: A toolbox to assess the accuracy of estimated parameters in environmental models

This paper presents a Matlab™ toolbox to assess the accuracy of the estimated parameters of environmental models, based on their approximate confidence regions. Before describing the application, the underlying theory is briefly recalled to familiarize the reader with the numerical methods involved. The software, named PEAS as an acronym for Parameter Estimation Accuracy Software, performs both the estimation and the accuracy analysis, using a user-friendly graphical interface to minimize the required programming. The user is required to specify the model structure according to the Matlab/Simulink™ syntax, supply the experimental data, provide an initial parameter guess and select an estimation method. PEAS provides several model assessment tools, in addition to parameter estimation, such as error function plotting, trajectory sensitivity, Monte Carlo analysis, all useful to assess the adequacy of the experimental data to the estimation problem. After the parameters have been estimated, the reliability assessment is performed: approximate and exact confidence regions are computed and a confidence test is produced. The Monte Carlo analysis is available for approximate accuracy assessment whenever the model structure prevents the application of the confidence regions method. The software, which is freely available for research purposes, is demonstrated here with two examples: a dynamical and an algebraic model. In both cases, software usage and outputs are presented and commented. The examples show how the user is guided through the application of the methods and how warning messages are returned if the estimation does not satisfy the accuracy criteria.