Simultaneous Interval Estimation Research Articles

Simultaneous interval estimation of actuator fault and state for a class of nonlinear systems by zonotope analysis

In this paper, an actuator fault and state interval estimation method for a class of nonlinear systems is proposed by integrating observer design and zonotope analysis. For the considered systems, we present a novel unknown input observer structure with broad applications. The design procedure is based on H∞ method to decrease the influence of unknown but bounded process disturbances and measurement noise. Moreover, a novel interval estimation method is presented based on zonotope analysis to obtain tighter intervals. Numerical simulations of a quadruple-tank system are conducted to assess the performance of the proposed approach.

Confidence Intervals for Low Dimensional Parameters in High Dimensional Linear Models

SummaryThe purpose of this paper is to propose methodologies for statistical inference of low dimensional parameters with high dimensional data. We focus on constructing confidence intervals for individual coefficients and linear combinations of several of them in a linear regression model, although our ideas are applicable in a much broader context. The theoretical results that are presented provide sufficient conditions for the asymptotic normality of the proposed estimators along with a consistent estimator for their finite dimensional covariance matrices. These sufficient conditions allow the number of variables to exceed the sample size and the presence of many small non-zero coefficients. Our methods and theory apply to interval estimation of a preconceived regression coefficient or contrast as well as simultaneous interval estimation of many regression coefficients. Moreover, the method proposed turns the regression data into an approximate Gaussian sequence of point estimators of individual regression coefficients, which can be used to select variables after proper thresholding. The simulation results that are presented demonstrate the accuracy of the coverage probability of the confidence intervals proposed as well as other desirable properties, strongly supporting the theoretical results.

Multiple testing and interval estimates of correlated correlations

The method of Cohen, Ma and Sackrowitz (2012) [3] is applied to the problems of multiple testing and simultaneous interval estimation of correlated correlations. A large sample method is assumed and thus the basic statistics considered are Fisher’s z statistics. The asymptotic joint distribution of the z statistics is derived and utilized in the procedure. An example using data from Westfall and Young (1993) [10] is provided.

Exact Nonparametric Confidence Bands for the Survivor Function

A method to produce exact simultaneous confidence bands for the empirical cumulative distribution function that was first described by Owen, and subsequently corrected by Jager and Wellner, is the starting point for deriving exact nonparametric confidence bands for the survivor function of any positive random variable. We invert a nonparametric likelihood test of uniformity, constructed from the Kaplan-Meier estimator of the survivor function, to obtain simultaneous lower and upper bands for the function of interest with specified global confidence level. The method involves calculating a null distribution and associated critical value for each observed sample configuration. However, Noe recursions and the Van Wijngaarden-Decker-Brent root-finding algorithm provide the necessary tools for efficient computation of these exact bounds. Various aspects of the effect of right censoring on these exact bands are investigated, using as illustrations two observational studies of survival experience among non-Hodgkin's lymphoma patients and a much larger group of subjects with advanced lung cancer enrolled in trials within the North Central Cancer Treatment Group. Monte Carlo simulations confirm the merits of the proposed method of deriving simultaneous interval estimates of the survivor function across the entire range of the observed sample. This research was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. It was begun while the author was visiting the Department of Statistics, University of Auckland, and completed during a subsequent sojourn at the Medical Research Council Biostatistics Unit in Cambridge. The support of both institutions, in addition to that of NSERC and the University of Waterloo, is greatly appreciated.

Subgroup specific incremental value of new markers for risk prediction

In many clinical applications, understanding when measurement of new markers is necessary to provide added accuracy to existing prediction tools could lead to more cost effective disease management. Many statistical tools for evaluating the incremental value (IncV) of the novel markers over the routine clinical risk factors have been developed in recent years. However, most existing literature focuses primarily on global assessment. Since the IncVs of new markers often vary across subgroups, it would be of great interest to identify subgroups for which the new markers are most/least useful in improving risk prediction. In this paper we provide novel statistical procedures for systematically identifying potential traditional-marker based subgroups in whom it might be beneficial to apply a new model with measurements of both the novel and traditional markers. We consider various conditional time-dependent accuracy parameters for censored failure time outcome to assess the subgroup-specific IncVs. We provide non-parametric kernel-based estimation procedures to calculate the proposed parameters. Simultaneous interval estimation procedures are provided to account for sampling variation and adjust for multiple testing. Simulation studies suggest that our proposed procedures work well in finite samples. The proposed procedures are applied to the Framingham Offspring Study to examine the added value of an inflammation marker, C-reactive protein, on top of the traditional Framingham risk score for predicting 10-year risk of cardiovascular disease.

Analysis of randomized comparative clinical trial data for personalized treatment selections

Suppose that under the conventional randomized clinical trial setting, a new therapy is compared with a standard treatment. In this article, we propose a systematic, 2-stage estimation procedure for the subject-level treatment differences for future patient's disease management and treatment selections. To construct this procedure, we first utilize a parametric or semiparametric method to estimate individual-level treatment differences, and use these estimates to create an index scoring system for grouping patients. We then consistently estimate the average treatment difference for each subgroup of subjects via a nonparametric function estimation method. Furthermore, pointwise and simultaneous interval estimates are constructed to make inferences about such subgroup-specific treatment differences. The new proposal is illustrated with the data from a clinical trial for evaluating the efficacy and toxicity of a 3-drug combination versus a standard 2-drug combination for treating HIV-1-infected patients.

Pairwise comparisons for proportions estimated by pooled testing

When estimating the prevalence of a rare trait, pooled testing can confer substantial benefits when compared to individual testing. In addition to screening experiments for infectious diseases in humans, pooled testing has also been exploited in other applications such as drug testing, epidemiological studies involving animal disease, plant disease assessment, and screening for rare genetic mutations. Within a pooled-testing context, we consider situations wherein different strata or treatments are to be compared with the goals of assessing significant and practical differences between strata and ranking strata in terms of prevalence. To achieve these goals, we first present two simultaneous pairwise interval estimation procedures for use with pooled data. Our procedures rely on asymptotic results, so we investigate small-sample behavior and compare the two procedures in terms of simultaneous coverage probability and mean interval length. We then present a unified approach to determine pool sizes which deliver desired coverage properties while taking testing costs and interval precision into account. We illustrate our methods using data from an observational HIV study involving heterosexual males who use intravenous drugs.

Constructing intervals for the intracluster correlation coefficient using Bayesian modelling, and application in cluster randomized trials

Studies in health research are commonly carried out in clustered settings, where the individual response data are correlated within clusters. Estimation and modelling of the extent of between-cluster variation contributes to understanding of the current study and to design of future studies. It is common to express between-cluster variation as an intracluster correlation coefficient (ICC), since this measure is directly comparable across outcomes. ICCs are generally reported unaccompanied by confidence intervals. In this paper, we describe a Bayesian modelling approach to interval estimation of the ICC. The flexibility of this framework allows useful extensions which are not easily available in existing methods, for example assumptions other than Normality for continuous outcome data, adjustment for individual-level covariates and simultaneous interval estimation of several ICCs. There is also the opportunity to incorporate prior beliefs on likely values of the ICC. The methods are exemplified using data from a cluster randomized trial.

Semiparametric Box–Cox power transformation models for censored survival observations

The accelerated failure time model specifies that the logarithm of the failure time is linearly related to the covariate vector without assuming a parametric error distribution. In this paper, we consider the semiparametric Box–Cox transformation model, which includes the above regression model as a special case, to analyse possibly censored failure time observations. Inference procedures for the transformation and regression parameters are proposed via a resampling technique. Prediction of the survival function of future subjects with a specific covariate vector is also provided via pointwise and simultaneous interval estimates. All the proposals are illustrated with datasets from two clinical studies.

Simultaneous inferences on the contrast of two hazard functions with censored observations.

In survival analysis, often times the pattern of instantaneous risk over time is more interesting than that of the cumulative risk. For this case, a nonparametric hazard function estimate is more appropriate for summarizing the risk experience of a group of patients than the corresponding Kaplan-Meier estimate. In comparing a new treatment with a standard therapy, it is important to know if the treatment loses it potency during the follow-up period, and if it does, one would like to know when it becomes ineffective. Unfortunately, with a plot of the differences of two Kaplan-Meier curves, it is rather difficult to capture such temporal trends. In this article, we propose simple procedures for constructing confidence bands for the contrast of two hazard functions with censored data. The simultaneous interval estimates are quite useful for identifying possible values of the contrast over time with a certain degree of confidence. The new proposals are illustrated with an example and a small simulation study.

Sample Size Determination for Interval Estimation of Multinomial Probabilities

Abstract Sample size determination is an important component of study design and can frequently be addressed using the multinomial probability model. The sample size determination procedures of Tortora and also Thompson describe two approaches to the problem based upon simultaneous interval estimation of multinomial probabilities. This article compares these procedures and highlights their differences. An alternative procedure is also developed. The new procedure employs Tortora's approach but is based upon different confidence limits and results in smaller recommended sample sizes.

Using Common Random Numbers and Control Variates in Multiple-Comparison Procedures

This paper considers the determination of the relative merits of two or more system designs via stochastic simulation experiments by constructing simultaneous interval estimates of certain differences in expected performance. Tukey's all-pairwise-comparisons procedure, Hsu's multiple-comparisons-with-the-best procedure, and Dunnett's multiple-comparisons-with-a-control procedure are standard methods for making such comparisons. We propose refinements for all three procedures through the use of two variance reduction techniques: common random numbers and control variates. We show that the proposed procedures are better than the standard multiple-comparison procedures in the sense that they have a larger probability of containing the true difference and, at the same time, not containing zero when a difference exists.

Petroleum exploration models—Estimation and applications

Three statistical models of oil and gas exploration prevail in the literature: field size distribution, “exploration function,” and “effectiveness of exploration” models. They may have many functional forms and may be used for prediction and planning results of future exploration activity. A review of American and Soviet literature of exploration models is presented with special emphasis on functions suitable for countries which, like Poland, are not rich in oil and gas. Two new procedures are proposed: a method for assessing field size distribution and a method for simultaneous interval estimation of parameters of the exploration function. The methods complement existing ones as they are suited to areas that are not rich in hydrocarbons. Examples of application to exploration of two Polish petroliferous basins are given.

Simultaneous interval estimates of the odds ratio in studies with two or more comparisons

More than one odds ratio estimate will often arise from a single epidemiologic study. Examples of designs where this may occur include those where there is more than one case or control group, and investigations of several risk factors as part of the same study. Various methods for presenting multiple interval estimates are discussed, including: the naive method, the Bonferroni method, the Dunn method, the Scheffe method, and the Dunnett method. For rectangular regions the Dunnett method gives a region with the most appropriate confidence level, but this region contains a different set of odds ratio estimates than are implied by the usual significance tests. A confidence ellipse circumscribed by the Scheffe limits gives the best agreement with the significance tests. Each of these methods is illustrated with a numerical example.

The Efficiency of Simulation-Based Multiple Comparisons

A frequently encountered problem in practice is that of simultaneous interval estimation of p linear combinations of a parameter beta in the setting of (or equivalent to) a univariate linear model. This problem has been solved adequately only in a few settings when the covariance matrix of the estimator is diagonal; in other cases, conservative solutions can be obtained by the methods of Scheffé, Bonferroni, or Sidák (1967, Journal of the American Statistical Association 62, 626-633). Here we investigate the efficiency of using a simulated critical point for exact intervals, which has been suggested before but never put to serious test. We find the simulation-based method to be completely reliable and essentially exact. Sample size savings are substantial (in our settings): 3-19% over the Sidák method, 4-37% over the Bonferroni method, and 27-33% over the Scheffé method. We illustrate the efficiency and flexibility of the simulation-based method with case studies in physiology and marine ecology.

A monte carlo sampling plan for estimating reliability parameters and related functions

AbstractConsider an undirected network G with node set V and arc set E = {1, …, n} where arcs fail randomly and independently. Let T be a subset of V and let Mk denote the number of ways that all the nodes of T are connected (T‐connectivity) with exactly k operating arcs and n ‐ k failed arcs. This paper describes a sampling plan for estimating {Mk} and linear functions of these parameters, including the T‐connectedness reliability function g(p) for common failure probability 1 ‐ p. Point and simultaneous interval estimates are derived for {Mk/(}, where the interval estimates meet a fixed width criterion in O(n log n) time as the size of the network n grows. Whereas all previously proposed Monte Carlo sampling plans enable one to estimate g(p) for a fixed p, the proposed method allows one to estimate the entire function {g(p), p ∈ P}, where either P = P* = {qi: O ≦ qi ≦ 1, 1 ≦ i = ≦ v} or P = P** = {[a, b]: 0 ≦ a ≦ b ≦ 1}. Here P* denotes a finite set of points in [0, 1] and P** an interval in [0, 1]. Simultaneous interval estimates are derived that meet a fixed width criterion in O(n) time for P = P* and in O(n2) time for P = P** as n → ∞. A priori bounds are also derived for g (p) and it is shown how these can be used to give guidance on the performance of the sampling plan. An example based on a network with 44 nodes and 85 arcs illustrates the proposed method.

The Multivariate Heteroscedastic Method: Distributions of Statistics and an Application

SYNOPTIC ABSTRACTAsymptotic approximations to the distributions of the basic variate, and of some statistics constructed by the heteroscedastic method are derived for the multivariate case. The results are numerically compared with exact ones in the case of one and/or two dimensions. As an application, simultaneous confidence interval estimation with a given length, for the difference of two normal population mean vectors, is considered and a numerical example given using Fisher's iris data.

Sidak-type simultaneous confidence intervals for the risk measures rsmr, based on proportional mortality measures in epidemiologic 1 studies involving several competing risks of death

In the present paper, simultaneous confidence interval estimates are constructed for the mortality measures RSMR. based on propor¬tional mortality measures SPMR. in epidemiologic studies for several competing risks of death to which the individuals in the study are exposed. It is demonstrated that, under a reasonable assumption, the joint sampling distribution of the statistics X. = RSMR./SPMR. for M competing risks9 may be approximated by means of a multi-variafe normal distribution, Sidak's (1967, 1968) mulfivariate normal probability inequalities are applied to construct the simultaneous confidence intervals for the measures RSMR., i=l3 2, ..., M. These are valid regardless of the covariance structure among the risks, As a particular case if the risks may be assumed as independent, our confidence intervals reduce to those for a single measure RSMR., which are narrower than those of Kupper et al., (1978), In this sense, our paper generalizes the results presently available in the literature in two direc...

Rizvi and Sobel Type Nonparametric Selection Procedures: Review and Extensions

SYNOPTIC ABSTRACTThis paper reviews the area of nonparametric selection procedures based on sample quantiles; the underlying distributions are assumed to be continuous. Both the indifference zone and the (random-size) subset selection formulations are surveyed. The problem of comparing several experimental populations with a control is reviewed and some new results are obtained. The three-decision problem which allows nonselection with the assertion of equivalence and the problem of simultaneous selection and interval estimation of the largest α-quantile are also discussed.

An Extension of the T-Method to General Unbalanced Models of Fixed Effects

1. Introduction and Summary Among the various procedures for simultaneous interval estimation, Tukey's T-method gives the shortest confidence intervals for pairwise contrasts when it is applicable. (Cf. Scheffé, 1959, Ch. 3; Miller, 1966, Ch. 2.) However, the T-method in its exact formulation (excluding one way anova, see Spjøtvoll and Stoline, 1973) is restricted to well-balanced designs. In this paper we discuss a generalized T-type procedure for simultaneous interval estimation of all estimable parametric functions in general fixed effects, univariate linear models. The new method, which we subsequently abbreviate as the GT method, is an extension of a generalized T-procedure for the unbalanced one-way anova model given in Spjøtvoll and Stoline (1973). In Section 2 we derive the new procedure in full-rank and in less than full-rank models. Some important features of the procedure are discussed in Section 3. In order to gain insight into the types of models where the new procedure is expected to produce satisfactory confidence intervals for pairwise contrasts, we illustrate, in Section 4, the application of the GT method in some unbalanced designs. In Section 5 we discuss a modified GT procedure which constitutes a combination of the GT1 in Hochberg (1974) and the GT methods. The proofs of some of the theorems are given in an Appendix for purposes of a convenient reading.