A note on confidence interval estimation following curtailed binomial tests

Techniques of Armitage (1958) for finding confidence intervals after sequential tests (SCI) are applied to curtailed binomial test boundaries. The form of the exact randomized SCI is given. We also show that the conservative confidence interval calculated as though a fixed sample size procedure had been used (FCI) remains conservative when used after stopping on a curtailed boundary. Numerical results are obtained to assess the potential gains that may be obtained by using the conservative SCI or exact SCI over the conservative FCI for these boundaries.     

Exact unconditional tests for a 2 x 2 matched-pairs design

The problem of comparing two proportions in a 2 x 2 matched-pairs design with binary responses is considered. We consider one-sided null and alternative hypotheses. The problem has two nuisance parameters. Using the monotonicity of the multinomial distribution, four exact unconditional tests based on p-values are proposed by reducing the dimension of the nuisance parameter space from two to one in computation. The size and power of the four exact tests and two other tests, the exact conditional binomial test and the asymptotic McNemar's test, are considered. It is shown that the tests based on the confidence interval p-value are more powerful than the tests based on the standard p-value. In addition, it is found that the exact conditional binomial test is conservative and not powerful for testing the hypothesis. Moreover, the asymptotic McNemar's test is shown to have incorrect size; that is, its size is larger than the nominal level of the test. Overall, the test based on McNemar's statistic and the confidence interval p-value is found to be the most powerful test with the correct size among the tests in this comparison.     

A simple procedure for computing improved prediction intervals for autoregressive models

Abstract.  This article concerns the construction of prediction intervals for time series models. The estimative or plug-in solution is usually not entirely adequate, since the (conditional) coverage probability may differ substantially from the nominal value. Prediction intervals with improved (conditional) coverage probability can be defined by adjusting the estimative ones, using rather complicated asymptotic procedures or suitable simulation techniques. This article extends to Markov process models a recent result by Vidoni, which defines a relatively simple predictive distribution function, giving improved prediction limits as quantiles. This new solution is fruitfully considered in the challenging context of prediction for time-series models, with particular regard to AR and ARCH processes.     

Comparison of non-parametric confidence intervals for the area under the ROC curve of a continuous-scale diagnostic test

The accuracy of a diagnostic test with continuous-scale results is of high importance in clinical medicine. It is often summarised by the area under the ROC curve (AUC). In this article, we discuss and compare nine non-parametric confidence intervals of the AUC for a continuous-scale diagnostic test. Simulation studies are conducted to evaluate the relative performance of the confidence intervals for the AUC in terms of coverage probability and average interval length. A real example is used to illustrate the application of the recommended methods.     

Efficient interval estimation for age-adjusted cancer rates

The age-adjusted cancer rates are defined as the weighted average of the age-specific cancer rates, where the weights are positive, known, and normalized so that their sum is 1. Fay and Feuer developed a confidence interval for a single age-adjusted rate based on the gamma approximation. Fay used the gamma approximations to construct an F interval for the ratio of two age-adjusted rates. Modifications of the gamma and F intervals are proposed and a simulation study is carried out to show that these modified gamma and modified F intervals are more efficient than the gamma and F intervals, respectively, in the sense that the proposed intervals have empirical coverage probabilities less than or equal to their counterparts, and that they also retain the nominal level. The normal and beta confidence intervals for a single age-adjusted rate are also provided, but they are shown to be slightly liberal. Finally, for comparing two correlated age-adjusted rates, the confidence intervals for the difference and for the ratio of the two age-adjusted rates are derived incorporating the correlation between the two rates. The proposed gamma and F intervals and the normal intervals for the correlated age-adjusted rates are recommended to be implemented in the Surveillance, Epidemiology and End Results Program of the National Cancer Institute.     

Estimating a minimum set of physically based dynamic parameters to enhance statistical inference in block-oriented modeling

In process identification (i.e., dynamic model development) information on the precision and reliability of a parameter estimate is conveyed by a confidence interval. The best confidence interval is the one with the shortest width for a given level of confidence. Confidence intervals widen as the standard error increases or as the number of estimated parameters increases. When the value of a parameter is needed for physical understanding of process characteristics, its precision and reliability, i.e., certainty, is crucial. Parameter certainty increases as the number of estimated parameters decreases because this causes confidence intervals to shorten and confidence levels to increase. Hence, this article focuses on maximizing parameter certainty of physically interpretable dynamic parameters under block-oriented modeling by obtaining accurate values for all the dynamic parameters from a minimum set of estimated parameters. This objective is accomplished by the development of a procedure that identifies equivalent sets of parameters and estimates one parameter for each set. For a seven (7) input, five (5) output, simulated CSTR, its 84 physically based dynamic parameters were accurately determined from 23 estimated parameters that resulted in an increase in confidence level from 50% to 99.9% for a fixed interval width.     

A symptotic robustness of sequential confidence intervals and the connection with influence functions

The robustness of sequential confidence intervals is studied by considering contamination with probability ε of the basic underlying distribution in a so-called gross errors model. Asymptotic theory is considered when d → 0, where the prescribed length of the interval is 2d, and simultaneously ε ? ε(d) → 0. A general theorem, in a distribution free setting, is given which provides expressions for the asymptotic coverage probability and the asymptotic distribution of the stopping variable. The results depend on the rate of ε(d)/d as d → 0 and on the contaminating distribution. If the latter distribution is degenerate, it turns out that the influence functions of the above mentioned two estimators used in the construction of the procedure, appear in the expressions for the asymptotic coverage probability and the asymptotic distribution of the stopping variable respectively. This shows how the sequential procedure inherits the robustness properties of the estimators concerned and how this is quantified. The general theorem is specialized to two procedures for the estimation of the mean of a symmetric distribution. Results of Monte Carlo studies indicate agreement between the asymptotic theory and the actual behavior of the procedures.     

Fixed-width confidence intervals for a function of normal parameters

This paper considers sequantial producers to construct fixed-width confidence intervals for some function θ of mean μ and variance σ² of normal distribution.Consideration is devoted to θ=exp( μ + σ²/2 ) and θ=μ/σ.Nonlinear renewal theory is used to drive asymptotic expansion of expectation of the stopping time and the estimate as the width of confidence interval decreases to zero.An improvement of the coverage probability is also discussed.     

Inference for Autocorrelations in the Possible Presence of a Unit Root

Abstract.  We consider the problem of making inference for the autocorrelations of a time series in the possible presence of a unit root. Even when the underlying series is assumed to be strictly stationary, the robustness against a unit root is a desirable property to ensure good finite-sample coverage in case the series has a near unit root. In addition to discussing a confidence interval for the autocorrelation at a given lag, we also consider a simultaneous confidence band for the first k autocorrelations. We suggest the use of the subsampling method applied to properly studentized statistics, which results in confidence intervals and bands with asymptotically correct coverage probability. An application to practical model selection is given, while a simulation study examines finite-sample performance.     

On Prediction Intervals for Conditionally Heteroscedastic Processes

Kabaila (1999) argues that the standard 1−α prediction intervals for a broad class of conditionally heteroscedastic processes are justified by their possession of what he calls the 'relevance property'. He considers both the case that the parameters of the process are known and that these parameters are unknown. We consider the former case and ask whether these prediction intervals can, alternatively, be deduced from the requirements of both (a) unconditional coverage probability 1−α and (b) minimum unconditional expected length. We show that the answer to this question is no, by presenting a counterexample. This counterexample concerns the standard 95% one-step-ahead prediction interval in the context of a simple Markovian bilinear process.     

On stopping times for fixed-width confidence regions

Analogous to the usual stopping time for fixed-width confidence interval estimation of a real-valued parameter θ(F), a stopping time for fixed-width confidence band estimation of, a statistical functional process R^F is considered. Techniques for showing that the resulting sequential confidence region has the correct asymptotic coverage probability are illustrated in several examples. Sufficient conditions for asymptotic efficiency and normality of the stopping time are given. Some implications of using the stopping time in conjunction with a bootstrap confidence band procedure are investigated in the context of estimating a dstribution function.     

Non-asymptotic confidence estimation of the parameters in stochastic regression models with Gaussian noises

The article considers the problem of estimating linear parameters in stochastic regression models with Gaussian noises, such as an autoregression of the first order, threshold autoregression, and some others. We propose the non-asymptotic technique for constructing a fixed-size confidence region for unknown parameters with any prescribed coverage probability. The construction makes use of some new properties of the sequential point estimates known in the literature. The results of Monte Carlo simulations for AR(1) and TAR(1) models are given. A new version of the sequential point estimate is proposed.     

Weighted confidence interval construction for binomial parameters

Confidence intervals, in general, have become an important aspect of reporting statistical results. In particular, interval estimators for binomial proportions have been studied extensively in recent literature. The large-sample Wald intervals are known to perform poorly, but the Wilson intervals have been shown to perform well in a variety of situations. One criticism is the relative difficulty of computing the Wilson or quadratic intervals in comparison to the Wald intervals. We offer a computational formula for the Wilson intervals that is a weighted estimator of the observed proportion, p, and that based on an uninformative prior, 1/2. This contribution enhances our understanding of the coverage behavior of the Wilson intervals. In addition, we contrast the Wilson intervals with other well-known intervals for the case of zero successes.     

Statistical analysis of masked data in a hybrid system based on copula theory under progressive hybrid censoring

This article considers the reliability analysis of a hybrid system with dependent components, which are linked by a copula function. Based on Type I progressive hybrid censored and masked system lifetime data, we drive some probability results for the hybrid system and then the maximum likelihood estimates as well as the asymptotic confidence intervals and bootstrap confidence intervals of the unknown parameters are obtained. The effects of different dependence structures on the estimates of the parameter and the reliability function are investigated. Finally, Monte Carlo simulations are implemented to compare the performances of the estimates when the components are dependent with those when the components are independent.     

Fixed-width confidence interval of log odds ratio for joint binomial and inverse binomial sampling

This article provides a two-stage procedure to develop a fixed-width confidence interval of log odds ratio in a joint binomial and inverse binomial setting where the stopping rule is obtained by adopting a two-stage procedure on the number of index subjects for the inverse sampling. A purely sequential version of this procedure is also studied. Different asymptotic results associated with the procedures are obtained. The findings are supported by detailed simulation study followed by one data example.     

Two-stage and sequential estimation of parameter N of binomial distribution when p is known

In this article, we propose two-stage and purely sequential procedures to construct bounded width and prescribed proportional closeness confidence intervals for the unknown parameter N of B(N,p) distribution where the parameter p is assumed to be known. The exact distributions of the stopping variables and the estimators of N at stopping are derived for all cases. The coverage probabilities of the proposed interval estimator are computed exactly and are shown to be nearly the same as the prescribed level.     

ROBUSTNESS PROPERTIES OF A SEQUENTIAL CONFIDENCE INTERVAL FOR THE COMMON LOCATION PARAMETER OF A NUMBER OF EXPONENTIAL POPULATIONS

The robustness of a sequential confidence interval for the common location parameter of a number k of exponential populations, with possibly different scale parameters, is studied. Asymptotic theory is considered where the underlying exponential distributions are contaminated in a so-called gross errors model with contamination probabilities ε₁, …, ε _k and where simultaneously d → 0, where d is the length of the confidence interval, and ε₁(d) → 0,…, ε _k (d) → 0 where the contamination probabilities are taken to be functions of d. In the case k = 1, considered first, the procedure considered reduces to the procedure of Mukhopadyay (1974) and asymptotic results are derived for the coverage probability and the sample size, for contamination to the right and to the left. The relationship with the influence functions of the estimators used is pointed out. The theoretical results for general k are derived using many of the results for the case k = 1. Results of Monte Carlo studies indicate agreement between the asymptotic theory and the actual behavior of the procedures and throw light on rates of ε(d) → 0 not covered by the theory.     

A decision theoretic approach to change point estimation for binomial CUSUM control charts

Detecting when the process has changed is a classical problem in sequential analysis and is an important practical issue in statistical process control. This article is concerned about the binomial cumulative sum (CUSUM) control chart, which is extensively applied to industrial process control, health care, public health surveillance, and other fields. For the binomial CUSUM, a maximum likelihood estimator has been proposed to estimate the change point. In our article, following a decision theoretic approach, we develop a new estimator that aims to improve the existing methods. For interval estimation, we propose a parametric bootstrap procedure to construct the confidence set of the change point. We compare our proposed method with the maximum likelihood estimator and Page's last zero estimator in terms of mean squared error by simulations. We find that the proposed method gives more unbiased and robust results than the existing procedures under various parameter designs. We analyze jewelry manufacturing data for illustration.     

基于岭估计的化工过程测量置信水平研究(英文)

Ordinary least squares （OLS） algorithm is widely applied in process measurement, because the sensor model used to estimate unknown parameters can be approximated through multivariate linear model. However, with few or noisy data or multi-collinearity, unbiased OLS leads to large variance. Biased estimators, especially ridge es-timator, have been introduced to improve OLS by trading bias for variance. Ridge estimator is feasible as an esti-mator with smaller variance. At the same confidence level, with additive noise as the normal random variable, the less variance one estimator has, the shorter the two-sided symmetric confidence interval is. However, this finding is limited to the unbiased estimator and few studies analyze and compare the confidence levels between ridge estima-tor and OLS. This paper derives the matrix of ridge parameters under necessary and sufficient conditions based on which ridge estimator is superior to OLS in terms of mean squares error matrix, rather than mean squares error. Then the confidence levels between ridge estimator and OLS are compared under the condition of OLS fixed sym-metric confidence interval, rather than the criteria for evaluating the validity of different unbiased estimators. We conclude that the confidence level of ridge estimator can not be directly compared with that of OLS based on the criteria available for unbiased estimators, which is verified by a simulation and a laboratory scale experiment on a single parameter measurement.     

Gas emission source term estimation with 1-step nonlinear partial swarm optimization-Tikhonov regularization hybrid method

Source term identification is very important for the contaminant gas emission event.Thus,it is necessary to study the source parameter estimation method with high computation efficiency,high estimation accuracy and reasonable confidence interval.Tikhonov regularization method is a potential good tool to identify the source parameters.However,it is invalid for nonlinear inverse problem like gas emission process.2-step nonlinear and linear PSO (partial swarm optimization)-Tikhonov regularization method proposed previously have estimated the emission source parameters successfully.But there are still some problems in computation efficiency and confidence interval.Hence,a new 1-step nonlinear method combined Tikhonov regularization and PSO algorithm with nonlinear forward dispersion model was proposed.First,the method was tested with simulation and experiment cases.The test results showed that 1-step nonlinear hybrid method is able to estimate multiple source parameters with reasonable confidence interval.Then,the estimation performances of different methods were compared with different cases.The estimation values with 1-step nonlinear method were close to that with 2-step nonlinear and linear PSO-Tikhonov regularization method.1-step nonlinear method even performs better than other two methods in some cases,especially for source strength and downwind distance estimation.Compared with 2-step nonlinear method,1-step method has higher computation efficiency.On the other hand,the confidence intervals with the method proposed in this paper seem more reasonable than that with other two methods.Finally,single PSO algorithm was compared with 1-step nonlinear PSO-Tikhonov hybrid regularization method.The results showed that the skill scores of 1-step nonlinear hybrid method to estimate source parameters were close to that of single PSO method and even better in some cases.One more important property of 1-step nonlinear PSO-Tikhonov regularization method is its reasonable confidence interval,which is not obtained by single PSO algorithm.Therefore,1-step nonlinear hybrid regularization method proposed in this paper is a potential good method to estimate contaminant gas emission source term.