Confidence interval for residual mean absolute deviation in regression models

The classic confidence interval for a residual variance is hypersensitive to minor violations of the normality assumption and its robustness does not improve with increasing sample size. An approximate confidence interval for a residual mean absolute deviation is proposed and shown to be robust to moderate violations of the normality assumption with robustness that improves with increasing sample size.     

Sample size needed to get given ratio of endpoints for confidence interval of standard deviation in a normal distribution

Confidence interval (CI) for a standard deviation in a normal distribution, based on pivotal quantity with a Chi-square distribution, is considered. As a measure of CI quality, the ratio of its endpoints is taken. There are given formulas for sample sizes so that this ratio does not exceed a fixed value. Both equally tailed and minimum ratio of endpoint CIs are considered.     

Robust simultaneous confidence interval estimation of principal component loadings

We propose a method that integrates bootstrap into the forward search algorithm in the construction of robust confidence intervals for elements of the eigenvectors of the correlation matrix in the presence of outliers. Coverage probability of the bootstrap simultaneous confidence intervals was compared to the coverage probabilities of regular asymptotic confidence region and asymptotic confidence region based on the minimum covariance determinant (MCD) approach through a simulation study. The method produced more stable coverage probabilities for datasets with or without outliers and across several sample sizes compared to approaches based on asymptotic confidence regions.     

Robust interval forecasting algorithm based on a probabilistic cluster model

For substantiation of managerial decisions the forecasting results of dynamic indicators are used. Therefore, forecasting accuracy of these indicators must be acceptable. Consequently, forecasting algorithms are constantly improved to get the acceptable accuracy. This paper considers a variant of the method of forecasting binary outcomes. This method allows prediction of whether or not a future value of the indicator exceeds a predetermined value. This method ‘interval forecasting’ was named. In this paper a robust interval forecasting algorithm based on a probabilistic cluster model is proposed. The algorithm’s accuracy was compared with an algorithm based on logistic regression. The indicators with different statistical properties were chosen. The obtained results have shown the accuracy of both the algorithms is approximately similar in most cases. However, the cases when the algorithm based on logistic regression demonstrated unacceptable accuracy, unlike the presented algorithm have been identified. Thus, this new algorithm is more accurate.     

Robust confidence bounds for extreme upper quantiles

Four related methods are discussed for obtaining robust confidence bounds for extreme upper quantiles of the unknown distribution of a positive random variable. These methods are designed to work when the upper tail of the distribution is neither too heavy nor too light in comparison to the exponential distribution. An extensive simulated study is described, which compares the performance of nominal 90% upper confidence bounds corresponding to the four methods over a wide variety of distributions having light to heavy upper tails, ranging from a half-normal distribution to a heavy-tailed lognormal distribution.     

A nonparametric confidence interval for slope based on spearman's rho

An algorithm is presented for computing an exact nonparametric interval estimate of the slope parameter in a simple linear regression model. The confidence interval is obtained by inverting the hypothesis test for slope that uses Spearman's rho. This method is compared to an exact procedure based on Kendall's tau. The Spearman rho procedure will generally give exact levels of confidence closer to desired levels, especially in small samples. Monte carlo results comparing these two methods with the parametric procedure are given     

Testing for lack of fit in regression - a review

Assessment of the adequacy of a proposed linear regression model is necessarily subjective. However, the following three criteria may warrant investigation whether the distributional assumptions for the stochastic portion of the model are satisfied, whether the predictive capability of the model is satisfactory, and whether the deterministic portion of the model is adejuate in a statistical sense. The first two criteria have been reviewed in the literature to some extent. This paper reviews statistical tests and procedures which aid the experimenter in deterrmining lack of fit or functional misspecification associated with the deterministic portion of a proposed linear regression model.     

A generalized score confidence interval for a binomial proportion

Constructing a confidence interval for a binomial proportion is one of the most basic problems in statistics. The score interval as well as the Wilson interval with some modified forms have been broadly investigated and suggested by many statisticians. In this paper, a generalized score interval CI_G(a) is proposed by replacing the coefficient 1/4 in the score interval with parameter a. Based on analyzing and comparing various confidence intervals, we recommend the generalized score interval CI_G(0.3) for the nominal confidence levels 0.90, 0.95 and 0.99, which improves the spike phenomenon of the score interval and behaves better and computes more easily than most of other approximate intervals such as the Agresti-Coull interval and the Jeffreys interval to estimate a binomial proportion.     

Robust confidence regions for multinomial probabilities

Generally, confidence regions for the probabilities of a multinomial population are constructed based on the Pearson χ² statistic. Morales et al. (Bootstrap confidence regions in multinomial sampling. Appl Math Comput. 2004;155:295–315) considered the bootstrap and asymptotic confidence regions based on a broader family of test statistics known as power-divergence test statistics. In this study, we extend their work and propose penalized power-divergence test statistics-based confidence regions. We only consider small sample sizes where asymptotic properties fail and alternative methods are needed. Both bootstrap and asymptotic confidence regions are constructed. We consider the percentile and the bias corrected and accelerated bootstrap confidence regions. The latter confidence region has not been studied previously for the power-divergence statistics much less for the penalized ones. Designed simulation studies are carried out to calculate average coverage probabilities. Mean absolute deviation between actual and nominal coverage probabilities is used to compare the proposed confidence regions.     

Confidence sets in a linear regression model for interval data

The construction of confidence sets for the parameters of a flexible simple linear regression model for interval-valued random sets is addressed. For that purpose, the asymptotic distribution of the least-squares estimators is analyzed. A simulation study is conducted to investigate the performance of those confidence sets. In particular, the empirical coverages are examined for various interval linear models. The applicability of the procedure is illustrated by means of a real-life case study.     

A note on the satterthwaite confidence interval for a variance

When using a Satterthwaite chi-squared approximation, it is generally thought that the approximation is satisfactory when it is applied to a positive linear combination of mean squares. In this note, we describe how the Williams - Tukey idea for getting a confidence interval for the among groups variance in a random one-way model can be incorporated into Satterthwaite’s procedure for getting a confidence interval for a variance. This adjusted Satterthwaite procedure insures that his chi-squared approximation is always applied to positive linear combinations of mean squares. A small simulation is included which suggests that the adjustment to the Satterthwaite procedure is effective.     

Bootstrap methods for bias correction and confidence interval estimation for nonlinear quantile regression of longitudinal data

This paper examines the use of bootstrapping for bias correction and calculation of confidence intervals (CIs) for a weighted nonlinear quantile regression estimator adjusted to the case of longitudinal data. Different weights and types of CIs are used and compared by computer simulation using a logistic growth function and error terms following an AR(1) model. The results indicate that bias correction reduces the bias of a point estimator but fails for CI calculations. A bootstrap percentile method and a normal approximation method perform well for two weights when used without bias correction. Taking both coverage and lengths of CIs into consideration, a non-bias-corrected percentile method with an unweighted estimator performs best.     

Robust confidence intervals for trend estimation in meta-analysis with publication bias

Confidence interval (CI) is very useful for trend estimation in meta-analysis. It provides a type of interval estimate of the regression slope as well as an indicator of the reliability of the estimate. Thus a precise calculation of confidence interval at an expected level is important. It is always difficult to explicitly quantify the CIs when there is publication bias in meta-analysis. Various CIs have been proposed, including the most widely used DerSimonian–Laird CI and the recently proposed Henmi–Copas CI. The latter provides a robust solution when there are non-ignorable missing data due to publication bias. In this paper we extended the idea into meta-analysis for trend estimation. We applied the method in different scenarios and showed that this type of CI is more robust than the others.     

Asymptotic normality of the lengths of a class of nonparametric confidence intervals for a regression parameter

In the linear regression model, the asymptotic distributions of certain functions of confidence bounds of a class of confidence intervals for the regression parameter arc investigated. The class of confidence intervals we consider in this paper are based on the usual linear rank statistics (signed as well as unsigned). Under suitable assumptions, if the confidence intervals are based on the signed linear rank statistics, it is established that the lengths, properly normalized, of the confidence intervals converge in law to the standard normal distributions; if the confidence intervals arc based on the unsigned linear rank statistics, it is then proved that a linear function of the confidence bounds converges in law to a normal distribution.     

Robust generalized confidence intervals

Most interval estimates are derived from computable conditional distributions conditional on the data. In this article, we call the random variables having such conditional distributions confidence distribution variables and define their finite-sample breakdown values. Based on this, the definition of breakdown value of confidence intervals is introduced, which covers the breakdowns in both the coverage probability and interval length. High-breakdown confidence intervals are constructed by the structural method in location-scale families. Simulation results are presented to compare the traditional confidence intervals and their robust analogues.     

A small sample confidence interval for autoregressive parameters

This paper is concerned with interval estimation of an autoregressive parameter when the parameter space allows for magnitudes outside the unit interval. In this case, intervals based on the least-squares estimator tend to require a high level of numerical computation and can be unreliable for small sample sizes. Intervals based on the asymptotic distribution of instrumental variable estimators provide an alternative. If the instrument is taken to be the sign function, the interval is centered at the Cauchy estimator and a large sample interval can be created by estimating the standard error of this estimator. The interval proposed in this paper avoids estimating this standard error and results in a small sample improvement in coverage probability. In fact, small sample coverage is exact when the innovations come from a normal distribution.     

Bias of maximum likelihood estimators of the response standard deviation in regression

The bias of maximum likelihood estimators of the standard deviation of the response in location/scale regression models is considered. Results are obtained for a very wide family of densities for the response variable. These are used to propose point estimators with improved mean square error properties and to demonstrate the importance of bias correction in statistical inference when samples are moderately small.     

Robust confidence intervals for contrasts based upon a likelihood ratio test

This paper describes how to compute robust confidence intervals for differences of the effects using the likelihood ratio testF _M in the two-way analysis of variance. The probability for the α-error and the average length of the confidence intervals withF _m and the quadratic formQ _M are investigated and compared with the classical confidence intervals fort-distributed and lognormal errors. We also give a warning of building confidence intervals withF _M andQ _M in the presence of heterogeneous scale parameters, because these tests which do not regard heteroscedasticity are then much too liberal.