Refining binomial confidence intervals

A method for refining an equivariant binomial confidence procedure is presented which, when applied to an existing procedure, produces a new set of equivariant intervals that are uniformly superior. The family of procedures generated from this method constitute a complete class within the class of all equivariant procedures. In certain cases it is shown that this class is also minimal complete. Also, an optimally property, monotone minimaxity, is investigated, and monotone minimax procedures are constructed.     

Estimation of regression parameters in a semiparametric transformation model

A semiparametric approach to model skewed/heteroscedastic regression data is discussed. We work with a semiparametric transform-both-sides regression model, which contains a parametric regression function and a nonparametric transformation. This model is adequate when the relationship between the median response and the explanatory variable has been specified by a theoretical result or a previous empirical study. The transform-both-sides model with a parametric transformation has been studied extensively and applied successfully to a number data sets. Allowing a nonparametric transformation function increases the flexibility of the model. In this article, we estimate the nonparametric transformation function by the conditional kernel density approach developed by Wang and Ruppert (1995), and then use a pseudo-maximum likelihood estimator to estimate the regression parameters. This estimate of the regression parameters has not been studied previously. In this article, the asymptotic distribution of this pseudo-MLE is derived. We also show that when σ, the standard deviation of the error, goes to zero (small σ asymptotics), this estimator is adaptive. Adaptive means that the regression parameters are estimated as precisely as when the transformation is known exactly. A similar result holds in the parametric approaches of Carroll and Ruppert (1984) and Ruppert and Aldershof (1989). Simulated and real examples are provided to illustrate the performance of the proposed estimator for finite sample size.     

A score test for monotonic trend in hazard rates for grouped survival data in stratified populations

Models for monotone trends in hazard rates for grouped survival data in stratified populations are introduced, and simple closed form score statistics for testing the significance of these trends are presented. The test statistics for some of the models understudy are shown to be independent of the assumed form of the function which relates the hazard rates to the sets of monotone scores assigned to the time intervals. The procedure is applied to test monotone trends in the recovery rates of erythematous response among skin cancer patients and controls that have been irradiated with a ultraviolent challenge.     

Monotone Nonparametric Regression and Confidence Intervals

Several variations of monotone nonparametric regression have been developed over the past 30 years. One approach is to first apply nonparametric regression to data and then monotone smooth the initial estimates to “iron out” violations to the assumed order. Here, such estimators are considered, where local polynomial regression is first used, followed by either least squares isotonic regression or a monotone method using simple averages. The primary focus of this work is to evaluate different types of confidence intervals for these monotone nonparametric regression estimators through Monte Carlo simulation. Most of the confidence intervals use bootstrap or jackknife procedures. Estimation of a response variable as a function of two continuous predictor variables is considered, where the estimation is performed at the observed values of the predictors (instead of on a grid). The methods are then applied to data involving subjects that worked at plants that use beryllium metal who have developed chronic beryllium disease.     

Monotone fitting for developmental variables

In order to study developmental variables, for example, neuromotor development of children and adolescents, monotone fitting is typically needed. Most methods, to estimate a monotone regression function non-parametrically, however, are not straightforward to implement, a difficult issue being the choice of smoothing parameters. In this paper, a convenient implementation of the monotone B-spline estimates of Ramsay [Monotone regression splines in action (with discussion), Stat. Sci. 3 (1988), pp. 425–461] and Kelly and Rice [Montone smoothing with application to dose-response curves and the assessment of synergism, Biometrics 46 (1990), pp. 1071–1085] is proposed and applied to neuromotor data. Knots are selected adaptively using ideas found in Friedman and Silverman [Flexible parsimonous smoothing and additive modelling (with discussion), Technometrics 31 (1989), pp. 3–39] yielding a flexible algorithm to automatically and accurately estimate a monotone regression function. Using splines also simultaneously allows to include other aspects in the estimation problem, such as modeling a constant difference between two groups or a known jump in the regression function. Finally, an estimate which is not only monotone but also has a ‘levelling-off’ (i.e. becomes constant after some point) is derived. This is useful when the developmental variable is known to attain a maximum/minimum within the interval of observation.     

On the variance estimate of a wilcoxon-mann whitney statistic for group ordered data

Two consistent estimators for the non-null variance of Wil-coxon-Mann-Whitney’s statistic applied to grouped ordered data, are considered. The first is based on U-statistics and the sec-ond is obtained by the Delta method. Some examples are given to demonstrate the extent of error when using a null variance esti-mate for constructing confidence intervals. It appears that the two consistent estimates are very close, but may both be disting-uishably larger or smaller than the null variance estimate.     

Simultaneous Confidence Intervals for the Population Cell Means,for Two-by-Two Factorial Data,that Utilize Uncertain Prior Information

Consider a two-by-two factorial experiment with more than one replicate. Suppose that we have uncertain prior information that the two-factor interaction is zero. We describe new simultaneous frequentist confidence intervals for the four population cell means, with simultaneous confidence coefficient 1 ? α, that utilize this prior information in the following sense. These simultaneous confidence intervals define a cube with expected volume that (a) is relatively small when the two-factor interaction is zero and (b) has maximum value that is not too large. Also, these intervals coincide with the standard simultaneous confidence intervals obtained by Tukey’s method, with simultaneous confidence coefficient 1 ? α, when the data strongly contradict the prior information that the two-factor interaction is zero. We illustrate the application of these new simultaneous confidence intervals to a real data set.     

A Measure for Peakedness

The kurtosis has often been used to characterize the peakedness of a density. Unfortunately, the kurtosis does not exist for all densities. This article defines a measure of peakedness that exists for all symmetric unimodal densities.     

Simultaneous Confidence Intervals for Monotone Contrasts of Normal Means

The one-way ANOVA model is considered. The variances are assumed to be either equal but unknown or unequal and unknown. Two-stage procedures for generating simultaneous confidence intervals (SCI) for the class of monotone contrasts of the means are presented. The intervals are of fixed length and independent of the unknown variances.     

IG-symmetry and R-symmetry: Interrelations and applications to the inverse Gaussian theory

The two parameter inverse Gaussian (IG) distribution is often more appropriate and convenient for modelling and analysis of nonnegative right skewed data than the better known and now ubiquitous Gaussian distribution. Its convenience stems from its analytic simplicity and the striking similarities of its methodologies with those employed with the normal theory models. These, known as the G–IG analogies, include the concepts and measures of IG-symmetry, IG-skewness and IG-kurtosis, the IG-analogues of the corresponding classical notions and measures. The new IG-associated entities, although well defined and mathematically transparent, are intuitively and conceptually opaque. In this paper, we first elaborate the importance of the IG distribution and of the G–IG analogies. Then we consider the IG-related root-reciprocal IG (RRIG) distribution and introduce a physically transparent, conceptually clear notion of reciprocal symmetry (R-symmetry) and use it to explain the IG-symmetry. We study the moments and mixture properties of the R-symmetric distributions and the relationship of R-symmetry with IG-symmetry and note that RRIG distribution provides a link, in addition to Tweedie's Laplace transform link, between the Gaussian and inverse Gaussian distributions. We also give a structural characterization of the unimodal R-symmetric distributions. This work further expands the long list of G–IG analogies. Several applications including product convolution, monotonicity of power functions, peakedness and monotone limit theorems of R-symmetry are outlined.     

Simultaneous confidence sets and confidence intervals for multiple ratios

This paper deals with the problem of simultaneously estimating multiple ratios. In the simplest case of only one ratio parameter, Fieller's theorem (J. Roy. Statist. Soc. Ser. B 16 (1954) 175) provides a confidence interval for the single ratio. For multiple ratios, there is no method available to construct simultaneous confidence intervals that exactly satisfy a given familywise confidence level. Many of the methods in use are conservative since they are based on probability inequalities. In this paper, first we consider exact simultaneous confidence sets based on the multivariate t-distribution. Two approaches of determining the exact simultaneous confidence sets are outlined. Second, approximate simultaneous confidence intervals based on the multivariate t-distribution with estimated correlation matrix and a resampling approach are discussed. The methods are applied to ratios of linear combinations of the means in the one-way layout and ratios of parameter combinations in the general linear model. Extensive Monte Carlo simulation is carried out to compare the performance of the various methods with respect to the stability of the estimated critical points and of the coverage probabilities.     

Simultaneous confidence intervals for comparing several inverse Gaussian means under heteroscedasticity

Recently, Zhang [Simultaneous confidence intervals for several inverse Gaussian populations. Stat Probab Lett. 2014;92:125–131] proposed simultaneous pairwise confidence intervals (SPCIs) based on the fiducial generalized pivotal quantity concept to make inferences about the inverse Gaussian means under heteroscedasticity. In this paper, we propose three new methods for constructing SPCIs to make inferences on the means of several inverse Gaussian distributions when scale parameters and sample sizes are unequal. One of the methods results in a set of classic SPCIs (in the sense that it is not simulation-based inference) and the two others are based on a parametric bootstrap approach. The advantages of our proposed methods over Zhang’s (2014) method are: (i) the simulation results show that the coverage probability of the proposed parametric bootstrap approaches is fairly close to the nominal confidence coefficient while the coverage probability of Zhang’s method is smaller than the nominal confidence coefficient when the number of groups and the variance of groups are large and (ii) the proposed set of classic SPCIs is conservative in contrast to Zhang’s method.     

Some results on simultaneous confidence intervals for monotone contrasts in one-way anova model

The one-way ANOVA model with common variance is considered. Simultaneous confidence Intervals (SCI) for monotone contrasts in the means are derived and compared to alternative intervals gene¬rated by Williams (1977)     

Peakedness for weighted sums of symmetric random variables

In this paper, we provide sufficient conditions on weighted coefficients under which the peakedness comparison between weighted sums of independent random variables can be carried out. These results extend and enrich the existing peakedness results in the literature including those presented by Proschan (1965) and Ma (1998).     

On the linear combinations of the means of inverse gaussian distributions

Consider the two parameter Inverse Gaussian distribution with mean μ and scale parameter λ.

Suppose one is interested in testing a problem on a linear combination for the means of Inverse Gaussian distributions. For this problem a test and confidence intervals are proposed when: (1) λ’s are known and; (2) λ’s are unknown.

Finally an application of the procedures is illustrated with a data set of failure times of high-speed turbine bearings.     

Best monotone M‐estimators

The author shows how to find M‐estimators of location whose generating function is monotone and which are optimal or close to optimal. It is easy to identify a consistent sequence of estimators in this class. In addition, it contains simple and efficient approximations in cases where the likelihood function is difficult to obtain. In some neighbourhoods of the normal distribution, the loss of efficiency due to the approximation is quite small. Optimal monotone M‐estimators can also be determined in cases when the underlying distribution is known only up to a certain neighbourhood. The author considers the e‐contamination model and an extension thereof that allows the distributions to be arbitrary outside compact intervals. His results also have implications for distributions with monotone score functions. The author illustrates his methodology using Student and stable distributions.     

Empirical likelihood‐based inferences for a low income proportion

Low income proportion is an important index in comparisons of poverty in countries around the world. The stability of a society depends heavily on this index. An accurate and reliable estimation of this index plays an important role for government's economic policies. In this paper, the authors study empirical likelihood‐based inferences for a low income proportion under the simple random sampling and stratified random sampling designs. It is shown that the limiting distributions of the empirical likelihood ratios for the low income proportion are the scaled chi‐square distributions. The authors propose various empirical likelihood‐based confidence intervals for the low income proportion. Extensive simulation studies are conducted to evaluate the relative performance of the normal approximation‐based interval, bootstrap‐based intervals, and the empirical likelihood‐based intervals. The proposed methods are also applied to analyzing a real economic survey income dataset. The Canadian Journal of Statistics 39: 1–16; 2011 ©2011 Statistical Society of Canada     

Copula, marginal distributions and model selection: a Bayesian note

Copula functions and marginal distributions are combined to produce multivariate distributions. We show advantages of estimating all parameters of these models using the Bayesian approach, which can be done with standard Markov chain Monte Carlo algorithms. Deviance-based model selection criteria are also discussed when applied to copula models since they are invariant under monotone increasing transformations of the marginals. We focus on the deviance information criterion. The joint estimation takes into account all dependence structure of the parameters’ posterior distributions in our chosen model selection criteria. Two Monte Carlo studies are conducted to show that model identification improves when the model parameters are jointly estimated. We study the Bayesian estimation of all unknown quantities at once considering bivariate copula functions and three known marginal distributions.     

Comparisons Between Local Linear Estimator and Kernel Smooth Estimator for a Smooth Distribution Based on MSE Under Right Censoring

We describe a method for estimating the coefficients in a logistic regression model when the predictors are subject to measurement error and an instrumental variable is present. The proposed method is based upon the theory of factor scores taken from factor analysis. Two versions of the proposed method, a simple one and an extended one, are compared to the methods referred to by Carrol, Ruppert and Stefanski (1995) through simulation studies. Our conclusion is that the simple version performs as well as the methods from Carrol et al. (1995), and the extended version performs betterwith respect to MSE, due to a reduction of bias.     

Fitting the additive model by recursion on dimension

We consider estimation for the homoscedastic additive model for multiple regression. A recursion is proposed in Opsomer (1999), and independently by the authors, for obtaining the estimators that solve the normal equations given by Hastie and Tibshirani (1990). The recursion can be exploited to obtain the asymptotic bias and variance expressions of the estimators for any p > 2 (Opsomer 1999) using repeated application of Opsomer and Ruppert (1997). Opsomer and Ruppert (1997) provide asymptotic bias and variance for the estimators when p = 2. Opsomer (1999) also uses the recursion to provide sufficient conditions for convergence of the backfitting algorithm to a unique solution of the normal equations. However, since explicit expressions for the solution to the normal equations are not given, he states, “The lemma does not provide a practical way of evaluating the existence and uniqueness of the backfitting estimators … ”. In this paper, explicit expressions for the estimators are derived. The explicit solution requires inverses of n × n matrices to solve the np × np system of normal equations. These matrix inverses are feasible to implement for moderate sample sizes and can be used in place of the backfitting algorithm.