Inferring the Predictability Induced by a Persistent Regressor in a Predictive Threshold Model

We develop tests for detecting possibly episodic predictability induced by a persistent predictor. Our framework is that of a predictive regression model with threshold effects and our goal is to develop operational and easily implementable inferences when one does not wish to impose à priori restrictions on the parameters of the model other than the slopes corresponding to the persistent predictor. Differently put our tests for the null hypothesis of no predictability against threshold predictability remain valid without the need to know whether the remaining parameters of the model are characterized by threshold effects or not (e.g., shifting versus nonshifting intercepts). One interesting feature of our setting is that our test statistics remain unaffected by whether some nuisance parameters are identified or not. We subsequently apply our methodology to the predictability of aggregate stock returns with valuation ratios and document a robust countercyclicality in the ability of some valuation ratios to predict returns in addition to highlighting a strong sensitivity of predictability based results to the time period under consideration.     

On Distributions of Exceedances Based on Generalized Order Statistics

Distributions of exceedance statistics based on generalized order statistics are obtained for a random threshold model. The ordinary order statistics, progressively Type-II right censored order statistics and record values are considered as special cases. The results obtained in the article imply many results on exceedance statistics for the variety of models of ordered random variables.     

Robust multivariate Bayesian forecasting under functional distortions in the -metric

This paper investigates robustness of multivariate forecasting in the Bayesian framework. The minimax approach is used to construct robust statistical procedures under deviations from hypothetical assumptions. The deviations are defined as functional distortions using the χ²-pseudo-metric. Two cases of deviations are considered: distortions of parameter distribution and distortions of joint distribution of observations and parameters. Explicit forms for the guaranteed upper risk functional are obtained and integral equations for robust prediction statistics are given for both cases.     

Inference in a two-parameter generalized order statistics model

As a submodel of generalized order statistics with two unknown model parameters, m-generalized order statistics may serve as a simple model for ordered quantities in a given application. It is shown that the joint distribution of m-generalized order statistics has a representation as a regular exponential family in the model parameters, as it is the case for the comprising model. Utilizing this finding, a minimal sufficient and complete statistic is obtained along with distributional properties. Joint maximum likelihood estimation of the parameters is considered, and strong consistency and asymptotic efficiency of the estimator are established. A test is provided to decide whether a restriction to the submodel is reasonable.     

Selecting between gamma and Weibull distributions—an approach based on predictions of order statistics

A method based on the prediction of order statistics is proposed to select the underlying parent distribution. A cross-validatory predictor and the best linear unbiased predictor are considered in choosing between gamma and Weibull models when shape parameters are only known to lie within a range. The proposed approach is evaluated using a large-scale Monte Carlo study. The results clearly show that the cross-validatory predictor performs well as a robust procedure in selecting between probability densities. Two well-known data sets are used to illustrate the procedure.     

A unified approach to rank tests for mixed models

The nonparametric version of the classical mixed model is considered and the common hypotheses of (parametric) main effects and interactions are reformulated in a nonparametric setup. To test these nonparametric hypotheses, the asymptotic distributions of quadratic forms of rank statistics are derived in a general framework which enables the derivation of the statistics for the nonparametric hypotheses of the fixed treatment effects and interactions in an arbitrary mixed model. The procedures given here are not restricted to semiparametric models or models with additive effects. Moreover, they are robust to outliers since only the ranks of the observations are needed. They are also applicable to pure ordinal data and since no continuity of the distribution functions is assumed, they can also be applied to data with ties. Some approximations for small sample sizes are suggested and analyzed in a simulation study. The application of the statistics and the interpretation of the results is demonstrated in several worked-out examples where some data sets given in the literature are re-analyzed.     

Modeling the presence of immunes by using the exponentiated-Weibull model

In this paper the exponentiated-Weibull model is modified to model the possibility that long-term survivors are present in the data. The modification leads to an exponentiated-Weibull mixture model which encompasses as special cases the exponential and Weibull mixture models typically used to model such data. Inference for the model parameters is considered via maximum likelihood and also via Bayesian inference by using Markov chain Monte Carlo simulation. Model comparison is considered by using likelihood ratio statistics and also the pseudo Bayes factor, which can be computed by using the generated samples. An example of a data set is considered for which the exponentiated-Weibull mixture model presents a better fit than the Weibull mixture model. Results of simulation studies are also reported, which show that the likelihood ratio statistics seems to be somewhat deficient for small and moderate sample sizes.     

A new look at the inverse Gaussian distribution with applications to insurance and economic data

Insurance and economic data are often positive, and we need to take into account this peculiarity in choosing a statistical model for their distribution. An example is the inverse Gaussian (IG), which is one of the most famous and considered distributions with positive support. With the aim of increasing the use of the IG distribution on insurance and economic data, we propose a convenient mode-based parameterization yielding the reparametrized IG (rIG) distribution; it allows/simplifies the use of the IG distribution in various branches of statistics, and we give some examples. In nonparametric statistics, we define a smoother based on rIG kernels. By construction, the estimator is well-defined and does not allocate probability mass to unrealistic negative values. We adopt likelihood cross-validation to select the smoothing parameter. In robust statistics, we propose the contaminated IG distribution, a heavy-tailed generalization of the rIG distribution to accommodate mild outliers. Finally, for model-based clustering and semiparametric density estimation, we present finite mixtures of rIG distributions. We use the EM algorithm to obtain maximum likelihood estimates of the parameters of the mixture and contaminated models. We use insurance data about bodily injury claims, and economic data about incomes of Italian households, to illustrate the models.     

Empirical phi-divergence test statistics for testing simple and composite null hypotheses

The main purpose of this paper is to introduce first a new family of empirical test statistics for testing a simple null hypothesis when the vector of parameters of interest is defined through a specific set of unbiased estimating functions. This family of test statistics is based on a distance between two probability vectors, with the first probability vector obtained by maximizing the empirical likelihood (EL) on the vector of parameters, and the second vector defined from the fixed vector of parameters under the simple null hypothesis. The distance considered for this purpose is the phi-divergence measure. The asymptotic distribution is then derived for this family of test statistics. The proposed methodology is illustrated through the well-known data of Newcomb's measurements on the passage time for light. A simulation study is carried out to compare its performance with that of the EL ratio test when confidence intervals are constructed based on the respective statistics for small sample sizes. The results suggest that the ‘empirical modified likelihood ratio test statistic’ provides a competitive alternative to the EL ratio test statistic, and is also more robust than the EL ratio test statistic in the presence of contamination in the data. Finally, we propose empirical phi-divergence test statistics for testing a composite null hypothesis and present some asymptotic as well as simulation results for evaluating the performance of these test procedures.     

Robust Tests in Semiparametric Partly Linear Models

Abstract.  This paper focuses on the problem of testing the null hypothesis that the regression parameter equals a fixed value under a semiparametric partly linear regression model by using a three-step robust estimate for the regression parameter and the regression function. Two families of tests statistics are considered and their asymptotic distributions are studied under the null hypothesis and under contiguous alternatives. A Monte Carlo study is performed to compare the finite sample behaviour of the proposed tests with the classical one.     

On runs of length exceeding a threshold: normal approximation

Run statistics denoting number of runs and sum of run lengths are defined on binary sequences and their asymptotic normality is established by a simple unified way for Bernoulli sequences. All the considered statistics share a common feature; they refer to runs of length exceeding a specific length (a threshold). Asymptotic results of associated statistics denoting run lengths and waiting times are derived as well. Specific probabilities of the examined statistics are used in applications in the fields of system reliability and molecular biology. The study is illustrated by an extensive numerical experimentation.     

A review of model selection procedures relevant to the weibull distribution

A number of goodness-of-fit and model selection procedures related to the Weibull distribution are reviewed. These procedures include probability plotting, correlation type goodness-of-fit tests, and chi-square goodness-of-fit tests. Also the Kolmogorow-Smirniv, Kuiper, and Cramer-Von Mises test statistics for completely specified hypothesis based on censored data are reviewed, and these test statistics based on complete samples for the unspecified parameters case are considered. Goodness-of-fit tests based on sample spacings, and a goodness-of-fit test for the Weibull process, is also discussed.

Model selection procedures for selecting between a Weibull and gamma model, a Weibull and lognormal model, and for selecting from among all three models are considered. Also tests of exponential versus Weibull and Weibull versus generalized gamma are mentioned.     

Robust slippage rank tests for k location parameters in the presence of gross errors

A robust slippage test problem of k location parameters in the presence of gross errors is formulated from the point of view of Huber's robust test theory. Under an asymptotic model of the robust slippage test problem an asymptotic level α slippage rank test based on k linear rank statistics is constructed by applying majorization methods and its asymptotic minimum power is evaluated by applying weak majorization methods. It is also shown that the slippage rank test is asymptotically unbiased.     

Robust estimation for boundary correction in wavelet regression

For boundary problems present in wavelet regression, two common methods are usually considered: polynomial wavelet regression (PWR) and hybrid local polynomial wavelet regression (LPWR). Normality assumption played a key role for making such choices for the order of the low-order polynomial, the wavelet thresholding value and other calculations involved in LPWR. However, in practice, the normality assumption may not be valid. In this paper, for PWR, we propose three automatic robust methods based on: MM-estimator, bootstrap and robust threshold procedure. For LPWR, the use of a robust local polynomial (RLP) estimator with a robust threshold procedure has been investigated. The proposed methods do not require any knowledge of noise distribution, are easy to implement and achieve high performances when only a small amount of data is in hand. A simulation study is conducted to assess the numerical performance of the proposed methods.     

Interval estimation of the population mean under model uncertainty: Robust versus empirical statistics

With reference to the problem of interval estimation of a population mean under model uncertainty, we compare approaches based on robust and empirical statistics via expected lengths of the associated confidence intervals. An explicit expression for confidence intervals arising from a general class of robust statistics is worked out and this is employed to obtain a higher order asymptotic formula for the expected lengths of such intervals. Comparative theoretical results, as well as a simulation study, are then presented.     

A new three-parameter extension of the log-logistic distribution with applications to survival data

In this article, a new three-parameter extension of the two-parameter log-logistic distribution is introduced. Several distributional properties such as moment-generating function, quantile function, mean residual lifetime, the Renyi and Shanon entropies, and order statistics are considered. The estimation of the model parameters for complete and right-censored cases is investigated competently by maximum likelihood estimation (MLE). A simulation study is conducted to show that these MLEs are consistent in moderate samples. Two real datasets are considered; one is a right-censored data to show that the proposed model has a superior performance over several existing popular models.     

Hausman-type tests for individual and time effects in the panel regression model with incomplete data

By comparing estimators of the variance of idiosyncratic error at different robust levels, two Hausman-type test statistics are respectively constructed for the existence of individual and time effects in the panel regression model with incomplete data. The resultant test statistics have several desired properties. Firstly, they are robust to the presence of one effect when the other is tested. Secondly, they are immune to the non-normal distribution of the disturbances since the distributional conditions are not needed in the construction of the statistics. Thirdly, they have more robust performances than the main competitors in the literature when the covariates are correlated with the effects. Additionally, they are very simple and have no heavy computational burden. Joint tests for both of the two effects are also discussed. Monte Carlo evidence shows that the proposed tests have desired finite sample properties, and a real data analysis gives further support.     

Bounded Estimation in the Presence of Nuisance Parameters

The aim of this paper is to extend in a natural fashion the results on the treatment of nuisance parameters from the profile likelihood theory to the field of robust statistics. Similarly to what happens when there are no nuisance parameters, the attempt is to derive a bounded estimating function for a parameter of interest in the presence of nuisance parameters. The proposed method is based on a classical truncation argument of the theory of robustness applied to a generalized profile score function. By means of comparative studies, we show that this robust procedure for inference in the presence of a nuisance parameter can be used successfully in a parametric setting.     

Robust testing with generalized partial linear models for longitudinal data

By approximating the nonparametric component using a regression spline in generalized partial linear models (GPLM), robust generalized estimating equations (GEE), involving bounded score function and leverage-based weighting function, can be used to estimate the regression parameters in GPLM robustly for longitudinal data or clustered data. In this paper, score test statistics are proposed for testing the regression parameters with robustness, and their asymptotic distributions under the null hypothesis and a class of local alternative hypotheses are studied. The proposed score tests reply on the estimation of a smaller model without the testing parameters involved, and perform well in the simulation studies and real data analysis conducted in this paper.