Nonparametric comparison of curves with dependent errors

Suppose that data {(x _l,i,n , y _l,i,n ): l?=?1, …, k; i?=?1, …, n} are observed from the regression models: Y _l,i,n ?=?m _l (x _l,i,n )?+?? _l,i,n , l?=?1, …, k, where the regression functions {m _l } _l=1 ^k are unknown and the random errors {? _l,i,n } are dependent, following an MA(∞) structure. A new test is proposed for testing the hypothesis H ₀: m ₁?=?·?·?·?=?m _k , without assuming that {m _l } _l=1 ^k are in a parametric family. The criterion of the test derives from a Crámer-von-Mises-type functional based on different distances between {[mcirc]} _l and {[mcirc]} _s , l?≠?s, l, s?=?1, …, k, where {[mcirc] _l } _l=1 ^k are nonparametric Gasser–Müller estimators of {m _l } _l=1 ^k . A generalization of the test to the case of unequal design points, with different sample sizes {n _l } _l=1 ^k and different design densities {f _l } _l=1 ^k , is also considered. The asymptotic normality of the test statistic is obtained under general conditions. Finally, a simulation study and an analysis with real data show a good behavior of the proposed test.     

Parameter estimation in nonlinear regression models 1

Fisher information contained in record values, inter-record times and their concomitants from a sample of fixed size is derived in general and explicit expressions are deduced for some specific known bivariate classes of distributions. A comparison between fixed sampling and inverse sampling schemes with equal number of records and concomitants is also carried out. We also consider parameter estimation based on bivariate records and a small simulation study is done.     

A robust test for asymptotic independence of bivariate extremes

We show that the existing tests for asymptotic independence are sensitive to outliers. A robust test is proposed. The new test is made stable under contamination through a shrinkage scheme. Simulations show that the new test performs well in the presence of contaminated data while maintaining good properties when there is no contamination. An application to real data shows the added value of our new robust approach.     

Reliability estimation in stress–strength models: an MCMC approach

In this paper, we consider the estimation of the stress–strength parameter R=P(Y<X) when X and Y are independent and both are modified Weibull distributions with the common two shape parameters but different scale parameters. The Markov Chain Monte Carlo sampling method is used for posterior inference of the reliability of the stress–strength model. The maximum-likelihood estimator of R and its asymptotic distribution are obtained. Based on the asymptotic distribution, the confidence interval of R can be obtained using the delta method. We also propose a bootstrap confidence interval of R. The Bayesian estimators with balanced loss function, using informative and non-informative priors, are derived. Different methods and the corresponding confidence intervals are compared using Monte Carlo simulations.     

Non-linear models 1

This is the first application of a new method for testing stationary random point processes. Consider the class of all stationary ergodic point processes on the real line with arbitrary dependences among the inter–point distances (spacing).The hypothesis is :The observed process φ is a homogeneous Poisson process or more (resp.less) regular than a Poisson process.The sample is the vector of the first n points t₁, …,t_n.There is a close relation between our method for testing and queueing theory: For finding an appropriate test statistic, we observe the behaviour of a single server queue with the input φ.A table of critical values is given.     

A Goodness-of-Fit Test for Location-Scale Max-Stable Distributions

In this article, a technique based on the sample correlation coefficient to construct goodness-of-fit tests for max-stable distributions with unknown location and scale parameters and finite second moment is proposed. Specific details to test for the Gumbel distribution are given, including critical values for small sample sizes as well as approximate critical values for larger sample sizes by using normal quantiles. A comparison by Monte Carlo simulation shows that the proposed test for the Gumbel hypothesis is substantially more powerful than some other known tests against some alternative distributions with positive skewness coefficient.     

CUSUM Hypothesis Tests and Alternative Response Probabilities for Finite Poisson Trials

This article develops a new cumulative sum statistic to identify aberrant behavior in a sequentially administered multiple-choice standardized examination. The examination responses can be described as finite Poisson trials, and the statistic can be used for other applications which fit this framework. The standardized examination setting uses a maximum likelihood estimate of examinee ability and an item response theory model. Aberrant and non aberrant probabilities are computed by an odds ratio analogous to risk adjusted CUSUM schemes. The significance level of a hypothesis test, where the null hypothesis is non-aberrant examinee behavior, is computed with Markov chains. A smoothing process is used to spread probabilities across the Markov states. The practicality of the approach to detect aberrant examinee behavior is demonstrated with results from both simulated and empirical data.     

Testing the Granger Noncausality Hypothesis in Stationary Nonlinear Models of Unknown Functional Form

In this article, we propose a general method for testing the Granger noncausality hypothesis in stationary nonlinear models of unknown functional form. These tests are based on a Taylor expansion of the nonlinear model around a given point in the sample space. We study the performance of our tests by a Monte Carlo experiment and compare these to the most widely used linear test. Our tests appear to be well-sized and have reasonably good power properties.     

A Nonstochastic Interpretation of Reported Significance Levels

Tests of significance are often made in situations where the standard assumptions underlying the probability calculations do not hold. As a result, the reported significance levels become difficult to interpret. This article sketches an alternative interpretation of a reported significance level, valid in considerable generality. This level locates the given data set within the spectrum of other data sets derived from the given one by an appropriate class of transformations. If the null hypothesis being tested holds, the derived data sets should be equivalent to the original one. Thus, a small reported significance level indicates an unusual data set. This development parallels that of randomization tests, but there is a crucial technical difference: our approach involves permuting observed residuals; the classical randomization approach involves permuting unobservable, or perhaps nonexistent, stochastic disturbance terms.     

Bad Probability,Good Statistics,and Group Testing for Binomial Estimation

This article examines some improperly stated but often used textbook probability problems. Moving from a probabilistic to a statistical setting provides insight into group testing (i.e., observing only whether one or more of a group responds and not the response of each individual). Exact methods are used to construct tables showing (i) that group testing n times to estimate p can be more efficient than n individual tests even for small n and large p, (ii) optimal grouping strategies for various (n, p) combinations, and (iii) the efficiencies and biases achieved.