Authors:O. V. Chernoyarov; S. Dachian; Yu. A. Kutoyants Pages: 39 - 62 Abstract: Abstract We consider the problem of parameter estimation by continuous time observations of a deterministic signal in white Gaussian noise. It is supposed that the signal has a cusp-type singularity. The properties of the maximum-likelihood and Bayesian estimators are described in the asymptotics of small noise. Special attention is paid to the problem of parameter estimation in the situation of misspecification in regularity, i.e., when the statistician supposes that the observed signal has this singularity, but the real signal is smooth. The rate and the asymptotic distribution of the maximum-likelihood estimator in this situation are described. PubDate: 2018-02-01 DOI: 10.1007/s10463-016-0581-x Issue No:Vol. 70, No. 1 (2018)

Authors:Prajamitra Bhuyan; Murari Mitra; Anup Dewanji Pages: 63 - 81 Abstract: Abstract In many real-life scenarios, system reliability depends on dynamic stress–strength interference, where strength degrades and stress accumulates concurrently over time. In some other cases, shocks appear at random time points, causing damage which is only effective at the instant of shock arrival. In this paper, we consider the identifiability problem of a system under deterministic strength degradation and stochastic damage due to shocks arriving according to a homogeneous Poisson process. We provide conditions under which the models are identifiable with respect to lifetime data only. We also consider current status data and suggest to collect additional information and discuss the issues of model identifiability under different data configurations. PubDate: 2018-02-01 DOI: 10.1007/s10463-016-0579-4 Issue No:Vol. 70, No. 1 (2018)

Authors:Matteo Ruggiero; Matteo Sordello Pages: 83 - 98 Abstract: Abstract Normalised generalised gamma processes are random probability measures that induce nonparametric prior distributions widely used in Bayesian statistics, particularly for mixture modelling. We construct a class of dependent normalised generalised gamma priors induced by a stationary population model of Moran type, which exploits a generalised Pólya urn scheme associated with the prior. We study the asymptotic scaling for the dynamics of the number of clusters in the sample, which in turn provides a dynamic measure of diversity in the underlying population. The limit is formalised to be a positive non-stationary diffusion process which falls outside well-known families, with unbounded drift and an entrance boundary at the origin. We also introduce a new class of stationary positive diffusions, whose invariant measures are explicit and have power law tails, which approximate weakly the scaling limit. PubDate: 2018-02-01 DOI: 10.1007/s10463-016-0583-8 Issue No:Vol. 70, No. 1 (2018)

Authors:D. Barden; H. Le; M. Owen Pages: 99 - 129 Abstract: Abstract As demonstrated in our previous work on \({\varvec{T}}_{\!4}\) , the space of phylogenetic trees with four leaves, the topological structure of the space plays an important role in the non-classical limiting behaviour of the sample Fréchet means in \({\varvec{T}}_{\!4}\) . Nevertheless, the techniques used in that paper cannot be adapted to analyse Fréchet means in the space \({\varvec{T}}_{\!m}\) of phylogenetic trees with \(m(\geqslant \!5)\) leaves. To investigate the latter, this paper first studies the log map of \({\varvec{T}}_{\!m}\) . Then, in terms of a modified version of this map, we characterise Fréchet means in \({\varvec{T}}_{\!m}\) that lie in top-dimensional or co-dimension one strata. We derive the limiting distributions for the corresponding sample Fréchet means, generalising our previous results. In particular, the results show that, although they are related to the Gaussian distribution, the forms taken by the limiting distributions depend on the co-dimensions of the strata in which the Fréchet means lie. PubDate: 2018-02-01 DOI: 10.1007/s10463-016-0582-9 Issue No:Vol. 70, No. 1 (2018)

Authors:Sijia Xiang; Weixin Yao Pages: 131 - 154 Abstract: Abstract In this article, we propose and study a new class of semiparametric mixture of regression models, where the mixing proportions and variances are constants, but the component regression functions are smooth functions of a covariate. A one-step backfitting estimate and two EM-type algorithms have been proposed to achieve the optimal convergence rate for both the global parameters and the nonparametric regression functions. We derive the asymptotic property of the proposed estimates and show that both the proposed EM-type algorithms preserve the asymptotic ascent property. A generalized likelihood ratio test is proposed for semiparametric inferences. We prove that the test follows an asymptotic \(\chi ^2\) -distribution under the null hypothesis, which is independent of the nuisance parameters. A simulation study and two real data examples have been conducted to demonstrate the finite sample performance of the proposed model. PubDate: 2018-02-01 DOI: 10.1007/s10463-016-0584-7 Issue No:Vol. 70, No. 1 (2018)

Authors:Han-Ying Liang; Elias Ould Saïd Pages: 155 - 189 Abstract: Abstract Based on empirical likelihood method, we construct new weighted estimators of conditional density and conditional survival functions when the interest random variable is subject to random left-truncation; further, we define a plug-in weighted estimator of the conditional hazard rate. Under strong mixing assumptions, we derive asymptotic normality of the proposed estimators which permit to built a confidence interval for the conditional hazard rate. The finite sample behavior of the estimators is investigated via simulations too. PubDate: 2018-02-01 DOI: 10.1007/s10463-016-0587-4 Issue No:Vol. 70, No. 1 (2018)

Authors:Yanxin Wang; Qibin Fan; Li Zhu Pages: 191 - 214 Abstract: Abstract Variable selection problems are typically addressed under the regularization framework. In this paper, an exponential type penalty which very closely resembles the \(L_0\) penalty is proposed, we called it EXP penalty. The EXP penalized least squares procedure is shown to consistently select the correct model and is asymptotically normal, provided the number of variables grows slower than the number of observations. EXP is efficiently implemented using a coordinate descent algorithm. Furthermore, we propose a modified BIC tuning parameter selection method for EXP and show that it consistently identifies the correct model, while allowing the number of variables to diverge. Simulation results and data example show that the EXP procedure performs very well in a variety of settings. PubDate: 2018-02-01 DOI: 10.1007/s10463-016-0588-3 Issue No:Vol. 70, No. 1 (2018)

Authors:Nan Zheng; Brajendra C. Sutradhar Pages: 215 - 247 Abstract: Abstract This paper considers a semi-parametric mixed model for longitudinal counts under the assumption that for conditional on a common random effect over time the repeated count responses of an individual follow a Poisson AR(1) (auto-regressive order 1) non-stationary correlation structure. A step-by-step estimation approach is developed which provides consistent estimators for the non-parametric function, regression parameters, variance of the random effects, and auto-correlation structure of the model. Proofs for the consistency properties of the estimators along with their convergence rates are derived. A simulation study is conducted to examine first the estimation effects on parameters when the non-parametric function is ignored, and then an overall estimation study is carried out in the presence of the non-parametric function by including its estimation as well. PubDate: 2018-02-01 DOI: 10.1007/s10463-016-0590-9 Issue No:Vol. 70, No. 1 (2018)

Authors:Dennis Dobler Abstract: Abstract This article is concerned with proving the consistency of Efron’s bootstrap for the Kaplan–Meier estimator on the whole support of a survival function. While previous works address the asymptotic Gaussianity of the Kaplan–Meier estimator without restricting time, we enable the construction of bootstrap-based time-simultaneous confidence bands for the whole survival function. Other practical applications include bootstrap-based confidence bands for the mean residual lifetime function or the Lorenz curve as well as confidence intervals for the Gini index. Theoretical results are complemented with a simulation study and a real data example which result in statistical recommendations. PubDate: 2018-01-13 DOI: 10.1007/s10463-017-0634-9

Authors:Yan Gao; Xinyu Zhang; Shouyang Wang; Terence Tai-leung Chong; Guohua Zou Abstract: Abstract This paper develops a frequentist model averaging approach for threshold model specifications. The resulting estimator is proved to be asymptotically optimal in the sense of achieving the lowest possible squared errors. In particular, when combining estimators from threshold autoregressive models, this approach is also proved to be asymptotically optimal. Simulation results show that for the situation where the existing model averaging approach is not applicable, our proposed model averaging approach has a good performance; for the other situations, our proposed model averaging approach performs marginally better than other commonly used model selection and model averaging methods. An empirical application of our approach on the US unemployment data is given. PubDate: 2018-01-12 DOI: 10.1007/s10463-017-0642-9

Authors:Ursula U. Müller; Hanxiang Peng; Anton Schick Abstract: Abstract We present a new, efficient maximum empirical likelihood estimator for the slope in linear regression with independent errors and covariates. The estimator does not require estimation of the influence function, in contrast to other approaches, and is easy to obtain numerically. Our approach can also be used in the model with responses missing at random, for which we recommend a complete case analysis. This suffices thanks to results by Müller and Schick (Bernoulli 23:2693–2719, 2017), which demonstrate that efficiency is preserved. We provide confidence intervals and tests for the slope, based on the limiting Chi-square distribution of the empirical likelihood, and a uniform expansion for the empirical likelihood ratio. The article concludes with a small simulation study. PubDate: 2017-12-01 DOI: 10.1007/s10463-017-0632-y

Authors:Jiajuan Liang; Kai Wang Ng; Guoliang Tian Abstract: Abstract In this paper we employ the conditional probability integral transformation (CPIT) method to transform a d-dimensional sample from two classes of generalized multivariate distributions into a uniform sample in the unit interval \((0,\,1)\) or in the unit hypercube \([0,\,1]^{d-1}\) ( \(d\ge 2\) ). A class of existing uniform statistics are adopted to test the uniformity of the transformed sample. Monte Carlo studies are carried out to demonstrate the performance of the tests in controlling type I error rates and power against a selected group of alternative distributions. It is concluded that the proposed tests have satisfactory empirical performance and the CPIT method in this paper can serve as a general way to construct goodness-of-fit tests for many generalized multivariate distributions. PubDate: 2017-12-01 DOI: 10.1007/s10463-017-0630-0

Authors:Cuizhen Niu; Lixing Zhu Abstract: Abstract Unfortunately, original article has been published without acknowledgement section. PubDate: 2017-11-29 DOI: 10.1007/s10463-017-0633-x

Authors:Estate Khmaladze; Wolfgang Weil Abstract: Abstract We give a survey on fold-up derivatives, a notion which was introduced by Khmaladze (J Math Anal Appl 334:1055–1072, 2007) and extended by Khmaladze and Weil (J Math Anal Appl 413:291–310, 2014) to describe infinitesimal changes in a set-valued function. We summarize the geometric background and discuss in detail applications in statistics, in particular to the change-set problem of spatial statistics, and show how the notion of fold-up derivatives leads to the theory of testing statistical hypotheses about the change-set. We formulate Poisson limit theorems for the log-likelihood ratio in two versions of this problem and present also the route to a central limit theorem. PubDate: 2017-11-28 DOI: 10.1007/s10463-017-0628-7

Authors:Markos V. Koutras; Demetrios P. Lyberopoulos Abstract: Abstract The concept of pattern arises in many applications comprising experimental trials with two or more possible outcomes in each trial. A binary scan of type r / k is a special pattern referring to success–failure strings of fixed length k that contain at least r-successes, where r, k are positive integers with \(r\le k\) . The multiple scan statistic \(W_{t,k,r}\) is defined as the enumerating random variable for the overlapping moving windows occurring until trial t which include a scan of type r / k. In the present work, we consider a sequence of independent binary trials with not necessarily equal probabilities of success and develop upper bounds for the probability of the event that the multiple scan statistic will perform a jump from \(\ell \) to \(\ell +1\) (where \(\ell \) is a nonnegative integer) in a finite time horizon. PubDate: 2017-11-16 DOI: 10.1007/s10463-017-0621-1

Authors:Yueheng An; Yichuan Zhao Abstract: Abstract The volume under a surface (VUS) is an effective measure for evaluating the discriminating power of a diagnostic test with three ordinal diagnostic groups. In this paper, we investigate the difference of two correlated VUS’s to compare two treatments for discrimination of three-class classification data. A jackknife empirical likelihood (JEL) procedure is employed to avoid the variance estimation in the existing methods. We prove that the limiting distribution of the empirical log-likelihood ratio statistic follows a \(\chi ^2\) distribution. Extensive numerical studies show that the JEL confidence intervals outperform those based on the normal approximation method. The proposed method is also applied to the Alzheimer’s disease data. PubDate: 2017-11-15 DOI: 10.1007/s10463-017-0631-z

Authors:Robert E. Gaunt; Satish Iyengar; Adri B. Olde Daalhuis; Burcin Simsek Abstract: Abstract The Conway–Maxwell–Poisson distribution is a two-parameter generalization of the Poisson distribution that can be used to model data that are under- or over-dispersed relative to the Poisson distribution. The normalizing constant \(Z(\lambda ,\nu )\) is given by an infinite series that in general has no closed form, although several papers have derived approximations for this sum. In this work, we start by using probabilistic argument to obtain the leading term in the asymptotic expansion of \(Z(\lambda ,\nu )\) in the limit \(\lambda \rightarrow \infty \) that holds for all \(\nu >0\) . We then use an integral representation to obtain the entire asymptotic series and give explicit formulas for the first eight coefficients. We apply this asymptotic series to obtain approximations for the mean, variance, cumulants, skewness, excess kurtosis and raw moments of CMP random variables. Numerical results confirm that these correction terms yield more accurate estimates than those obtained using just the leading-order term. PubDate: 2017-11-15 DOI: 10.1007/s10463-017-0629-6

Authors:Cuizhen Niu; Lixing Zhu Abstract: Abstract This paper is devoted to test the parametric single-index structure of the underlying model when there are outliers in observations. First, a test that is robust against outliers is suggested. The Hampel’s second-order influence function of the test statistic is proved to be bounded. Second, the test fully uses the dimension reduction structure of the hypothetical model and automatically adapts to alternative models when the null hypothesis is false. Thus, the test can greatly overcome the dimensionality problem and is still omnibus against general alternative models. The performance of the test is demonstrated by both Monte Carlo simulation studies and an application to a real dataset. PubDate: 2017-11-02 DOI: 10.1007/s10463-017-0626-9

Authors:Benedikt Bauer; Felix Heimrich; Michael Kohler; Adam Krzyżak Abstract: Abstract Estimation of surrogate models for computer experiments leads to nonparametric regression estimation problems without noise in the dependent variable. In this paper, we propose an empirical maximal deviation minimization principle to construct estimates in this context and analyze the rate of convergence of corresponding quantile estimates. As an application, we consider estimation of computer experiments with moderately high dimension by neural networks and show that here we can circumvent the so-called curse of dimensionality by imposing rather general assumptions on the structure of the regression function. The estimates are illustrated by applying them to simulated data and to a simulation model in mechanical engineering. PubDate: 2017-11-02 DOI: 10.1007/s10463-017-0627-8

Authors:Kun-Lin Kuo; Yuchung J. Wang Abstract: Abstract Conditionally specified models offers a higher level of flexibility than the joint approach. Regression switching in multiple imputation is a typical example. However, reasonable-seeming conditional models are generally not coherent with one another. Gibbs sampler based on incompatible conditionals is called pseudo-Gibbs sampler, whose properties are mostly unknown. This article investigates the richness and commonalities among their stationary distributions. We show that Gibbs sampler replaces the conditional distributions iteratively, but keep the marginal distributions invariant. In the process, it minimizes the Kullback–Leibler divergence. Next, we prove that systematic pseudo-Gibbs projections converge for every scan order, and the stationary distributions share marginal distributions in a circularly fashion. Therefore, regardless of compatibility, univariate consistency is guaranteed when the orders of imputation are circularly related. Moreover, a conditional model and its pseudo-Gibbs distributions have equal number of parameters. Study of pseudo-Gibbs sampler provides a fresh perspective for understanding the original Gibbs sampler. PubDate: 2017-10-24 DOI: 10.1007/s10463-017-0625-x