Abstract: Abstract For testing a linear hypothesis about fixed effects in a normal mixed linear model, a popular approach is to use a Wald test, in which the test statistic is assumed to have a null distribution that is approximately chi-squared. This approximation is questionable, however, for small samples. In 1997 Kenward and Roger constructed a test that addresses this problem. They altered the Wald test in three ways: (a) adjusting the test statistic, (b) approximating the null distribution by a scaled F distribution, and (c) modifying the formulas to achieve an exact F test in two special cases. Alterations (a) and (b) lead to formulas that are somewhat complicated but can be explained by using Taylor series approximations and a few convenient assumptions. The modified formulas used in alteration (c), however, are more mysterious. Restricting attention to models with linear variance–covariance structure, we provide details of a derivation that justifies these formulas. We show that similar but different derivations lead to different formulas that also produce exact F tests in the two special cases and are equally justifiable. A simulation study was done for testing the equality of treatment effects in block-design models. Tests based on the different derivations performed very similarly. Moreover, the simulations confirm that alteration (c) is worthwhile. The Kenward–Roger test showed greater accuracy in its p values than did the unmodified version of the test. PubDate: 2019-03-01 DOI: 10.1007/s00184-018-0669-9

Abstract: Abstract The paper deals with three generalized dependent setups arising from a sequence of Bernoulli trials. Various distributional properties, such as probability generating function, probability mass function and moments are discussed for these setups and their waiting time. Also, explicit forms of probability generating function and probability mass function are obtained. Finally, two applications to demonstrate the relevance of the results are given. PubDate: 2019-03-01 DOI: 10.1007/s00184-018-0668-x

Abstract: Abstract We introduce an estimator for an unknown population size in a capture–recapture framework where the count of identifications follows a geometric distribution. This can be thought of as a Poisson count adjusted for exponentially distributed heterogeneity. As a result, a new Turing-type estimator under the geometric distribution is obtained. This estimator can be used in many real life situations of capture–recapture, in which the geometric distribution is more appropriate than the Poisson. The proposed estimator shows a behavior comparable to the maximum likelihood one, on both simulated and real data. Its asymptotic variance is obtained by applying a conditional technique and its empirical behavior is investigated through a large-scale simulation study. Comparisons with other well-established estimators are provided. Empirical applications, in which the population size is known, are also included to further corroborate the simulation results. PubDate: 2019-03-01 DOI: 10.1007/s00184-018-0695-7

Abstract: Abstract We consider a varying-coefficient partially linear proportional odds model with current status data. This model enables one to examine the extent to which some covariates interact nonlinearly with an exposure variable, while other covariates present linear effects. B-spline approach and sieve maximum likelihood estimation method are used to get an integrated estimate for the linear coefficients, the varying-coefficient functions and the baseline function. The proposed parameter estimators are proved to be semiparametrically efficient and asymptotically normal, and the estimators for the nonparametric functions achieve the optimal rate of convergence. Simulation studies and a real data analysis are used for assessment and illustration. PubDate: 2019-03-01 DOI: 10.1007/s00184-018-0698-4

Abstract: Abstract An empirical likelihood ratio testing method is proposed, in this paper, for semi-functional partial linear regression models. Two empirical likelihood ratio statistics are employed to test the linear hypothesis of parametric components, then we demonstrate that their asymptotic null distributions are standard Chi-square distributions with the degrees of freedom being independent of the nuisance parameters. We also verify the proposed statistics follow non-central Chi-square distributions under the alternative hypothesis, and their powers are derived. Furthermore, we apply the proposed method to test the significance of parametric components. In addition, a F-test statistic is introduced. Simulations are undertaken to demonstrate the proposed methodologies and the simulation results indicate that the proposed testing methods are workable. A real example is applied for illustration. PubDate: 2019-03-01 DOI: 10.1007/s00184-018-0680-1

Abstract: Abstract This paper deals with the reliability of a k-out-of-n:G system in the stress–strength setup with three different types (cold, warm or hot) of standby components. The switching time of the standby component to the k-out-of-n:G stress–strength system has been studied and how its effect on the stress–strength reliability and costs have been assessed. By taking into account the switching time of the standby component, some expressions for the stress–strength reliability and the mean remaining strength functions are obtained. The results for exponential and Weibull distributions are given in detail and the optimal time to activate the standby components to the working state is investigated. PubDate: 2019-03-01 DOI: 10.1007/s00184-018-0694-8

Abstract: Abstract The conditional copula of a random pair \((Y_1,Y_2)\) given the value taken by some covariate \(X \in {\mathbb {R}}\) is the function \(C_x:[0,1]^2 \rightarrow [0,1]\) such that \({\mathbb {P}}(Y_1 \le y_1, Y_2 \le y_2 X=x) = C_x \{ {\mathbb {P}}(Y_1\le y_1 X=x), {\mathbb {P}}(Y_2\le y_2 X=x) \}\) . In this note, the weak convergence of the two estimators of \(C_x\) proposed by Gijbels et al. (Comput Stat Data Anal 55(5):1919–1932, 2011) is established under \(\alpha \) -mixing. It is shown that under appropriate conditions on the weight functions and on the mixing coefficients, the limiting processes are the same as those obtained by Veraverbeke et al. (Scand J Stat 38(4):766–780, 2011) under the i.i.d. setting. The performance of these estimators in small sample sizes is investigated with simulations. PubDate: 2019-02-28 DOI: 10.1007/s00184-019-00711-y

Abstract: Abstract Various nonparametric test statistics have been proposed for censored data. Two-sample nonparametric testing plays an important role in biometry. While most of two-sample nonparametric tests intend to detect a shift in location or in scale, the two-sample Cucconi test statistic is suitable for the joint comparison of both parameters. The Cucconi test statistic is extended to the left- and right-censored data based on the theory of ties. We derive the limiting distribution of the Cucconi test statistic for censored data. We conduct simulation studies to investigate the convergence of the Cucconi test statistic to the limiting distribution and the power of the proposed statistic with various population distributions. The method is illustrated with an analysis using real data. PubDate: 2019-02-28 DOI: 10.1007/s00184-019-00712-x

Abstract: Abstract Consider an exponential family F on the set of non-negative integers indexed by the parameter a. The cumulative distribution function of an element of F estimated on k is both a function of a and k. Assume that the derivative of this function with respect to a is the product of three things: a function of k, a function of a and the function a to the power k. We show that this assumption implies that the exponential family is either a binomial, or the Poisson, or a negative binomial family. Next, we study an analogous property for continuous distributions and we find that it is satisfied if and only the families are either Gaussian or Gamma. Ultimately, the proofs rely on the fact that only Möbius functions preserve the cross ratio. PubDate: 2019-02-27 DOI: 10.1007/s00184-019-00710-z

Abstract: Abstract Often, experimenters are only interested in estimating a few factor specified effects. In this paper, we broadly call a design which can reach this target a compromise design. First, for assessing and selecting this kind of designs we introduce a partial aliased effect number pattern (P-AENP), then we use this pattern to study class one two-level compromise designs. Some theoretical results are obtained and a number of class one clear, strongly clear and general optimal \(2^{n-m}\) compromise designs with 8, 16, 32 and 64 runs are tabulated. PubDate: 2019-02-22 DOI: 10.1007/s00184-018-00705-2

Abstract: Abstract This paper is about models for a vector of probabilities whose elements must have a multiplicative structure and sum to 1 at the same time; in certain applications, like basket analysis, these models may be seen as a constrained version of quasi-independence. After reviewing the basic properties of the models, their geometric features as a curved exponential family are investigated. An improved algorithm for computing maximum likelihood estimates is introduced and new insights are provided on the underlying geometry. The asymptotic distribution of three statistics for hypothesis testing are derived and a small simulation study is presented to investigate the accuracy of asymptotic approximations. PubDate: 2019-02-01 DOI: 10.1007/s00184-019-00709-6

Abstract: Abstract We propose a class of weighted \(L^2\) -type tests of fit to the Gamma distribution. Our novel procedure is based on a fixed point property of a new transformation connected to a Steinian characterization of the family of Gamma distributions. We derive the weak limits of the statistic under the null hypothesis and under contiguous alternatives. The result on the limit null distribution is used to prove the asymptotic validity of the parametric bootstrap that is implemented to run the tests. Further, we establish the global consistency of our tests in this bootstrap setting, and conduct a Monte Carlo simulation study to show the competitiveness to existing test procedures. PubDate: 2019-01-29 DOI: 10.1007/s00184-019-00708-7

Abstract: Abstract In this paper, we discuss the existence and uniqueness of the maximum likelihood estimates (MLEs) of the parameters of Laplace Birnbaum–Saunders distribution based on Type-I, Type-II and hybrid censored samples. We first derive the relationship between the MLEs of the two parameters and then discuss the monotonicity property of the profile likelihood function. Numerical iterative procedure is then discussed for determining the MLEs of the parameters. Finally, for illustrative purpose, we analyze one real data from the literature and present some graphical illustrations of the approach. PubDate: 2019-01-29 DOI: 10.1007/s00184-019-00707-8

Abstract: Abstract The recently introduced weighted optimality criteria for experimental designs allow one to place various emphasis on different parameters or functions of parameters of interest. However, various emphasis on parameter functions can also be expressed by considering the well-developed optimality criteria for estimating a parameter system of interest (the partial optimality criteria). We prove that the approaches of weighted optimality and of partial optimality are in fact equivalent for any eigenvalue-based optimality criterion. This opens up the possibility to use the large body of existing theoretical and computational results for the partial optimality to derive theorems and numerical algorithms for the weighted optimality of experimental designs. We demonstrate the applicability of the proven equivalence on a few examples. We also propose a slight generalization of the weighted optimality so that it can represent the experimental objective consisting of any system of linear estimable functions. PubDate: 2019-01-22 DOI: 10.1007/s00184-019-00706-9

Abstract: Abstract We consider the problem of providing the exact distribution of the likelihood ratio test (LRT) statistic for testing the homogeneity of scale parameters of \( k \ge 2 \) two-parameter exponential distributions. To this end, we apply the Millen inverse transform and the Jacobi polynomial expansion to the moments of LRT statistic. We consider the problem when the data are either complete or censored under the different kinds of Type II censoring, such as the Type II right, progressively Type II right, and double Type II censoring schemes. We also discuss the exact null distribution of the LRT when the data are censored under the Type I censoring scheme. PubDate: 2019-01-05 DOI: 10.1007/s00184-018-00704-3

Abstract: Abstract In this paper, we consider composite quantile estimation for the partial functional linear regression model with errors from a short-range dependent and strictly stationary linear processes. The functional principal component analysis method is employed to estimate the slope function and the functional predictive variable, respectively. Under some regularity conditions, we obtain the optimal convergence rate of the slope function, and the asymptotic normality of the parameter vector. Simulation studies demonstrate that the proposed new estimation method is robust and works much better than the least squares based method when there are outliers in the dataset or the autoregressive error distribution follows a heavy-tailed distribution. Finally, we apply the proposed methodology to electricity consumption data. PubDate: 2019-01-02 DOI: 10.1007/s00184-018-0699-3

Abstract: Abstract Measure of uncertainty in past lifetime plays an important role in different areas such as information theory, reliability theory, survival analysis, economics, business, forensic science and other related fields. In this paper, we propose a cumulative Tsallis entropy in past lifetime based on quantile function. We obtain different characterizations based on the proposed measure and quantile-based reliability measures. We also study the quantile-based cumulative Tsallis entropy of order statistics in past lifetime. PubDate: 2019-01-01 DOI: 10.1007/s00184-018-0678-8

Abstract: This paper focuses on semi-functional partially linear regression model, where a scalar response variable with missing at random is explained by a sum of an unknown linear combination of the components of multivariate random variables and an unknown transformation of a functional random variable which takes its value in a semi-metric abstract space \({\mathscr {H}}\) with a semi-metric \(d\left( \cdot , \cdot \right) \) . The main purpose of this paper is to construct the estimators of unknown parameters and an unknown regression operator respectively. Then some asymptotic properties of the estimators such as almost sure convergence rates of the nonparametric component and asymptotic distribution of the parametric one are obtained under some mild conditions. Furthermore, a simulation study is carried out to evaluate the finite sample performances of the estimators. Finally, an application to real data analysis for food fat predictions shows the usefulness of the proposed methodology. PubDate: 2019-01-01 DOI: 10.1007/s00184-018-0688-6

Abstract: Abstract Recently new methods for measuring and testing dependence have appeared in the literature. One way to evaluate and compare these measures with each other and with classical ones is to consider what are reasonable and natural axioms that should hold for any measure of dependence. We propose four natural axioms for dependence measures and establish which axioms hold or fail to hold for several widely applied methods. All of the proposed axioms are satisfied by distance correlation. We prove that if a dependence measure is defined for all bounded nonconstant real valued random variables and is invariant with respect to all one-to-one measurable transformations of the real line, then the dependence measure cannot be weakly continuous. This implies that the classical maximal correlation cannot be continuous and thus its application is problematic. The recently introduced maximal information coefficient has the same disadvantage. The lack of weak continuity means that as the sample size increases the empirical values of a dependence measure do not necessarily converge to the population value. PubDate: 2019-01-01 DOI: 10.1007/s00184-018-0670-3

Abstract: Abstract Linear mixed models have become popular in many statistical applications during recent years. However design issues for multi-response linear mixed models are rarely discussed. The main purpose of this paper is to investigate D-optimal designs for multi-response linear mixed models. We provide two equivalence theorems to characterize the optimal designs for the estimation of the fixed effects and the prediction of random effects, respectively. Two examples of the D-optimal designs for multi-response linear mixed models are given for illustration. PubDate: 2019-01-01 DOI: 10.1007/s00184-018-0679-7