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This paper introduces a novel two-sample test for a broad class of orthogonally invariant positive definite symmetric matrix distributions. Our test is the first of its kind, and we derive its asymptotic distribution. To estimate the test power, we use a warp-speed bootstrap method and consider the most common matrix distributions. We provide several real data examples, including the data for main cryptocurrencies and stock data of major US companies. The real data examples demonstrate the applicability of our test in the context closely related to algorithmic trading. The popularity of matrix distributions in many applications and the need for such a test in the literature are reconciled by our findings. [ABSTRACT FROM AUTHOR]

We establish the asymptotic theory of least absolute deviation estimators for AR(1) processes with autoregressive parameter satisfying n (ρ n - 1) → γ for some fixed γ as n → ∞ , which is parallel to the results of ordinary least squares estimators developed by Andrews and Guggenberger (Journal of Time Series Analysis, 29, 203–212, 2008) in the case γ = 0 or Chan and Wei (Annals of Statistics, 15, 1050–1063, 1987) and Phillips (Biometrika, 74, 535–574, 1987) in the case γ ≠ 0 . Simulation experiments are conducted to confirm the theoretical results and to demonstrate the robustness of the least absolute deviation estimation. [ABSTRACT FROM AUTHOR]

In this paper, we consider possibly misspecified stochastic differential equation models driven by Lévy processes. Regardless of whether the driving noise is Gaussian or not, Gaussian quasi-likelihood estimator can estimate unknown parameters in the drift and scale coefficients. However, in the misspecified case, the asymptotic distribution of the estimator varies by the correction of the misspecification bias, and consistent estimators for the asymptotic variance proposed in the correctly specified case may lose theoretical validity. As one of its solutions, we propose a bootstrap method for approximating the asymptotic distribution. We show that our bootstrap method theoretically works in both correctly specified case and misspecified case without assuming the precise distribution of the driving noise. [ABSTRACT FROM AUTHOR]

For Gaussian stationary triangular arrays, it is well known that the extreme values may occur in clusters. Here we consider the joint behaviors of the point processes of clusters and the partial sums of bivariate stationary Gaussian triangular arrays. For a bivariate stationary Gaussian triangular array, we derive the asymptotic joint behavior of the point processes of clusters and prove that the point processes and partial sums are asymptotically independent. As an immediate consequence of the results, one may obtain the asymptotic joint distributions of the extremes and partial sums. We illustrate the theoretical findings with a numeric example. [ABSTRACT FROM AUTHOR]

For 1 ≤ i ≤ r , let F i be the cumulative incidence function (CIF) corresponding to the ith risk in an r-competing risks model. We assume a discrete or a grouped time framework and obtain the maximum likelihood estimators (m.l.e.) of these CIFs under the restriction that F i (t) / F i + 1 (t) is nondecreasing, 1 ≤ i ≤ r - 1. We also derive the likelihood ratio tests for testing for and against this restriction and obtain their asymptotic distributions. The theory developed here can also be used to investigate the association between a failure time and a discretized or ordinal mark variable that is observed only at the time of failure. To illustrate the applicability of our results, we give examples in the competing risks and the mark variable settings. [ABSTRACT FROM AUTHOR]

Wasserstein geometry and information geometry are two important structures to be introduced in a manifold of probability distributions. Wasserstein geometry is defined by using the transportation cost between two distributions, so it reflects the metric of the base manifold on which the distributions are defined. Information geometry is defined to be invariant under reversible transformations of the base space. Both have their own merits for applications. In this study, we analyze statistical inference based on the Wasserstein geometry in the case that the base space is one-dimensional. By using the location-scale model, we further derive the W-estimator that explicitly minimizes the transportation cost from the empirical distribution to a statistical model and study its asymptotic behaviors. We show that the W-estimator is consistent and explicitly give its asymptotic distribution by using the functional delta method. The W-estimator is Fisher efficient in the Gaussian case. [ABSTRACT FROM AUTHOR]

We consider a weighted local linear estimator based on the inverse selection probability for nonparametric regression with missing covariates at random. The asymptotic distribution of the maximal deviation between the estimator and the true regression function is derived and an asymptotically accurate simultaneous confidence band is constructed. The estimator for the regression function is shown to be oracally efficient in the sense that it is uniformly indistinguishable from that when the selection probabilities are known. Finite sample performance is examined via simulation studies which support our asymptotic theory. The proposed method is demonstrated via an analysis of a data set from the Canada 2010/2011 Youth Student Survey. [ABSTRACT FROM AUTHOR]

The paper presents a novel approach to solve a classical two-sample problem with right-censored data. As a result, an efficient procedure for verifying equality of the two survival curves is developed. It generalizes, in a natural manner, a well-known standard, that is, the log-rank test. Under the null hypothesis, the new test statistic has an asymptotic Chi-square distribution with one degree of freedom, while the corresponding test is consistent for a wide range of the alternatives. On the other hand, to control the actual Type I error rate when sample sizes are finite, permutation approach is employed for the inference. An extensive simulation study shows that the new test procedure improves upon classical solutions and popular recent developments in the field. An analysis of the real datasets is included. A routine, written in R, is attached as Supplementary Material. [ABSTRACT FROM AUTHOR]

We take a wavelet-based approach to the analysis of point processes and the estimation of the first-order intensity under a continuous-time setting. A Haar wavelet multiresolution analysis is formulated which motivates the definition of homogeneity at different scales of resolution, termed J-th level homogeneity. Further to this, the activity in a point process' first-order behaviour at different scales of resolution is also defined and termed L-th level innovation. Likelihood ratio tests for both these properties are proposed with asymptotic distributions provided, even when only a single realization is observed. The test for L-th level innovation forms the basis for a collection of statistical strategies for thresholding coefficients in a wavelet-based estimator of the intensity function. These thresholding strategies outperform the existing local hard thresholding strategy on a range of simulation scenarios. This methodology is applied to NetFlow data, characterizing multiscale behaviour on computer networks. [ABSTRACT FROM AUTHOR]

We initiate the study of goodness-of-fit testing for data consisting of positive definite matrices. Motivated by the appearance of positive definite matrices in numerous applications, including factor analysis, diffusion tensor imaging, volatility models for financial time series, wireless communication systems, and polarimetric radar imaging, we apply the method of Hankel transforms of matrix argument to develop goodness-of-fit tests for Wishart distributions with given shape parameter and unknown scale matrix. We obtain the limiting null distribution of the test statistic and a corresponding covariance operator, show that the eigenvalues of the operator satisfy an interlacing property, and apply our test to some financial data. We establish the consistency of the test against a large class of alternative distributions and derive the asymptotic distribution of the test statistic under a sequence of contiguous alternatives. We obtain the Bahadur and Pitman efficiency properties of the test statistic and establish a modified version of Wieand's condition. [ABSTRACT FROM AUTHOR]

We consider the estimation of an extreme conditional quantile. In a first part, we propose a new tail condition in order to establish the asymptotic distribution of an extreme conditional quantile estimator. Next, a general class of estimators is introduced, which encompasses, among others, kernel or nearest neighbors types of estimators. A unified theorem of the asymptotic normality for this general class of estimators is provided under the new tail condition and illustrated on the different well-known examples. A comparison between different estimators belonging to this class is provided on a small simulation study and illustrated on a real dataset on earthquake magnitudes. [ABSTRACT FROM AUTHOR]

Incomplete information on explanatory variables is commonly encountered in studies of possibly censored event times. A popular approach to deal with partially observed covariates is multiple imputation, where a number of completed data sets, that can be analyzed by standard complete data methods, are obtained by imputing missing values from an appropriate distribution. We show how the combination of multiple imputations from a compatible model with suitably estimated parameters and the usual Cox regression estimators leads to consistent and asymptotically Gaussian estimators of both the finite-dimensional regression parameter and the infinite-dimensional cumulative baseline hazard parameter. We also derive a consistent estimator of the covariance operator. Simulation studies and an application to a study on survival after treatment for liver cirrhosis show that the estimators perform well with moderate sample sizes and indicate that iterating the multiple-imputation estimator increases the precision. [ABSTRACT FROM AUTHOR]

The cross ratio function (CRF) is a commonly used tool to describe local dependence between two correlated variables. Being a ratio of conditional hazards, the CRF can be rewritten in terms of (first and second derivatives of) the survival copula of these variables. Bernstein estimators for (the derivatives of) this survival copula are used to define a nonparametric estimator of the cross ratio, and asymptotic normality thereof is established. We consider simulations to study the finite sample performance of our estimator for copulas with different types of local dependency. A real dataset is used to investigate the dependence between food expenditure and net income. The estimated CRF reveals that families with a low net income relative to the mean net income will spend less money to buy food compared to families with larger net incomes. This dependence, however, disappears when the net income is large compared to the mean income. [ABSTRACT FROM AUTHOR]

We are interested in the estimation of a parameter θ that maximizes a certain criterion function depending on an unknown, possibly infinite-dimensional nuisance parameter h. A common estimation procedure consists in maximizing the corresponding empirical criterion, in which the nuisance parameter is replaced by a nonparametric estimator. In the literature, this research topic, commonly referred to as semiparametric M-estimation, has received a lot of attention in the case where the criterion function satisfies certain smoothness properties. In certain applications, these smoothness conditions are, however, not satisfied. The aim of this paper is therefore to extend the existing theory on semiparametric M-estimators, in order to cover non-smooth M-estimators as well. In particular, we develop 'high-level' conditions under which the proposed M-estimator is consistent and has an asymptotic limit. We also check these conditions for a specific example of a semiparametric M-estimator coming from the area of classification with missing data. [ABSTRACT FROM AUTHOR]

In this paper, we develop an empirical likelihood-based test for the presence of stochastic ordering under censoring in the k-sample case. The proposed test statistic is formed by taking the supremum of localized empirical likelihood ratio test statistics. Its asymptotic null distribution has a simple representation in terms of a standard Brownian motion process. Through simulations, we show that it outperforms in terms of power existing methods for the same problem at all the distributions that we consider. A real-life example is used to illustrate the applicability of this new test. [ABSTRACT FROM AUTHOR]

Consider independent observations (X i , R i) with random or fixed ranks R i , while conditional on R i , the random variable X i has the same distribution as the R i -th order statistic within a random sample of size k from an unknown distribution function F. Such observation schemes are well known from ranked set sampling and judgment post-stratification. Within a general, not necessarily balanced setting we derive and compare the asymptotic distributions of three different estimators of the distribution function F: a stratified estimator, a nonparametric maximum-likelihood estimator and a moment-based estimator. Our functional central limit theorems generalize and refine previous asymptotic analyses. In addition, we discuss briefly pointwise and simultaneous confidence intervals for the distribution function with guaranteed coverage probability for finite sample sizes. The methods are illustrated with a real data example, and the potential impact of imperfect rankings is investigated in a small simulation experiment. [ABSTRACT FROM AUTHOR]

This paper describes a family of divergences, named herein as the C-divergence family, which is a generalized version of the power divergence family and also includes the density power divergence family as a particular member of this class. We explore the connection of this family with other divergence families and establish several characteristics of the corresponding minimum distance estimator including its asymptotic distribution under both discrete and continuous models; we also explore the use of the C-divergence family in parametric tests of hypothesis. We study the influence function of these minimum distance estimators, in both the first and second order, and indicate the possible limitations of the first-order influence function in this case. We also briefly study the breakdown results of the corresponding estimators. Some simulation results and real data examples demonstrate the small sample efficiency and robustness properties of the estimators. [ABSTRACT FROM AUTHOR]

In this paper, we consider testing the homogeneity for proportions in independent binomial distributions, especially when data are sparse for large number of groups. We provide broad aspects of our proposed tests such as theoretical studies, simulations and real data application. We present the asymptotic null distributions and asymptotic powers for our proposed tests and compare their performance with existing tests. Our simulation studies show that none of tests dominate the others; however, our proposed test and a few tests are expected to control given sizes and obtain significant powers. We also present a real example regarding safety concerns associated with Avandia (rosiglitazone) in Nissen and Wolski (New Engl J Med 356:2457–2471, 2007). [ABSTRACT FROM AUTHOR]

In the spirit of Bross (Biometrics 14:18-38, 1958), this paper considers ridit reliability functionals to develop test procedures for the equality of K(>2) treatment effects in nonparametric analysis of covariance (ANCOVA) model with d covariates based on two different methods. The procedures are asymptotically distribution free and are not based on the assumption that the distribution functions (d.f.'s) of the response variable and the associated covariates are continuous. By means of simulation study, the proposed methods are compared with the methods provided by Tsangari and Akritas (J Multivar Anal 88:298-319, 2004) and Bathke and Brunner (Recent advances and trends in nonparametric statistics, Elsevier, Amsterdam, 2003) under ANCOVA in terms of type I error rate and power. [ABSTRACT FROM AUTHOR]

In many situations, when dealing with several populations, equality of the covariance operators is assumed. An important issue is to study whether this assumption holds before making other inferences. In this paper, we develop a test for comparing covariance operators of several functional data samples. The proposed test is based on the Hilbert-Schmidt norm of the difference between estimated covariance operators. In particular, when dealing with two populations, the test statistic is just the squared norm of the difference between the two covariance operators estimators. The asymptotic behaviour of the test statistic under both the null hypothesis and local alternatives is obtained. The computation of the quantiles of the null asymptotic distribution is not feasible in practice. To overcome this problem, a bootstrap procedure is considered. The performance of the test statistic for small sample sizes is illustrated through a Monte Carlo study and on a real data set. [ABSTRACT FROM AUTHOR]

In this article, we focus on the problem of testing the equality of several high dimensional mean vectors with unequal covariance matrices. This is one of the most important problems in multivariate statistical analysis and there have been various tests proposed in the literature. Motivated by Bai and Saranadasa (Stat Sin 6:311-329, 1996) and Chen and Qin (Ann Stat 38:808-835, 2010), we introduce a test statistic and derive the asymptotic distributions under the null and the alternative hypothesis. In addition, it is compared with a test statistic recently proposed by Srivastava and Kubokawa (J Multivar Anal 115:204-216, 2013). It is shown that our test statistic performs better especially in the large dimensional case. [ABSTRACT FROM AUTHOR]

In this paper, we show that some of the most commonly used tests of symmetry do not have power which is reflective of the size of asymmetry. This is because the primary rationale for the test statistics that are proposed in the literature to test for symmetry is to detect the departure from symmetry, rather than the quantification of the asymmetry. As a result, tests of symmetry based upon these statistics do not necessarily generate power that is representative of the departure from the null hypothesis of symmetry. Recent research has produced new measures of asymmetry, which have been shown to do an admirable job of quantifying the amount of asymmetry. We propose several new tests based upon one such measure. We derive the asymptotic distribution of the test statistics and analyse the performance of these proposed tests through the use of a simulation study. [ABSTRACT FROM AUTHOR]

Distributions of runs of length at least k ( Type II runs) and overlapping runs of length k ( Type III runs) are derived in a unified way using a new generating function approach. A new and more compact formula is obtained for the probability mass function of the Type III runs. [ABSTRACT FROM AUTHOR]

For a truncated exponential family of distributions with a natural parameter $$\theta $$ and a truncation parameter $$\gamma $$ as a nuisance parameter, it is known that the maximum likelihood estimators (MLEs) $$\hat{\theta }_{\mathrm{ML}}^{\gamma }$$ and $$\hat{\theta }_{\mathrm{ML}}$$ of $$\theta $$ for known $$\gamma $$ and unknown $$\gamma $$ , respectively, and the maximum conditional likelihood estimator $$\hat{\theta }_{\mathrm{MCL}}$$ of $$\theta $$ are asymptotically equivalent. In this paper, the stochastic expansions of $$\hat{\theta }_{\mathrm{ML}}^{\gamma }$$ , $$\hat{\theta }_{\mathrm{ML}}$$ and $$\hat{\theta }_{\mathrm{MCL}}$$ are derived, and their second-order asymptotic variances are obtained. The second-order asymptotic loss of a bias-adjusted MLE $$\hat{\theta }_{\mathrm{ML}}^{*}$$ relative to $$\hat{\theta }_{\mathrm{ML}}^{\gamma }$$ is also given, and $$\hat{\theta }_{\mathrm{ML}}^{*}$$ and $$\hat{\theta }_{\mathrm{MCL}}$$ are shown to be second-order asymptotically equivalent. Further, some examples are given. [ABSTRACT FROM AUTHOR]

The ordinary Bayes estimator based on the posterior density can have potential problems with outliers. Using the density power divergence measure, we develop an estimation method in this paper based on the so-called ' $$R^{(\alpha )}$$ -posterior density'; this construction uses the concept of priors in Bayesian context and generates highly robust estimators with good efficiency under the true model. We develop the asymptotic properties of the proposed estimator and illustrate its performance numerically. [ABSTRACT FROM AUTHOR]

In a multivariate nonparametric regression problem with fixed, deterministic design asymptotic, uniform confidence bands for the regression function are constructed. The construction of the bands is based on the asymptotic distribution of the maximal deviation between a suitable nonparametric estimator and the true regression function which is derived by multivariate strong approximation methods and a limit theorem for the supremum of a stationary Gaussian field over an increasing system of sets. The results are derived for a general class of estimators which includes local polynomial estimators as a special case. The finite sample properties of the proposed asymptotic bands are investigated by means of a small simulation study. [ABSTRACT FROM AUTHOR]

In this paper, we study integrated regression techniques to check the adequacy of a given model in the context of selection-biased observations. We introduce integrated regression in this setting, providing not only a suitable statistic for enabling a model checking test, but also a bootstrap distributional approximation to carry out the test. We also address the behaviour of the test under different alternatives showing that this behaviour is asymptotically the same for both selection-biased and non selection-biased data. The technique is illustrated with a simulation study and a data analysis based on a real situation that shows the performance of the method and how selection bias affect both estimation and inference. [ABSTRACT FROM AUTHOR]

We determine the joint limiting distribution of adjacent spacings around a central, intermediate, or an extreme order statistic $$X_{k:n}$$ of a random sample of size $$n$$ from a continuous distribution $$F$$ . For central and intermediate cases, normalized spacings in the left and right neighborhoods are asymptotically i.i.d. exponential random variables. The associated independent Poisson arrival processes are independent of $$X_{k:n}$$ . For an extreme $$X_{k:n}$$ , the asymptotic independence property of spacings fails for $$F$$ in the domain of attraction of Fréchet and Weibull ( $$\alpha \ne 1$$ ) distributions. This work also provides additional insight into the limiting distribution for the number of observations around $$X_{k:n}$$ for all three cases. [ABSTRACT FROM AUTHOR]

The sinh-arcsinh transform is used to obtain a flexible four-parameter model that provides a natural framework with which to perform inference robust to wide-ranging departures from the logistic distribution. Its basic properties are established and its distribution and quantile functions, and properties related to them, shown to be highly tractable. Two important subfamilies are also explored. Maximum likelihood estimation is discussed, and reparametrisations designed to reduce the asymptotic correlations between the maximum likelihood estimates provided. A likelihood-ratio test for logisticness, which outperforms standard empirical distribution function based tests, follows naturally. The application of the proposed model and inferential methods is illustrated in an analysis of carbon fibre strength data. Multivariate extensions of the model are explored. [ABSTRACT FROM AUTHOR]

We consider the problem of estimating the location of a change point $$\theta _0$$ in a regression model. Most change point models studied so far were based on regression functions with a jump. However, we focus on regression functions, which are continuous at $$\theta _0$$ . The degree of smoothness $$q_0$$ has to be estimated as well. We investigate the consistency with increasing sample size $$n$$ of the least squares estimates $$(\hat{\theta }_n,\hat{q}_n)$$ of $$(\theta _0, q_0)$$ . It turns out that the rates of convergence of $$\hat{\theta }_n$$ depend on $$q_0$$ : for $$q_0$$ greater than $$1/2$$ we have a rate of $$\sqrt{n}$$ and the asymptotic normality property; for $$q_0$$ less than $$1/2$$ the rate is $$\displaystyle n^{1/(2q_0+1)}$$ and the change point estimator converges to a maximizer of a Gaussian process; for $$q_0$$ equal to $$1/2$$ the rate is $$\sqrt{n \cdot \mathrm{ln}(n)}$$ . Interestingly, in the last case the limiting distribution is also normal. [ABSTRACT FROM AUTHOR]

Asymptotic properties of the harmonic moment tail index Estimator are derived for distributions with regularly varying tails. The estimator shows good robustness properties and stands out for its simplicity. A tuning parameter allows for regulating the trade-off between robustness and efficiency. Small sample properties are illustrated by a simulation study. [ABSTRACT FROM AUTHOR]

For analysis of time-to-event data with incomplete information beyond right-censoring, many generalizations of the inference of the distribution and regression model have been proposed. However, the development of martingale approaches in this area has not progressed greatly, while for right-censored data such an approach has spread widely to study the asymptotic properties of estimators and to derive regression diagnosis methods. In this paper, focusing on doubly censored data, we discuss a martingale approach for inference of the nonparametric maximum likelihood estimator (NPMLE). We formulate a martingale structure of the NPMLE using a score function of the semiparametric profile likelihood. Finally, an expression of the asymptotic distribution of the NPMLE is derived more conveniently without depending on an infinite matrix expression as in previous research. A further useful point is that a variance-covariance formula of the NPMLE computable in a larger sample is obtained as an empirical version of the limit form presented here. [ABSTRACT FROM AUTHOR]

Large sample statistical analysis of threshold autoregressive models is usually based on the assumption that the underlying driving noise is uncorrelated. In this paper, we consider a model, driven by Gaussian noise with geometric correlation tail and derive a complete characterization of the asymptotic distribution for the Bayes estimator of the threshold parameter. [ABSTRACT FROM AUTHOR]

We derive the asymptotic distributions of degenerate $$U$$- and $$V$$-statistics of stationary and ergodic random variables. Statistics of these types naturally appear as approximations of test statistics. Since the limit variables are of complicated structure, typically depending on unknown parameters, quantiles can hardly be obtained directly. Therefore, we prove a general result on the consistency of model-based bootstrap methods for $$U$$- and $$V$$-statistics under easily verifiable conditions. Three applications to hypothesis testing are presented. Finally, the finite sample behavior of the bootstrap-based tests is illustrated by a simulation study. [ABSTRACT FROM AUTHOR]

We study the problem of parameter estimation for Ornstein-Uhlenbeck processes driven by symmetric α-stable motions, based on discrete observations. A least squares estimator is obtained by minimizing a contrast function based on the integral form of the process. Let h be the length of time interval between two consecutive observations. For both the case of fixed h and that of h → 0, consistencies and asymptotic distributions of the estimator are derived. Moreover, for both of the cases of h, the estimator has a higher order of convergence for the Ornstein-Uhlenbeck process driven by non-Gaussian α-stable motions (0 < α < 2) than for the process driven by the classical Gaussian case ( α = 2). [ABSTRACT FROM AUTHOR]

The statistic introduced in Fortiana and Grané (J R Stat Soc B 65(1):115-126, ) is modified so that it can be used to test the goodness-of-fit of a censored sample, when the distribution function is fully specified. Exact and asymptotic distributions of three modified versions of this statistic are obtained and exact critical values are given for different sample sizes. Empirical power studies show the good performance of these statistics in detecting symmetrical alternatives. [ABSTRACT FROM AUTHOR]

In this paper, we present results for testing main, simple and interaction effects in heteroscedastic two factor MANOVA models. In particular, we suggest modifications to the MANOVA sum of squares and cross product matrices to account for heteroscedasticity. Based on these modified matrices, we define some multivariate test statistics and derive their asymptotic distributions under non-normality for the null as well as non-null cases. Derivation of these results relies on the perturbation method and limit theorems for independently distributed random matrices. Based on the asymptotic distributions, we devise small sample approximations for the quantiles of the null distributions. The numerical accuracy of the large sample as well as small sample approximations are favorable. A real data set from a Smoking Cessation Trial is analyzed to illustrate the application of the methods. [ABSTRACT FROM AUTHOR]

For the regression parameter β in the Cox model, there have been several estimators constructed based on various types of approximated likelihood, but none of them has demonstrated small-sample advantage over Cox's partial likelihood estimator. In this article, we derive the full likelihood function for ( β, F), where F is the baseline distribution in the Cox model. Using the empirical likelihood parameterization, we explicitly profile out nuisance parameter F to obtain the full-profile likelihood function for β and the maximum likelihood estimator (MLE) for ( β, F). The relation between the MLE and Cox's partial likelihood estimator for β is made clear by showing that Taylor's expansion gives Cox's partial likelihood estimating function as the leading term of the full-profile likelihood estimating function. We show that the log full-likelihood ratio has an asymptotic chi-squared distribution, while the simulation studies indicate that for small or moderate sample sizes, the MLE performs favorably over Cox's partial likelihood estimator. In a real dataset example, our full likelihood ratio test and Cox's partial likelihood ratio test lead to statistically different conclusions. [ABSTRACT FROM AUTHOR]