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Given independent samples from P and Q, two-sample permutation tests allow one to construct exact level tests when the null hypothesis is P = Q. On the other hand, when comparing or testing particular parameters θ of P and Q, such as their means or medians, permutation tests need not be level α, or even approximately level α in large samples. Under very weak assumptions for comparing estimators, we provide a general test procedure whereby the asymptotic validity of the permutation test holds while retaining the exact rejection probability α in finite samples when the underlying distributions are identical. The ideas are broadly applicable and special attention is given to the k-sample problem of comparing general parameters, whereby a permutation test is constructed which is exact level α under the hypothesis of identical distributions, but has asymptotic rejection probability α under the more general null hypothesis of equality of parameters. A Monte Carlo simulation study is performed as well. A quite general theory is possible based on a coupling construction, as well as a key contiguity argument for the multinomial and multivariate hypergeometric distributions.

Although asymptotic analyses of undirected network models based on degree sequences have started to appear in recent literature, it remains an open problem to study statistical properties of directed network models. In this paper, we provide for the first time a rigorous analysis of directed exponential random graph models using the in-degrees and out-degrees as sufficient statistics with binary as well as continuous weighted edges. We establish the uniform consistency and the asymptotic normality for the maximum likelihood estimate, when the number of parameters grows and only one realized observation of the graph is available. One key technique in the proofs is to approximate the inverse of the Fisher information matrix using a simple matrix with high accuracy. Numerical studies confirm our theoretical findings.

This work is motivated by the challenge organized for the 10th International Conference on Extreme-Value Analysis (EVA2017) to predict daily precipitation quantiles at the 99.8%\\(99.8\\%\\) level for each month at observed and unobserved locations. Our approach is based on a Bayesian generalized additive modeling framework that is designed to estimate complex trends in marginal extremes over space and time. First, we estimate a high non-stationary threshold using a gamma distribution for precipitation intensities that incorporates spatial and temporal random effects. Then, we use the Bernoulli and generalized Pareto (GP) distributions to model the rate and size of threshold exceedances, respectively, which we also assume to vary in space and time. The latent random effects are modeled additively using Gaussian process priors, which provide high flexibility and interpretability. We develop a penalized complexity (PC) prior specification for the tail index that shrinks the GP model towards the exponential distribution, thus preventing unrealistically heavy tails. Fast and accurate estimation of the posterior distributions is performed thanks to the integrated nested Laplace approximation (INLA). We illustrate this methodology by modeling the daily precipitation data provided by the EVA2017 challenge, which consist of observations from 40 stations in the Netherlands recorded during the period 1972–2016. Capitalizing on INLA’s fast computational capacity and powerful distributed computing resources, we conduct an extensive cross-validation study to select the model parameters that govern the smoothness of trends. Our results clearly outperform simple benchmarks and are comparable to the best-scoring approaches of the other teams.

We study the asymptotic properties of bridge estimators in sparse, high-dimensional, linear regression models when the number of covariates may increase to infinity with the sample size. We are particularly interested in the use of bridge estimators to distinguish between covariates whose coefficients are zero and covariates whose coefficients are nonzero. We show that under appropriate conditions, bridge estimators correctly select covariates with nonzero coefficients with probability converging to one and that the estimators of nonzero coefficients have the same asymptotic distribution that they would have if the zero coefficients were known in advance. Thus, bridge estimators have an oracle property in the sense of Fan and Li [J. Amer. Statist. Assoc. 96 (2001) 1348-1360] and Fan and Peng [Ann. Statist. 32 (2004) 928-961]. In general, the oracle property holds only if the number of covariates is smaller than the sample size. However, under a partial orthogonality condition in which the covariates of the zero coefficients are uncorrelated or weakly correlated with the covariates of nonzero coefficients, we show that marginal bridge estimators can correctly distinguish between covariates with nonzero and zero coefficients with probability converging to one even when the number of covariates is greater than the sample size.

Zou [J. Amer. Statist. Assoc. 101 (2006) 1418—1429] proposed the Adaptive LASSO (ALASSO) method for simultaneous variable selection and estimation of the regression parameters, and established its oracle property. In this paper, we investigate the rate of convergence of the ALASSO estimator to the oracle distribution when the dimension of the regression parameters may grow to infinity with the sample size. It is shown that the rate critically depends on the choices of the penalty parameter and the initial estimator, among other factors, and that confidence intervals (CIs) based on the oracle limit law often have poor coverage accuracy. As an alternative, we consider the residual bootstrap method for the ALASSO estimators that has been recently shown to be consistent; cf. Chatterjee and Lahiri [J. Amer. Statist. Assoc. 106 (2011a) 608—625]. We show that the bootstrap applied to a suitable studentized version of the ALASSO estimator achieves second-order correctness, even when the dimension of the regression parameters is unbounded. Results from a moderately large simulation study show marked improvement in coverage accuracy for the bootstrap CIs over the oracle based CIs.

Generalized linear models (GLMs) are very widely used, but formal goodness-of-fit (GOF) tests for the overall fit of the model seem to be in wide use only for certain classes of GLMs. We develop and apply a new goodness-of-fit test, similar to the well-known and commonly used Hosmer–Lemeshow (HL) test, that can be used with a wide variety of GLMs. The test statistic is a variant of the HL statistic, but we rigorously derive an asymptotically correct sampling distribution using methods of Stute and Zhu (Scand J Stat 29(3):535–545, 2002) and demonstrate its consistency. We compare the performance of our new test with other GOF tests for GLMs, including a naive direct application of the HL test to the Poisson problem. Our test provides competitive or comparable power in various simulation settings and we identify a situation where a naive version of the test fails to hold its size. Our generalized HL test is straightforward to implement and interpret and an R package is publicly available.

Mazucheli et al. (J Appl Stat 46(4):700–714, 2019) introduced unit Lindley distribution by transforming Lindley (J Roy Stat Soc Ser B Stat Methodol 20(1):102–107, 1958) distribution for modelling proportion data. In this paper, we consider an exponential version of unit Lindley distribution. Various statistical and structural properties of the new distribution are discussed such as moments, hazard rate function, inequality measures, entropy etc. Different estimation methods are used to estimate the parameters of the model and their performances are demonstrated by Monte Carlo simulation. Finally the dominance of proposed distribution is embellished through real life data sets by comparing with some other unit distributions available in literature.

In real data analysis, the underlying model is frequently unknown. Hence, the modeling strategy plays a key role in the success of data analysis. Inspired by the idea of model averaging, we propose a novel semiparametric modeling strategy for the conditional quantile prediction, without assuming that the underlying model is any specific parametric or semiparametric model. Due to the optimality of the weights selected by leave-one-out cross-validation, the proposed modeling strategy provides a more precise prediction than those based on some commonly used semiparametric models such as the varying coefficient and additive models. Asymptotic properties are established in the proposed modeling strategy along with its estimation procedure. We conducted extensive simulations to compare our method with alternatives across various scenarios. The results show that our method provides more accurate predictions. Finally, we applied our approach to the Boston housing data, yielding more precise quantile predictions of house prices compared with commonly used methods, and thus offering a clearer picture of the Boston housing market.

A class of variable selection procedures for parametric models via non-concave penalized likelihood was proposed by Fan and Li to simultaneously estimate parameters and select important variables. They demonstrated that this class of procedures has an oracle property when the number of parameters is finite. However, in most model selection problems the number of parameters should be large and grow with the sample size. In this paper some asymptotic properties of the nonconcave penalized likelihood are established for situations in which the number of parameters tends to ∞ as the sample size increases. Under regularity conditions we have established an oracle property and the asymptotic normality of the penalized likelihood estimators. Furthermore, the consistency of the sandwich formula of the covariance matrix is demonstrated. Nonconcave penalized likelihood ratio statistics are discussed, and their asymptotic distributions under the null hypothesis are obtained by imposing some mild conditions on the penalty functions. The asymptotic results are augmented by a simulation study, and the newly developed methodology is illustrated by an analysis of a court case on the sexual discrimination of salary.

We compute the limiting distributions of the largest eigenvalue of a complex Gaussian sample covariance matrix when both the number of samples and the number of variables in each sample become large. When all but finitely many, say r, eigenvalues of the covariance matrix are the same, the dependence of the limiting distribution of the largest eigenvalue of the sample covariance matrix on those distinguished r eigenvalues of the covariance matrix is completely characterized in terms of an infinite sequence of new distribution functions that generalize the Tracy-Widom distributions of the random matrix theory. Especially a phase transition phenomenon is observed. Our results also apply to a last passage percolation model and a queueing model.