Central limit theorem for the empirical process of a linear sequence with long memory

We discuss the functional central limit theorem (FCLT) for the empirical process of a moving-average stationary sequence with long memory. The cases of one-sided and double-sided moving averages are discussed. In the case of one-sided (causal) moving average, the FCLT is obtained under weak conditions of smoothness of the distribution and the existence of (2+δ)-moment of i.i.d. innovations, by using the martingale difference decomposition due to Ho and Hsing (1996, Ann. Statist. 24, 992–1014). In the case of double-sided moving average, the proof of the FCLT is based on an asymptotic expansion of the bivariate probability density.     

M-Estimation in the partially linear model with Bernstein polynomials under shape constrains

We develope an M-estimator for partially linear models in which the nonparametric component is subject to various shape constraints. Bernstein polynomials are used to approximate the unknown nonparametric function, and shape constraints are imposed on the coefficients. Asymptotic normality of regression parameters and the optimal rate of convergence of the shape-restricted nonparametric function estimator are established under very mild conditions. Some simulation studies and a real data analysis are conducted to evaluate the finite sample performance of the proposed method.     

Some Estimates for the Rate of Convergence in the Multidimensional Central Limit Theorem for Homogeneous Markov Chains

This paper obtains some estimates for the rate of convergence in the multi-dimensional central limit theorem for vector-valued functions of a homogeneous Markov chain without assuming the finiteness of their absolute third moment. These estimates have a universal character and generalize the results that hold when the third moments are finite.     

An adaptive randomized design with application to estimation

When allocating observations to two populations for estimation or testing, the optimal proportion of the data that should be allocated to the first population, if it exists, often depends on unknown parameters. Adaptive designs have thus been proposed, in which allocation of the next observation is based on an estimate of the optimal proportion computed from the data already gathered. The authors introduce a simple randomized adaptive design and give some of its properties. Applications are given to estimating the difference of two success probabilities, and the difference of two normal means.     

Asymptotically optimal Berry-Esseen-type bounds for distributions with an absolutely continuous part

Recursive and closed form upper bounds are offered for the Kolmogorov and the total variation distance between the standard normal distribution and the distribution of a standardized sum of n independent and identically distributed random variables. The method employed is a modification of the method of compositions along with Zolotarev's ideal metric. The approximation error in the CLT obtained vanishes at a rate O(n^−k/2+1), provided that the common distribution of the summands possesses an absolutely continuous part, and shares the same k−1 (k?3) first moments with the standard normal distribution. Moreover, for the first time, these new uniform Berry-Esseen-type bounds are asymptotically optimal, that is, the ratio of the true distance to the respective bound converges to unity for a large class of distributions of the summands. Thus, apart from the correct rate, the proposed error estimates incorporate an optimal asymptotic constant (factor). Finally, three illustrative examples are presented along with numerical comparisons revealing that the new bounds are sharp enough even to be used in practical statistical applications.     

Estimation of extinction probability of a multitype markov branching process

Maximum likelihood estimation of the probability of ultimate extinction of a multitype Markov branching process is studied when the process is observed completely over a fixed time interval [0,t]. The asymptotic properties of the estimator are discussed.     

Testing exchangeability of copulas in arbitrary dimension

A test for exchangeability of copulas for arbitrary dimensions is proposed, generalising and extending a result by Genest et al. [(2012), ‘Tests of Symmetry for Bivariate Copulas’, Annals of the Institute of Statistical Mathematics, 64, 811–834]. Three test statistics together with some modifications are presented and their asymptotical behaviour is analysed. Empirical p-values are computed by using a bootstrap-procedure proposed by Rémillard and Scaillet [(2009), ‘Testing for Equality between Two Copulas’, Journal of Multivariate Analysis, 100, 377–386] and suggested by Bücher and Dette [(2010), ‘A Note on Bootstrap Approximations for the Empirical Copula Process’, Statistics & Probability Letters, 80, 1925–1932], based on a multiplier central limit theorem by van der Vaart and Wellner [(1996), Weak Convergence and Empirical Processes, Springer Series in Statistics, New York: Springer]. Finally a simulation study compares various versions of the proposed tests.     

Estimation of impulse response functions in two-output systems

Abstract

A single input–double output (SIDO) linear time-invariant (LTI) system is considered, whose impulse response function (IRF) is assumed to have one unknown component. The problem is to estimate this unknown component after observations of the second component. Both IRF’s components are supposed to be L₂-integrable, and the estimation is made by cross-correlating the outputs, given that the input is a standard Wiener process on R. Weak asymptotic normality of appropriately centred estimators is proved.     

Estimation of Correlation for Continuous Semimartingales

Abstract. In this study we are concerned with inference on the correlation parameter ρ of two Brownian motions, when only high‐frequency observations from two one‐dimensional continuous Itô semimartingales, driven by these particular Brownian motions, are available. Estimators for ρ are constructed in two situations: either when both components are observed (at the same time), or when only one component is observed and the other one represents its volatility process and thus has to be estimated from the data as well. In the first case it is shown that our estimator has the same asymptotic behaviour as the standard one for i.i.d. normal observations, whereas a feasible estimator can still be defined in the second framework, but with a slower rate of convergence.     

Testing for Bivariate Extreme Dependence Using Kendall's Process

Abstract. New tests for the hypothesis of bivariate extreme‐value dependence are proposed. All test statistics that are investigated are continuous functionals of either Kendall's process or its version with estimated parameters. The procedures considered are based on linear combinations of moments and on Cramér–von Mises distances. A suitably adapted version of the multiplier central limit theorem for Kendall's process enables the computation of asymptotically valid p‐values. The power of the tests is evaluated for small, moderate and large sample sizes, as well as asymptotically, under local alternatives. An illustration with a real data set is presented.     

A uniform central limit theorem for neural network-based autoregressive processes with applications to change-point analysis

We consider an autoregressive process with a nonlinear regression function that is modelled by a feedforward neural network. First, we derive a uniform central limit theorem which is useful in the context of change-point analysis. Then, we propose a test for a change in the autoregression function which – by the uniform central limit theorem – has asymptotic power one for a large class of alternatives including local alternatives not restricted to the correctly specified model.     

Proportional Trapping-Removal Model with a Known Ratio

ABSTRACT

Population size estimator is derived for a proportional trapping-removal model with a known ratio between two sub-population sizes, and the corresponding asymptotic properties is obtained. The performance of the proposed estimator is checked via simulation studies and an example.     

Model-based confidence bands for survival functions

This paper focuses on a novel method of developing one-sample confidence bands for survival functions from right censored data. The approach is model-based, relying on a parametric model for the conditional expectation of the censoring indicator given the observed minimum, and derives its strength from easy access to a good-fitting model among a plethora of choices available for binary response data. The substantive methodological contribution is in exploiting a semiparametric estimator of the survival function to produce improved simultaneous confidence bands. To obtain critical values for computing the confidence bands, a two-stage bootstrap approach that combines the classical bootstrap with the more recent model-based regeneration of censoring indicators is proposed and a justification of its asymptotic validity is also provided. Several different confidence bands are studied using the proposed approach. Numerical studies, including robustness of the proposed bands to misspecification, are carried out to check efficacy. The method is illustrated using two lung cancer data sets.     

Bayes and Empirical Bayes Two-Stage Procedures for Sampling Inspection

A batch of M items is inspected for defectives. Suppose there are d defective items in the batch. Let d ₀ be a given standard used to evaluate the quality of the population where 0 < d ₀ < M. The problem of testing H ₀: d < d ₀ versus H ₁: d ≥ d ₀ is considered. It is assumed that past observations are available when the current testing problem is considered. Accordingly, the empirical Bayes approach is employed. By using information obtained from the past data, an empirical Bayes two-stage testing procedure is developed. The associated asymptotic optimality is investigated. It is proved that the rate of convergence of the empirical Bayes two-stage testing procedure is of order O (exp(? c? n)), for some constant c? > 0, where n is the number of past observations at hand.     

Asymptotic results for random coefficient bifurcating autoregressive processes

The purpose of this paper is to study the asymptotic behaviour of the weighted least-squares estimators of the unknown parameters of random coefficient bifurcating autoregressive processes. Under suitable assumptions on the immigration and the inheritance, we establish the almost sure convergence of our estimators, as well as a quadratic strong law and central limit theorems. Our study mostly relies on limit theorems for vector-valued martingales.     

A test for randomness of the environments in a branching process

This paper discusses an approximate score test for testing randomness of environments in a branching process without observing the environments. Using an appropriate martingale central limit theorem the asymptotic null distribution of test statistic is shown to be normal. When the offspring distribution is Poisson, the detail derivation of asymptotic distribution of the test statistic is presented.