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The Nadaraya-Watson estimator is certainly the most popular nonparametric regression estimator. The asymptotic bias and variance of this estimator, say $\hat{m}(x)$ , are well known. Nevertheless, its higher moments are rarely mentioned in the literature. In this paper, explicit formulas for asymptotic higher moments, such as $E((\hat{m}(x)-m(x))^{\gamma})$ or $E((\hat{m}(x)-E(\hat{m}(x)))^{\gamma}...
The Generalized Cumulative Entropy for residual life time (GCRE), introduced by Kumar and Taneja (Statist Probab Lett 81:1072–1077, 2011), is viewed as a dynamic measure of uncertainty. The present paper addresses the question of extending this measure to higher dimensions and study its properties. It is shown that the proposed measure uniquely determines the distribution. We use this measure to characterize...
This paper develops simple and efficient procedures for sampling approximations of the two-Parameter Poisson-Dirichlet Process and the normalized inverse-Gaussian process. We compare the efficiency of the new approximations with the corresponding stick-breaking approximations, in which we demonstrate a substantial improvement.
The mean discount rate for a simple stochastic model behaves asymptotically roughly like in contrast to the usual geometric discounting in a deterministic model.
For a simple versus simple hypothesis testing, in the continuous case, there exists an exact ancillary statistic, which is a function of the likelihood ratio. This fact may be used to assess error probabilities in a conditional sense. A simulation strategy is proposed to compute conditional error probabilities when these are not available in a closed form. As an application, discriminating between...
Theorem 2.2 of Majumdar (Sankhyā 67:670–673, 2005) obtained two conditions that are necessary and jointly sufficient for weak convergence in $ {\mathcal{M}}\left({H}\right)$ . In this note we significantly relax the second condition while maintaining the joint sufficiency.
The problem of estimating the density-weighted average derivative of a regression function is considered. We present a new consistent estimator based on a plug-in approach and wavelet projections. Its performances are explored under various dependence structures on the observations: the independent case, the ρ-mixing case and the α-mixing case. More precisely, denoting n the number of observations,...
We consider the problem of estimating the sum of squared means when the data (x1,...,xn) are independent values with xi ∼ N(θi, 1) and θ1, θ2... are a priori i.i.d. N(0, σ2) with σ2 known. This example has posed difficulties for many approaches to inference. We examine the consistency properties of several estimators derived from Bayesian considerations. We prove that a particular Bayesian...
It is common practice to make assertions about the symmetry or asymmetry of a probability density function based on coefficients of skewness. Since most coefficients of skewness are designed to be zero for a symmetric density, they do, overall, provide an indication of symmetry. However, skewness, as opposed to asymmetry, is primarily influenced by the tail behavior of a density function. Therefore,...
The class of generalized s-selfdecomposable probability distributions can be viewed as an image, via the random integral mapping , of the class ID of all infinitely divisible measures. We prove that a composition of the mappings , is again...
Florens, Richard and Rolin (2003) proposed a specification test of a parametric hypothesis against a nonparametric one, in the framework of a Bayesian encompassing test. Building on that work, this paper elaborates the procedure under a condition of partial observability. The general procedure is illustrated with the case where only the sign is observable, and more generally when the available data...
This study focuses on the nonparametric estimation of the conditional density of a scalar response variable given a random variable taking values in separable Hilbert space. We establish under general conditions the almost complete convergence rates of the conditional density estimator under α-mixing dependence, based on the single-index structure. We also demonstrate the impact of this functional...
An important problem in statistics is the construction of confidence regions for unknown parameters. In most cases, asymptotic distribution theory is used to construct confidence regions, so any coverage probability claims only hold approximately, for large samples. This paper describes a new approach, using random sets, which allows users to construct exact confidence regions without appeal to asymptotic...
This paper considers the problem of making inferences on a linear combination of means from independent normal distributions with unknown variances. While standard inference procedures require the assumption that the variances of the normal distributions are all equal or have known ratios, in this paper procedures are developed which allow the variances to take any unknown values. Bounds are developed...
In this paper, we define and discuss a class of generalized Wishart distributions under elliptical models. We derive the non-central moments of the likelihood ratio statistic for testing the equality of two covariance matrices under elliptical models for the corresponding matrices. Known classical expressions for the Gaussian model are then deduced from these general results. Finally, the exact distribution...
In this second part of the paper, we make use of the distribution of the trace of a generalized Wishart matrix based on elliptical models to derive the moments of statistic V used for testing sphericity for a general elliptical model. From the general expressions, we derive specific expressions for the special case of the Kotz family, which includes the Gaussian subfamily. Finally, to illustrate the...
In this paper, we investigate the problem of the local linear estimation of the cumulative distribution function of a real random variable Y conditioned by a functional variable X (valued in an infinite dimensional space). The almost-complete and the mean square consistencies, with rates, of the constructed estimator are stated. We precise that the exact expression involved in the leading terms of...
In this paper a new technique for monitoring shifts in covariance matrices of Gaussian processes is developed. The processes we monitor are obtained from the covariance matrices estimated using a single observation. These processes follow independent Gaussian distribution in the in-control state, thus allowing for application of standard control charts. Furthermore, in contrary to the existing literature,...
D. Basu gave a striking bivariate normal example, N2(0, 0, 1, 1, ρ) with an unknown correlation coefficient ρ, − 1 < ρ < 1, where the jointly sufficient statistic (X1, X2) consists of two ancillary statistics X1, X2. We exhibit examples of ancillary statistics involving both X1, X2 followed by other variations. A situation is highlighted where a jointly minimal sufficient statistic (X...
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