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Том 28, № 2 (2019)
- Год: 2019
- Статей: 5
- URL: https://journals.rcsi.science/1066-5307/issue/view/13910
Article
Asymptotic Theory for Longitudinal Data with Missing Responses Adjusted by Inverse Probability Weights
Аннотация
In this article, we propose a new method for analyzing longitudinal data which contain responses that are missing at random. This method consists in solving the generalized estimating equation (GEE) of [8] in which the incomplete responses are replaced by values adjusted using the inverse probability weights proposed in [17]. We show that the root estimator is consistent and asymptotically normal, essentially under the some conditions on the marginal distribution and the surrogate correlation matrix as those presented in [15] in the case of complete data, and under minimal assumptions on the missingness probabilities. This method is applied to a real-life data set taken from [13], which examines the incidence of respiratory disease in a sample of 250 pre-school age Indonesian children which were examined every 3 months for 18 months, using as covariates the age, gender, and vitamin A deficiency.
83-103
The Empirical Process of Residuals from an Inverse Regression
Аннотация
In this paper we investigate an indirect regression model characterized by the Radon transformation. This model is useful for recovery of medical images obtained by computed tomography scans. The indirect regression function is estimated using a series estimator motivated by a spectral cutoff technique. Further, we investigate the empirical process of residuals from this regression, and show that it satisfies a functional central limit theorem.
104-126
On Predictive Density Estimation under α-Divergence Loss
Аннотация
Based on X ∼ Nd(θ, σX2Id), we study the efficiency of predictive densities under α-divergence loss Lα for estimating the density of Y ∼ Nd(θ, σY2Id). We identify a large number of cases where improvement on a plug-in density are obtainable by expanding the variance, thus extending earlier findings applicable to Kullback-Leibler loss. The results and proofs are unified with respect to the dimension d, the variances σX2 and σY2, the choice of loss Lα; α ∈ (−1, 1). The findings also apply to a large number of plug-in densities, as well as for restricted parameter spaces with θ ∈ Θ ⊂ ℝd. The theoretical findings are accompanied by various observations, illustrations, and implications dealing for instance with robustness with respect to the model variances and simultaneous dominance with respect to the loss.
127-143
On the Asymptotic Power of Tests of Fit under Local Alternatives in Autoregression
Аннотация
We consider a stationary AR(p) model. The autoregression parameters are unknown as well as the distribution of innovations. Based on the residuals from the parameter estimates, an analog of empirical distribution function is defined and the tests of Kolmogorov’s and ω2 type are constructed for testing hypotheses on the distribution of innovations. We obtain the asymptotic power of these tests under local alternatives.
144-154
A Multiple Hypothesis Testing Approach to Detection Changes in Distribution
Аннотация
Let X1, X2,... be independent random variables observed sequentially and such that X1,..., Xθ−1 have a common probability density p0, while Xθ, Xθ+1,... are all distributed according to p1 ≠ p0. It is assumed that p0 and p1 are known, but the time change θ ∈ ℤ+ is unknown and the goal is to construct a stopping time τ that detects the change-point θ as soon as possible. The standard approaches to this problem rely essentially on some prior information about θ. For instance, in the Bayes approach, it is assumed that θ is a random variable with a known probability distribution. In the methods related to hypothesis testing, this a priori information is hidden in the so-called average run length. The main goal in this paper is to construct stopping times that are free from a priori information about θ. More formally, we propose an approach to solving approximately the following minimization problem:
where α(θ; τ) = Pθ{τ < θ} is the false alarm probability and Δ(θ; τ) = Eθ(τ − θ)+ is the average detection delay computed for a given stopping time τ. In contrast to the standard CUSUM algorithm based on the sequential maximum likelihood test, our approach is related to a multiple hypothesis testing methods and permits, in particular, to construct universal stopping times with nearly Bayes detection delays.
155-167