Vol 28, No 2 (2019)

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.

Based on X ∼ N_d(θ, σ_X²I_d), we study the efficiency of predictive densities under α-divergence loss L_α for estimating the density of Y ∼ N_d(θ, σ_Y²I_d). 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 σ_X² and σ_Y², the choice of loss L_α; 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 ⊂ ℝ^d. 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.

Let X₁, X₂,... be independent random variables observed sequentially and such that X₁,..., X_θ−1 have a common probability density p₀, while X_θ, X_θ+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:

\(\Delta(\theta;{\tau^\alpha})\rightarrow\min_{\tau^\alpha}\;\;\text{subject}\;\text{to}\;\;\alpha(\theta;{\tau^\alpha})\leq\alpha\;\text{for}\;\text{any}\;\theta\geq1,\)

where α(θ; τ) = P_θ{τ < θ} is the false alarm probability and Δ(θ; τ) = E_θ(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.