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Let { Z (τ , s) , (τ , s) ∈ [ a , b ] × [ 0 , T ] } with some positive constants a, b, T be a centered Gaussian random field with variance function σ 2 (τ , s) satisfying σ 2 (τ , s) = σ 2 (τ) . We first derive the exact tail asymptotics (as u → ∞ ) for the probability that the maximum M H (T) = max (τ , s) ∈ [ a , b ] × [ 0 , T ] [ Z (τ , s) / σ (τ) ] exceeds a given level u, for any fixed 0 < a < b < ∞ and T > 0 ; and we further derive the extreme limit law for M H (T) . As applications of the main results, we derive the exact tail asymptotics and the extreme limit laws for Shepp statistics with stationary Gaussian process, fractional Brownian motion and Gaussian integrated process as inputs. [ABSTRACT FROM AUTHOR]

In this paper, we consider how to incorporate quantile information to improve estimator efficiency for regression model with missing covariates. We combine the quantile information with least-squares normal equations and construct an unbiased estimating equations (EEs). The lack of smoothness of the objective EEs is overcome by replacing them with smooth approximations. The maximum smoothed empirical likelihood (MSEL) estimators are established based on inverse probability weighted (IPW) smoothed EEs and their asymptotic properties are studied under some regular conditions. Moreover, we develop two novel testing procedures for the underlying model. The finite-sample performance of the proposed methodology is examined by simulation studies. A real example is used to illustrate our methods. [ABSTRACT FROM AUTHOR]