We study the asymptotic behavior of the marginal expected shortfall when the two random variables are asymptotic independent but positively associated, which is modeled by the so-called tail dependent coefficient. We construct an estimator of the marginal expected shortfall, which is shown to be asymptotically normal. The finite sample performance of the estimator is investigated in a small simulation study. The method is also applied to estimate the expected amount of rainfall at a weather station given that there is a once every 100 years rainfall at another weather station nearby.

We consider Grenander-type estimators for a monotone function (Formula presented.), obtained as the slope of a concave (convex) estimate of the primitive of λ. Our main result is a central limit theorem for the Hellinger loss, which applies to estimation of a probability density, a regression function or a failure rate. In the case of density estimation, the limiting variance of the Hellinger loss turns out to be independent of λ.

We consider the smoothed maximum likelihood estimator and the smoothed Grenander-type estimator for a monotone baseline hazard rate 0 in the Cox model. We analyze their asymptotic behaviour and show that they are asymptotically normal at rate nm=.2mC1/, when 0 is m 2 times continuously differentiable, and that both estimators are asymptotically equivalent. Finally, we present numerical results on pointwise confidence intervals that illustrate the comparable behaviour of the two methods.

We consider the process Λ̂_n−Λ_n, where Λ_n is a cadlag step estimator for the primitive Λ of a nonincreasing function λ on [0,1], and Λ̂_n is the least concave majorant of Λ_n. We extend the results in Kulikov and Lopuhaä (2006, 2008) to the general setting considered in Durot (2007). Under this setting we prove that a suitably scaled version of Λ̂_n−Λ_n converges in distribution to the corresponding process for two-sided Brownian motion with parabolic drift and we establish a central limit theorem for the L_p-distance between Λ̂_n and Λ_n.

We consider two isotonic smooth estimators for a monotone baseline hazard in the Cox model, a maximum smooth likelihood estimator and a Grenander-type estimator based on the smoothed Breslow estimator for the cumulative baseline hazard. We show that they are both asymptotically normal at rate n^m∕(2m+1), where m≥2 denotes the level of smoothness considered, and we relate their limit behavior to kernel smoothed isotonic estimators studied in Lopuhaä and Musta (2016). It turns out that the Grenander-type estimator is asymptotically equivalent to the kernel smoothed isotonic estimators, while the maximum smoothed likelihood estimator exhibits the same asymptotic variance but a different bias. Finally, we present numerical results on pointwise confidence intervals that illustrate the comparable behavior of the two methods.