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Let Y = m ( X )+ ϵ be a regression model with a dichotomous output Y and a step function m with exact one jump at a point θ and two different levels a and b . In the applied sciences the parameter θ is interpreted as a split-point whereas b and 1- a are known as positive and negative predictive value, respectively. We prove n -consistency and a weak convergence type result for a two-step plug-in maximum likelihood estimator of θ . The limit variable is not normal, but a maximizing point of a compound Poisson process on the real line. Estimation of ( a,b ) yields the usual √ n -consistency with normal limit. Both results can be extended to a multivariate weak limit theorem. It allows for the construction of asymptotic confidence intervals for ( θ , a,b ). The theory is applied to real life data of a large epidemiological study.

We work with fractional Brownian motion with Hurst index H > 1/2. We show that the pricing model based on geometric fractional Brownian motion behaves to certain extend as a process with bounded variation. This observation is based on a new change of variables formula for a convex function composed with fractional Brownian motion. The stochastic integral in the change of variables formula is a Riemann–Stieltjes integral. We apply the change of variables formula to hedging of convex payoffs in this pricing model. It turns out that the hedging strategy is as if the pricing model was driven by a continuous process with bounded variation. This in turn allows us to construct new arbitrage strategies in this model. On the other hand our findings may be useful in connection to the corresponding pricing model with transaction costs.

We investigate the potential Bayesianity of maximum likelihood estimators (MLE), under absolute value error loss, for estimating the location parameter θ of symmetric and unimodal density functions in the presence of (i) a lower (or upper) bounded constraint, and (ii) an interval constraint, for θ . With these problems being expressed in terms of integral equations, we establish for logconcave densities: the generalized Bayesianity of the MLE in (i) ; and the proper Bayesianity and admissibility of the MLE in (ii) which extends the normal model result of Iwasa and Moritani. In (i) , a key feature concerns a correspondence with a Riemann–Hilbert problem, while in (ii) we use Fredholm´s technique and a contraction mapping argument. We demonstrate that logconcavity is a critical condition with sufficient conditions for non-Bayesianity and, accordingly, with a class of counterexamples. Note that the Bayesianity of the MLE under absolute value loss in the restricted location parameter case is in marked counterdistinction to that under quadratic loss, where, typically, a generalized Bayes estimator must be a smooth function. Finally, various other remarks, illustrations and numerical evaluations are provided.

We introduce a generalised subgradient for law-invariant closed convex risk measures on L 1 and establish its relationship with optimal risk allocations and equilibria. Our main result gives sufficient conditions ensuring a non-empty generalised subgradient.