Shape constrained kernel density estimation

In this paper, a method for estimating monotone, convex and log-concave densities is proposed. The estimation procedure consists of an unconstrained kernel estimator which is modi?ed in a second step with respect to the desired shape constraint by using monotone rearrangements. It is shown that the resulting estimate is a density itself and shares the asymptotic properties of the unconstrained estimate. A short simulation study shows the ?nite sample behavior. --Convexity,log-concavity,monotone rearrangements,monotonicity,nonparametric density estimation

Central limit theorems for the integrated squared error of derivative estimators

A central limit theorem for the weighted integrated squared error of kernel type estimators of the first two derivatives of a nonparametric regression function is proved by using results for martingale differences and U-statistics. The results focus on the setting of the Nadaraya-Watson estimator but can also be transfered to local polynomial estimates. --central limit theorem,integrated squared error,kernel estimates,local polynomial estimate,Nadaraya-Watson estimate,nonparametric regression

Shape constrained estimators in inverse regression models with convolution-type operator

In this paper we are concerned with shape restricted estimation in inverse regression problems with convolution-type operator. We use increasing rearrangements to compute increasingand convex estimates from an (in principle arbitrary) unconstrained estimate of the unknown regression function. An advantage of our approach is that it is not necessary that prior shape information is known to be valid on the complete domain of the regression function. Instead, it is sufficient if it holds on some compact interval. A simulation study shows that the shape restricted estimate on the respective interval is significantly less sensitive to moderate undersmoothing than the unconstrained estimate, which substantially improves applicability of estimates based on data-driven bandwidth estimators. Finally, we demonstrate the application of the increasing estimator by the estimation of the luminosity profile of an elliptical galaxy. Here, a major interest is in reconstructing the central peak of the profile, which, due to its small size, requires to select the bandwidth as small as possible. --convexity,increasing rearrangements,image reconstruction,inverse problems,monotonicity,order restricted inference,regression estimation,shape restrictions

Nonparametric option pricing with no-arbitrage constraints

We propose a completely kernel based method of estimating the call price function or the state price density of options. The new estimator of the call price function fulfills the constraints like monotonicity and convexity given in Breeden and Litzenberger (1978) without necessarily estimating the state price density for an underlying asset price from its option prices. It can be shown that the estimator is pointwise consistent and asymptotically normal. In a simulation study we compare the new estimator to the unconstrained kernel estimator and to the estimator given in Aït-Sahalia and Duarte (2003). --call pricing function b,constrained nonparametric estimation,monotone rearrangements,state price density

Neuroscience and Settlement: An Examination of Scientific Innovations and Practical Applications

Published in cooperation with the American Bar Association Section of Dispute Resolutio

Central limit theorems for the integrated squared error of derivative estimators

A central limit theorem for the weighted integrated squared error of kernel type estimators of the first two derivatives of a nonparametric regression function is proved by using results for martingale differences and U-statistics. The results focus on the setting of the Nadaraya- Watson estimator but can also be transfered to local polynomial estimates

Estimating a convex function in nonparametric regression

A new nonparametric estimate of a convex regression function is proposed and its stochastic properties are studied. The method starts with an unconstrained estimate of the derivative of the regression function, which is firstly isotonized and then integrated. We prove asymptotic normality of the new estimate and show that it is first order asymptotically equivalent to the initial unconstrained estimate if the regression function is in fact convex. If convexity is not present the method estimates a convex function whose derivative has the same Lp-norm as the derivative of the (non-convex) underlying regression function. The finite sample properties of the new estimate are investigated by means of a simulation study and the application of the new method is demonstrated in two data examples