A comparative study of monotone nonparametric kernel estimates

In this paper we present a detailed numerical comparison of three monotone nonparametric kernel regression estimates, which isotonize a nonparametric curve estimator. The first estimate is the classical smoothed isotone estimate of Brunk (1958). The second method has recently been proposed by Hall and Huang (2001) and modifies the weights of a commonly used kernel estimate such that the resulting estimate is monotone. The third estimate was recently proposed by Dette, Neumeyer and Pilz (2003) and combines density and regression estimation techniques to obtain a monotone curve estimate of the inverse of the isotone regression function. The three concepts are briefly reviewed and their finite sample properties are studied by means of a simulation study. Although all estimates are first order asymptotically equivalent (provided that the unknown regression function is isotone) some differences for moderate samples are observed. --isotonic regression,order restricted inference,Nadaraya-Watson estimator,local linear regression,monte carlo simulation

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

A note on nonparametric estimation of the effective dose in quantal bioassay

For the common binary response model we propose a direct method for the nonparametric estimation of the effective dose level ED? (0Binary response model,effective dose level,nonparametric regression,isotonic regression,order restricted inference,local linear regression

A simple nonparametric estimator of a monotone regression function

In this paper a new method for monotone estimation of a regression function is proposed. The estimator is obtained by the combination of a density and a regression estimate and is appealing to users of conventional smoothing methods as kernel estimators, local polynomials, series estimators or smoothing splines. The main idea of the new approach is to construct a density estimate from the estimated values ˆm(i/N) (i = 1, . . . ,N) of the regression function to use these “data” for the calculation of an estimate of the inverse of the regression function. The final estimate is then obtained by a numerical inversion. Compared to the conventially used techniques for monotone estimation the new method is computationally more efficient, because it does not require constrained optimization techniques for the calculation of the estimate. We prove asymptotic normality of the new estimate and compare the asymptotic properties with the unconstrained estimate. In particular it is shown that for kernel estimates or local polynomials the monotone estimate is first order asymptotically equivalent to the unconstrained estimate. We also illustrate the performance of the new procedure by means of a simulation study. --isotonic regression,order restricted inference,Nadaraya-Watson estimator,local linear regression

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 asymptot- ically normal. In a simulation study we compare the new estimator to the unconstrained kernel estimator and to the estimator given in Aıt-Sahalia and Duarte (2003)

Parameterized Neural Networks for Finance

We discuss and analyze a neural network architecture, that enables learning a model class for a set of different data samples rather than just learning a single model for a specific data sample. In this sense, it may help to reduce the overfitting problem, since, after learning the model class over a larger data sample consisting of such different data sets, just a few parameters need to be adjusted for modeling a new, specific problem. After analyzing the method theoretically and by regression examples for different one-dimensional problems, we finally apply the approach to one of the standard problems asset managers and banks are facing: the calibration of spread curves. The presented results clearly show the potential that lies within this method. Furthermore, this application is of particular interest to financial practitioners, since nearly all asset managers and banks which are having solutions in place may need to adapt or even change their current methodologies when ESG ratings additionally affect the bond spreads.Comment: 24 pages, 17 figure

Flip Distance Between Triangulations of a Simple Polygon is NP-Complete

Let T be a triangulation of a simple polygon. A flip in T is the operation of removing one diagonal of T and adding a different one such that the resulting graph is again a triangulation. The flip distance between two triangulations is the smallest number of flips required to transform one triangulation into the other. For the special case of convex polygons, the problem of determining the shortest flip distance between two triangulations is equivalent to determining the rotation distance between two binary trees, a central problem which is still open after over 25 years of intensive study. We show that computing the flip distance between two triangulations of a simple polygon is NP-complete. This complements a recent result that shows APX-hardness of determining the flip distance between two triangulations of a planar point set.Comment: Accepted versio