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

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

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)

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

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? (0 < ? < 1). The estimator is obtained by the composition of a nonparametric estimate of the quantile response curve and a classical density estimate. The new method yields a simple and reliable monotone estimate of the effective dose level curve ? ? ED? and is appealing to users of conventional smoothing methods as kernel estimators, local polynomials, series estimators or smoothing splines. Moreover it is computationally very efficient, because it does not require a numerical inversion of a monotonized estimate of the quantile dose response curve. We prove asymptotic normality of the new estimate and compare it with an available alternative estimate (based on a monotonized nonparametric estimate of the dose response curve and the calculation of the inverse function) by means of a simulation study