Very fast optimal bandwidth selection for univariate kernel density estimation

Most automatic bandwidth selection procedures for kernel density estimates require estimation of quantities involving the density derivatives. Estimation of modes and inflexion points of densities also require derivative estimates. The computational complexity of evaluating the density derivative at M evaluation points given N sample points from the density is O(MN). In this paper we propose a computationally efficient $\epsilon$-exact approximation algorithm for univariate, Gaussian kernel based, density derivative estimation that reduces the computational complexity from O(MN) to linear order (O(N+M)). The constant depends on the desired arbitrary accuracy, $\epsilon$. We apply the density derivative evaluation procedure to estimate the optimal bandwidth for kernel density estimation, a process that is often intractable for large data sets. For example for N = M = 409,600 points while the direct evaluation of the density derivative takes around 12.76 hours the fast evaluation requires only 65 seconds with an error of around 10^{-12). Algorithm details, error bounds, procedure to choose the parameters and numerical experiments are presented. We demonstrate the speedup achieved on the bandwidth selection using the ``solve-the-equation plug-in method'' [18]. We also demonstrate that the proposed procedure can be extremely useful for speeding up exploratory projection pursuit techniques

Mode-Seeking Clustering and Density Ridge Estimation via Direct Estimation of Density-Derivative-Ratios

Modes and ridges of the probability density function behind observed data are useful geometric features. Mode-seeking clustering assigns cluster labels by associating data samples with the nearest modes, and estimation of density ridges enables us to find lower-dimensional structures hidden in data. A key technical challenge both in mode-seeking clustering and density ridge estimation is accurate estimation of the ratios of the first- and second-order density derivatives to the density. A naive approach takes a three-step approach of first estimating the data density, then computing its derivatives, and finally taking their ratios. However, this three-step approach can be unreliable because a good density estimator does not necessarily mean a good density derivative estimator, and division by the estimated density could significantly magnify the estimation error. To cope with these problems, we propose a novel estimator for the density-derivative-ratios. The proposed estimator does not involve density estimation, but rather directly approximates the ratios of density derivatives of any order. Moreover, we establish a convergence rate of the proposed estimator. Based on the proposed estimator, novel methods both for mode-seeking clustering and density ridge estimation are developed, and the respective convergence rates to the mode and ridge of the underlying density are also established. Finally, we experimentally demonstrate that the developed methods significantly outperform existing methods, particularly for relatively high-dimensional data.Peer reviewe

Nonparametric Estimation of Risk-Neutral Densities

This chapter deals with nonparametric estimation of the risk neutral density. We present three different approaches which do not require parametric functional assumptions on the underlying asset price dynamics nor on the distributional form of the risk neutral density. The first estimator is a kernel smoother of the second derivative of call prices, while the second procedure applies kernel type smoothing in the implied volatility domain. In the conceptually different third approach we assume the existence of a stochastic discount factor (pricing kernel) which establishes the risk neutral density conditional on the physical measure of the underlying asset. Via direct series type estimation of the pricing kernel we can derive an estimate of the risk neutral density by solving a constrained optimization problem. The methods are compared using European call option prices. The focus of the presentation is on practical aspects such as appropriate choice of smoothing parameters in order to facilitate the application of the techniques.Risk neutral density, Pricing kernel, Kernel smoothing, Local polynomials, Series methods

Uniform confidence bands for pricing kernels

Pricing kernels implicit in option prices play a key role in assessing the risk aversion over equity returns. We deal with nonparametric estimation of the pricing kernel (Empirical Pricing Kernel) given by the ratio of the risk-neutral density estimator and the subjective density estimator. The former density can be represented as the second derivative w.r.t. the European call option price function, which we estimate by nonparametric regression. The subjective density is estimated nonparametrically too. In this framework, we develop the asymptotic distribution theory of the EPK in the L1 sense. Particularly, to evaluate the overall variation of the pricing kernel, we develop a uniform confidence band of the EPK. Furthermore, as an alternative to the asymptotic approach, we propose a bootstrap confidence band. The developed theory is helpful for testing parametric specifications of pricing kernels and has a direct extension to estimating risk aversion patterns. The established results are assessed and compared in a Monte-Carlo study. As a real application, we test risk aversion over time induced by the EPK.Empirical Pricing Kernel, Confidence band, Bootstrap; Kernel Smoothing; Nonparametric

Optimal change-point estimation from indirect observations

We study nonparametric change-point estimation from indirect noisy observations. Focusing on the white noise convolution model, we consider two classes of functions that are smooth apart from the change-point. We establish lower bounds on the minimax risk in estimating the change-point and develop rate optimal estimation procedures. The results demonstrate that the best achievable rates of convergence are determined both by smoothness of the function away from the change-point and by the degree of ill-posedness of the convolution operator. Optimality is obtained by introducing a new technique that involves, as a key element, detection of zero crossings of an estimate of the properly smoothed second derivative of the underlying function.Comment: Published at http://dx.doi.org/10.1214/009053605000000750 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org