Density estimation in the uniform deconvolution model

We consider the problem of estimating a probability density function based on data that are corrupted by noise from a uniform distribution. The (nonparametric) maximum likelihood estimator for the corresponding distribution function is well defined. For the density function this is not the case. We study two nonparametric estimators for this density. The first is a type of kernel density estimate based on the empirical distribution function of the observable data. The second is a kernel density estimate based on the MLE of the distribution function of the unobservable (uncorrupted) data.     

Combining kernel estimators in the uniform deconvolution problem

We construct a density estimator and an estimator of the distribution function in the uniform deconvolution model. The estimators are based on inversion formulas and kernel estimators of the density of the observations and its derivative. Initially the inversions yield two different estimators of the density and two estimators of the distribution function. We construct asymptotically optimal convex combinations of these two estimators. We also derive pointwise asymptotic normality of the resulting estimators, the pointwise asymptotic biases and an expansion of the mean integrated squared error of the density estimator. It turns out that the pointwise limit distribution of the density estimator is the same as the pointwise limit distribution of the density estimator introduced by Groeneboom and Jongbloed (Neerlandica, 57, 2003, 136), a kernel smoothed nonparametric maximum likelihood estimator of the distribution function.     

Estimation of dynamic models with nonparametric simulated maximum likelihood

We propose an easy-to-implement simulated maximum likelihood estimator for dynamic models where no closed-form representation of the likelihood function is available. Our method can handle any simulable model without latent dynamics. Using simulated observations, we nonparametrically estimate the unknown density by kernel methods, and then construct a likelihood function that can be maximized. We prove that this nonparametric simulated maximum likelihood (NPSML) estimator is consistent and asymptotically efficient. The higher-order impact of simulations and kernel smoothing on the resulting estimator is also analyzed; in particular, it is shown that the NPSML does not suffer from the usual curse of dimensionality associated with kernel estimators. A simulation study shows good performance of the method when employed in the estimation of jump-diffusion models.     

Density estimates of low bias

Two methods are given for adapting a kernel density estimate to obtain an estimate of a density function with bias O(h ^p ) for any given p, where h = h(n) is the bandwidth and n is the sample size. The first method is standard. The second method is new and involves use of Bell polynomials. The second method is shown to yield smaller biases and smaller mean squared errors than classical kernel density estimates and those due to Jones et al. (Biometrika 82:327–338, 1995).     

Remarks on the strong law of large numbers for a triangular array of associated random variables

A strong law of large numbers for a triangular array of strictly stationary associated random variables is proved. It is used to derive the pointwise strong consistency of kernel type density estimator of the one-dimensional marginal density function of a strictly stationary sequence of associated random variables, and to obtain an improved version of a result by Van Ryzin (1969) on the strong consistency of density estimator for a sequence of independent and identically distributed random variables.     

Implied risk aversion and pricing kernel in the FTSE 100 index

This paper studies the estimation of the pricing kernel and explains the pricing kernel puzzle found in the FTSE 100 index. We use prices of options and futures on the FTSE 100 index to derive the risk neutral density (RND). The option-implied RND is inverted by using two nonparametric methods: the implied-volatility surface interpolation method and the positive convolution approximation (PCA) method. The actual density distribution is estimated from the historical data of the FTSE 100 index by using the threshold GARCH (TGARCH) model. The results show that the RNDs derived from the two methods above are relatively negatively skewed and fat-tailed, compared to the actual probability density, that is consistent with the phenomenon of “volatility smile.” The derived risk aversion is found to be locally increasing at the center, but decreasing at both tails asymmetrically. This is the so-called pricing kernel puzzle. The simulation results based on a representative agent model with two state variables show that the pricing kernel is locally increasing with the wealth at the level of 1 and is consistent with the empirical pricing kernel in shape and magnitude.     

Kernel density estimation for time series data

A time-varying probability density function, or the corresponding cumulative distribution function, may be estimated nonparametrically by using a kernel and weighting the observations using schemes derived from time series modelling. The parameters, including the bandwidth, may be estimated by maximum likelihood or cross-validation. Diagnostic checks may be carried out directly on residuals given by the predictive cumulative distribution function. Since tracking the distribution is only viable if it changes relatively slowly, the technique may need to be combined with a filter for scale and/or location. The methods are applied to data on the NASDAQ index and the Hong Kong and Korean stock market indices.     

Nonparametric model validations for hidden Markov models with applications in financial econometrics

We address the nonparametric model validation problem for hidden Markov models with partially observable variables and hidden states. We achieve this goal by constructing a nonparametric simultaneous confidence envelope for transition density function of the observable variables and checking whether the parametric density estimate is contained within such an envelope. Our specification test procedure is motivated by a functional connection between the transition density of the observable variables and the Markov transition kernel of the hidden states. Our approach is applicable for continuous-time diffusion models, stochastic volatility models, nonlinear time series models, and models with market microstructure noise.     

Direct density estimation as an ill-posed inverse estimation problem

A possible definition of ill-posedness in statistical estimation is the lack of qualitative robustness. In this sense direct density estimation shares ill-posedness with the more obviously ill-posed indirect density estimation models, of which it is a special case. A general construction pattern for estimators is proposed, based on suitable preconditioning, that works for both direct and indirect density estimation. Special emphasis is on its application to the direct case, where in general it yields delta-sequence estimators. More specifically both kernel and series type estimators are included depending on the choice of preconditioning operator. In particular sinc and other flattop kernel estimators emerge in a natural way.     

Equilibrium selection in games: the mollifier method

This paper introduces an embedding of a Nash equilibrium into a sequence of perturbed games, which achieves continuous differentiability of best responses by mollifying them over a continuously differentiable density with compact support (window size). Along any sequence with shrinking window size, the equilibria are single-valued whenever the function has a regular Jacobian and the set of equilibria where it is singular has measure zero. We achieve a further reduction of the equilibrium set by inserting within the embedding a procedure that eliminates the strict interior of equilibrium sets. The mollifier can be approximated consistently using kernel density regression, and we sketch a non-stationary stochastic optimization algorithm that uses this approximation and converges with probability one to an equilibrium of the original game.     

Robust estimators for estimating discontinuous functions

We study the asymptotic behavior of a wide class of kernel estimators for estimating an unknown regression function. In particular we derive the asymptotic behavior at discontinuity points of the regression function. It turns out that some kernel estimators based on outlier robust estimators are consistent at jumps.     

Fuzzy density estimation

A new approach to density estimation with fuzzy random variables (FRV) is developed. In this approach, three methods (histogram, empirical c.d.f., and kernel methods) are extended for density estimation based on α-cuts of FRVs.     

Three Sides of Smoothing: Categorical Data Smoothing, Nonparametric Regression, and Density Estimation

The past forty years have seen a great deal of research into the construction and properties of nonparametric estimates of smooth functions. This research has focused primarily on two sides of the smoothing problem: nonparametric regression and density estimation. Theoretical results for these two situations are similar, and multivariate density estimation was an early justification for the Nadaraya-Watson kernel regression estimator.
A third, less well-explored, strand of applications of smoothing is to the estimation of probabilities in categorical data. In this paper the position of categorical data smoothing as a bridge between nonparametric regression and density estimation is explored. Nonparametric regression provides a paradigm for the construction of effective categorical smoothing estimates, and use of an appropriate likelihood function yields cell probability estimates with many desirable properties. Such estimates can be used to construct regression estimates when one or more of the categorical variables are viewed as response variables. They also lead naturally to the construction of well-behaved density estimates using local or penalized likelihood estimation, which can then be used in a regression context. Several real data sets are used to illustrate these points.     

Robust normal reference bandwidth for kernel density estimation

Bandwidth selection is the main problem of kernel density estimation, the most popular method of density estimation. The classical normal reference bandwidth usually oversmoothes the density estimate. The existing hi-tech bandwidths have computational problems (even may not exist) and are not robust against outliers in the sample. A highly robust normal reference bandwidth is proposed, which adapts to different types of densities.     

A goodness-of-fit test for GARCH innovation density

We prove asymptotic normality of a suitably standardized integrated square difference between a kernel type error density estimator based on residuals and the expected value of the error density estimator based on innovations in GARCH models. This result is similar to that of Bickel–Rosenblatt under i.i.d. set up. Consequently the goodness-of-fit test for the innovation density of GARCH processes based on this statistic is asymptotically distribution free, unlike the tests based on the residual empirical process. A simulation study comparing the finite sample behavior of this test with Kolmogorov–Smirnov test and the test based on integrated square difference between the kernel density estimate and null density shows some superiority of the proposed test.     

Iterative Density Estimation from Contaminated Observations

We introduce an iterative procedure for estimating the unknown density of a random variable X from n independent copies of Y=X+ɛ, where ɛ is normally distributed measurement error independent of X. Mean integrated squared error convergence rates are studied over function classes arising from Fourier conditions. Minimax rates are derived for these classes. It is found that the sequence of estimators defined by the iterative procedure attains the optimal rates. In addition, it is shown that the sequence of estimators converges exponentially fast to an estimator within the class of deconvoluting kernel density estimators. The iterative scheme shows how, in practice, density estimation from indirect observations may be performed by simply correcting an appropriate ordinary density estimator. This allows to assess the effect that the perturbation due to contamination by ɛ has on the density to be estimated. We also suggest a method to select the smoothing parameter required by the iterative approach and, utilizing this method, perform a simulation study.     

Measure preserving derivatives and the pricing kernel puzzle

Recent empirical studies have found evidence of nonmonotonicity in the pricing kernels for a variety of market indices. This phenomenon is known as the pricing kernel puzzle. The payoff distribution pricing model of Dybvig predicts that the payoff distribution of a direct investment of $1 in a market index may be replicated by investing less than $1 in some derivative written on that market index whenever the associated pricing kernel is nondecreasing. Using the Hardy–Littlewood rearrangement inequality, we obtain an explicit solution for the cheapest replicating derivative, which we refer to as the optimal measure preserving derivative. The optimal measure preserving derivative is the permutation appearing in Ryff’s decomposition of the pricing kernel with respect to the market payoff measure. We compute optimal measure preserving derivatives corresponding to the estimated physical and risk neutral distributions in the paper by Jackwerth (2000) that first brought attention to the pricing kernel puzzle.     

金融资产厚尾分布及常用的风险度量

本文首先运用正态分布、带有位置-尺度参数的t分布、logistic分布、极值分布、-stable分布和核密度估计对上证综指收益率分布进行拟合,结果表明核密度估计优于其他分布。其次,在进行尾部风险拟合和度量风险方面,通过设定相关指标,在显著性水平为1%时,-stable分布更适合衡量风险程度,在此基础上提出了调和-stable分布,并得到一个同构表示解。最后,本文给出了蒙特卡洛-stable分布模拟和经验值下的MDD、DaR和CDaR,并得到了模型值和经验值之间的乘离率。     

Local multiplicative bias correction for asymmetric kernel density estimators

We consider semiparametric asymmetric kernel density estimators when the unknown density has support on [0,∞)$[0,\infty )$. We provide a unifying framework which relies on a local multiplicative bias correction, and contains asymmetric kernel versions of several semiparametric density estimators considered previously in the literature. This framework allows us to use popular parametric models in a nonparametric fashion and yields estimators which are robust to misspecification. We further develop a specification test to determine if a density belongs to a particular parametric family. The proposed estimators outperform rival non- and semiparametric estimators in finite samples and are easy to implement. We provide applications to loss data from a large Swiss health insurer and Brazilian income data.     

Semiparametric efficient adaptive estimation of asymmetric GARCH models

In this paper we derive a semiparametric efficient adaptive estimator of an asymmetric GARCH model. Applying some general results from Drost et al. [1997. The Annals of Statistics 25, 786–818], we first estimate the unknown density function of the disturbances by kernel methods, then apply a one-step Newton–Raphson method to obtain a more efficient estimator than the quasi-maximum likelihood estimator. The proposed semiparametric estimator is adaptive for parameters appearing in the conditional standard deviation model with respect to the unknown distribution of the disturbances.