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Bert van Es

Publications and source records attributed to Bert van Es.

18 recordsLinked to original sources
arXiv2023

Inference via the Skewness-Kurtosis Set

Kurtosis minus squared skewness is bounded from below by 1, but for unimodal distributions this parameter is bounded by 189/125. In some applications it is natural to compare distributions by comparing their kurtosis-minus-squared-skewness parameters. The asymptotic behavior of the empirical version of this parameter is studied here for i.i.d. random variables. The result may be used to test the hypothesis of unimodality against the alternative that the kurtosis-minus-squared-skewness parameter is less than 189/125. However, such a test has to be applied with care, since this parameter can take arbitrarily large values, also for multimodal distributions. Numerical results are presented and for three classes of distributions the skewness-kurtosis sets are described in detail.

math.ST↗
arXiv2018

Strength of forensic evidence for composite hypotheses: An empirical Bayes view with a fixed prior quantile

Motivated by the forensic problem of determining the strength of evidence of a continuously distributed measurement of evidence, in the situation of composite hypotheses of the prosecutor and the defence concerning a parameter of a parametric model, we consider empirical Bayes methods with a prescribed quantile value for the prior distribution. Firstly we derive the strength of evidence for nonparametric priors. It turns out that we get the by now more or less accepted strength of evidence as the ratio of two suprema, $\sup_{θ\geqθ_0}f(x|θ)/\sup_{θ<θ_0}f(x|θ)$. Here the hypotheses of the prosecutor and defence are given by $H_p: θ\geq θ_0$ and $H_d:θ<θ_0$. The evidence is seen as a measurement $x$ which is a realization of a random variable with a density $f(x|θ)$. Secondly we consider a similar parametric empirical Bayes method with a quantile restriction on the prior where the prior distribution is assumed to be normal. Some interesting strength of evidence functions are derived for this situation.

stat.AP↗
arXiv2017

Nonparametric Kernel Density Estimation for Univariate Curent Status Data

We derive estimators of the density of the event times of current status data. The estimators are derived for the situations where the distribution of the observation times is known and where this distribution is unknown. The density estimators are constructed from kernel estimators of the density of transformed current status data, which have a distribution similar to uniform deconvolution data. Expansions of the expectation and variance as well as asymptotic normality are derived. A reference density based bandwidth selection method is proposed. A simulated example is presented.

math.ST↗
arXiv2011

Bivariate Uniform Deconvolution

We construct a density estimator in the bivariate uniform deconvolution model. For this model we derive four inversion formulas to express the bivariate density that we want to estimate in terms of the bivariate density of the observations. By substituting a kernel density estimator of the density of the observations we then get four different estimators. Next we construct an asymptotically optimal convex combination of these four estimators. Expansions for the bias, variance, as well as asymptotic normality, are derived. Some simulated examples are presented.

stat.ME↗
arXiv2011

Deconvolution for an atomic distribution: rates of convergence

Let $X_1,..., X_n$ be i.i.d.\ copies of a random variable $X=Y+Z,$ where $ X_i=Y_i+Z_i,$ and $Y_i$ and $Z_i$ are independent and have the same distribution as $Y$ and $Z,$ respectively. Assume that the random variables $Y_i$'s are unobservable and that $Y=AV,$ where $A$ and $V$ are independent, $A$ has a Bernoulli distribution with probability of success equal to $1-p$ and $V$ has a distribution function $F$ with density $f.$ Let the random variable $Z$ have a known distribution with density $k.$ Based on a sample $X_1,...,X_n,$ we consider the problem of nonparametric estimation of the density $f$ and the probability $p.$ Our estimators of $f$ and $p$ are constructed via Fourier inversion and kernel smoothing. We derive their convergence rates over suitable functional classes. By establishing in a number of cases the lower bounds for estimation of $f$ and $p$ we show that our estimators are rate-optimal in these cases.

math.ST↗
arXiv2011

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. Asymptotic normality and the asymptotic biases are derived.

math.ST↗
arXiv2009

Nonparametric methods for volatility density estimation

Stochastic volatility modelling of financial processes has become increasingly popular. The proposed models usually contain a stationary volatility process. We will motivate and review several nonparametric methods for estimation of the density of the volatility process. Both models based on discretely sampled continuous time processes and discrete time models will be discussed. The key insight for the analysis is a transformation of the volatility density estimation problem to a deconvolution model for which standard methods exist. Three type of nonparametric density estimators are reviewed: the Fourier-type deconvolution kernel density estimator, a wavelet deconvolution density estimator and a penalized projection estimator. The performance of these estimators will be compared. Key words: stochastic volatility models, deconvolution, density estimation, kernel estimator, wavelets, minimum contrast estimation, mixing

stat.ME↗
arXiv2009

Multivariate Nonparametric Volatility Density Estimation

We consider a continuous-time stochastic volatility model. The model contains a stationary volatility process, the multivariate density of the finite dimensional distributions of which we aim to estimate. We assume that we observe the process at discrete instants in time. The sampling times will be equidistant with vanishing distance. A multivariate Fourier-type deconvolution kernel density estimator based on the logarithm of the squared processes is proposed to estimate the multivariate volatility density. An expansion of the bias and a bound on the variance are derived. Key words: stochastic volatility models, multivariate density estimation, kernel estimator, deconvolution, mixing

math.ST↗
arXiv2009

Asymptotic normality of the deconvolution kernel density estimator under the vanishing error variance

Let $X_1,...,X_n$ be i.i.d. observations, where $X_i=Y_i+σ_n Z_i$ and the $Y$'s and $Z$'s are independent. Assume that the $Y$'s are unobservable and that they have the density $f$ and also that the $Z$'s have a known density $k.$ Furthermore, let $σ_n$ depend on $n$ and let $σ_n\to 0$ as $n\to\infty.$ We consider the deconvolution problem, i.e. the problem of estimation of the density $f$ based on the sample $X_1,...,X_n.$ A popular estimator of $f$ in this setting is the deconvolution kernel density estimator. We derive its asymptotic normality under two different assumptions on the relation between the sequence $σ_n$ and the sequence of bandwidths $h_n.$ We also consider several simulation examples which illustrate different types of asymptotics corresponding to the derived theoretical results and which show that there exist situations where models with $σ_n\to 0$ have to be preferred to the models with fixed $σ.$

math.ST↗
arXiv2008

Deconvolution for an atomic distribution

Let $X_1,...,X_n$ be i.i.d. observations, where $X_i=Y_i+σZ_i$ and $Y_i$ and $Z_i$ are independent. Assume that unobservable $Y$'s are distributed as a random variable $UV,$ where $U$ and $V$ are independent, $U$ has a Bernoulli distribution with probability of zero equal to $p$ and $V$ has a distribution function $F$ with density $f.$ Furthermore, let the random variables $Z_i$ have the standard normal distribution and let $σ>0.$ Based on a sample $X_1,..., X_n,$ we consider the problem of estimation of the density $f$ and the probability $p.$ We propose a kernel type deconvolution estimator for $f$ and derive its asymptotic normality at a fixed point. A consistent estimator for $p$ is given as well. Our results demonstrate that our estimator behaves very much like the kernel type deconvolution estimator in the classical deconvolution problem.

math.ST↗
arXiv2008

Some thoughts on the asymptotics of the deconvolution kernel density estimator

Via a simulation study we compare the finite sample performance of the deconvolution kernel density estimator in the supersmooth deconvolution problem to its asymptotic behaviour predicted by two asymptotic normality theorems. Our results indicate that for lower noise levels and moderate sample sizes the match between the asymptotic theory and the finite sample performance of the estimator is not satisfactory. On the other hand we show that the two approaches produce reasonably close results for higher noise levels. These observations in turn provide additional motivation for the study of deconvolution problems under the assumption that the error term variance $σ^2\to 0$ as the sample size $n\to\infty.$

stat.ME↗
arXiv2007

A kernel type nonparametric density estimator for decompounding

Given a sample from a discretely observed compound Poisson process, we consider estimation of the density of the jump sizes. We propose a kernel type nonparametric density estimator and study its asymptotic properties. An order bound for the bias and an asymptotic expansion of the variance of the estimator are given. Pointwise weak consistency and asymptotic normality are established. The results show that, asymptotically, the estimator behaves very much like an ordinary kernel estimator.

math.ST↗
arXiv2006

A Comparison of Information Concerning the Regression Parameter in The Accelerated Failure Time Model under Current Duration and Length Biased Sampling: Does it Pay to be Patient?

Longitudinal observations are sometimes costly or not available. Cross sectional sampling can be an alternative. Observations are drawn then at a specific point in time from a population of durations whose distributions satisfy a {\em core model}. Subsequently, one has a choice. One may process the data immediately, obtaining so called current duration data. Or one waits until the sampled durations are known completely obtaining the full durations via length biased sampling. We compare the Fisher information for the Euclidean parameter corresponding to an Accelerated Failure Time core model when the observations are obtained by either current duration or length biased sampling.

math.ST↗
arXiv2003

Efficient estimation in the accelerated failure time model under cross sectional sampling

Consider estimation of the regression parameter in the accelerated failure time model, when data are obtained by cross sectional sampling. It is shown that it is possible under regularity of the model to construct an efficient estimator of the unknown Euclidean regression parameter if the distribution of the covariate vector is known and also if it is unknown with vanishing mean.

math.ST↗
arXiv2002

Nonparametric Volatility Density Estimation

We consider two kinds of stochastic volatility models. Both kinds of models contain a stationary volatility process, the density of which, at a fixed instant in time, we aim to estimate. We discuss discrete time models where for instance a log price process is modeled as the product of a volatility process and i.i.d. noise. We also consider samples of certain continuous time diffusion processes. The sampled time instants will be be equidistant with vanishing distance. A Fourier type deconvolution kernel density estimator based on the logarithm of the squared processes is proposed to estimate the volatility density. Expansions of the bias and bounds on the variances are derived.

math.ST↗
arXiv2002

Nonparametric volatility density estimation for discrete time models

We consider discrete time models for asset prices with a stationary volatility process. We aim at estimating the multivariate density of this process at a set of consecutive time instants. A Fourier type deconvolution kernel density estimator based on the logarithm of the squared process is proposed to estimate the volatility density. Expansions of the bias and bounds on the variance are derived.

math.ST↗
arXiv2002

Estimating a structural distribution function by grouping

By the method of Poissonization we confirm some existing results concerning consistent estimation of the structural distribution function in the situation of a large number of rare events. Inconsistency of the so called natural estimator is proved. The method of grouping in cells of equal size is investigated and its consistency derived. A bound on the mean squared error is derived.

math.ST↗