Transformation kernel density estimation of actuarial loss functions

A transformation kernel density estimator that is suitable for heavy-tailed distributions is discussed. Using a truncated Beta transformation, the choice of the bandwidth parameter becomes straightforward. An application to insurance data and the calculation of the value-at-risk are presented.non-parametric methods, heavy-tailed distributions, value at risk

Shortcomings of a parametric VaR approach and nonparametric improvements based on a non-stationary return series model

A non-stationary regression model for financial returns is examined theoretically in this paper. Volatility dynamics are modelled both exogenously and deterministic, captured by a nonparametric curve estimation on equidistant centered returns. We prove consistency and asymptotic normality of a symmetric variance estimator and of a one-sided variance estimator analytically, and derive remarks on the bandwidth decision. Further attention is paid to asymmetry and heavy tails of the return distribution, implemented by an asymmetric version of the Pearson type VII distribution for random innovations. By providing a method of moments for its parameter estimation and a connection to the Student-t distribution we offer the framework for a factor-based VaR approach. The approximation quality of the non-stationary model is supported by simulation studies. --heteroscedastic asset returns,non-stationarity,nonparametric regression,volatility,innovation modelling,asymmetric heavy-tails,distributional forecast,Value at Risk (VaR)

Implementing Loss Distribution Approach for Operational Risk

To quantify the operational risk capital charge under the current regulatory framework for banking supervision, referred to as Basel II, many banks adopt the Loss Distribution Approach. There are many modeling issues that should be resolved to use the approach in practice. In this paper we review the quantitative methods suggested in literature for implementation of the approach. In particular, the use of the Bayesian inference method that allows to take expert judgement and parameter uncertainty into account, modeling dependence and inclusion of insurance are discussed

On a class of distributions with simple exponential tails

A simple general construction is put forward which covers many unimodal univariate distributions with simple exponentially decaying tails (e.g. asymmetric Laplace, log F and hyperbolic distributions as well as many new models). The proposed family is a special subset of a regular exponential family, and many properties flow therefrom. Two main practical points are made in the context of maximum likelihood fitting of these distributions to data. The first of these is that three, rather than an apparent four, parameters of the distributions suffice. The second is that maximum likelihood estimation of location in the new distributions is precisely equivalent to a standard form of kernel quantile estimation, choice of kernel being equivalent to specific choice of model within the class. This leads to a maximum likelihood method for bandwidth selection in kernel quantile estimation, but its practical performance is shown to be somewhat mixed. Further distribution theoretical aspects are also pursued, particularly distributions related to the main construction as special cases, limiting cases or by simple transformation

Non-parametric estimation of extreme risk measures from conditional heavy-tailed distributions

International audienceIn this paper, we introduce a new risk measure, the so-called Conditional Tail Moment. It is the moment of order a>0 of the loss distribution above the upper alpha-quantile. Estimating the Conditional Tail Moment permits to estimate all risk measures based on conditional moments such as Conditional Tail Expectation, Conditional Value-at-Risk or Conditional Tail Variance. Here, we focus on the estimation of these risk measures in case of extreme losses (where alpha converges to 0). It is moreover assumed that the loss distribution is heavy-tailed and depends on a covariate. The estimation method thus combines nonparametric kernel methods with extreme-value statistics. The asymptotic distribution of the estimators is established and their finite sample behavior is illustrated both on simulated data and on a real data set of daily rainfalls in the Cévennes-Vivarais region (France)

Quantile spectral processes: Asymptotic analysis and inference

Quantile- and copula-related spectral concepts recently have been considered by various authors. Those spectra, in their most general form, provide a full characterization of the copulas associated with the pairs $(X_t,X_{t-k})$ in a process $(X_t)_{t\in\mathbb{Z}}$, and account for important dynamic features, such as changes in the conditional shape (skewness, kurtosis), time-irreversibility, or dependence in the extremes that their traditional counterparts cannot capture. Despite various proposals for estimation strategies, only quite incomplete asymptotic distributional results are available so far for the proposed estimators, which constitutes an important obstacle for their practical application. In this paper, we provide a detailed asymptotic analysis of a class of smoothed rank-based cross-periodograms associated with the copula spectral density kernels introduced in Dette et al. [Bernoulli 21 (2015) 781-831]. We show that, for a very general class of (possibly nonlinear) processes, properly scaled and centered smoothed versions of those cross-periodograms, indexed by couples of quantile levels, converge weakly, as stochastic processes, to Gaussian processes. A first application of those results is the construction of asymptotic confidence intervals for copula spectral density kernels. The same convergence results also provide asymptotic distributions (under serially dependent observations) for a new class of rank-based spectral methods involving the Fourier transforms of rank-based serial statistics such as the Spearman, Blomqvist or Gini autocovariance coefficients.Comment: Published at http://dx.doi.org/10.3150/15-BEJ711 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm