Nonparametric construction of probability maps under local stationarity

The environmental contamination risk can be evaluated in a specific area by approximating the probability that the pollutant under study exceeds a critical value. This issue requires the estimation of the distribution function involved, which can be addressed by applying the indicator kriging methodology or by approximating the sill of the variogram of the underlying indicator process. These approaches demand an appropriate characterization of the indicator variogram, which in turn requires a previous specification of the trend function, if the latter is suspected to be non-constant. Since accuracy of the results will be strongly dependent on the adequate approximation of both functions, we suggest proceeding in a different way to avoid these requirements. Thus, in the current paper, two kerneltype estimators are proposed, based on first approximating the distribution at the sampled sites and then obtaining a weighted average of the resulting values, to derive a valid estimator at each (sampled or unsampled) location. Consistency of the kernel approaches is proved under rather general conditions, such as local stationarity and the existence of derivatives up to the second order of the distribution function. Numerical studies have been carried out to illustrate the performance of our proposals when compared to those procedures requiring the approximation of the indicator variogram. In a final step, the kernel-type estimation of the distribution function has been applied to map the risk of contamination by arsenic in the Central Region of Portugal. With this aim, biomonitoring data of arsenic concentrations were used to detect those zones with higher risk of arsenic accumulation, which is mainly located on the northern part of the region.The authors would like to thank the helpful suggestions and comments from the Editor, the Associate Editor, and the Reviewers. The authors are also grateful to Karen J. Duncan for her contribution in the language revision. The first author’s work has been partially supported by the Spanish National Research and Development Program project [TEC2015-65353-R], by the European Regional Development Fund (ERDF), and by the Galician Regional Government under project GRC 2015/018 and under agreement for funding AtlantTIC (Atlantic Research Center for Information and Communication Technologies). The second author acknowledges financial support from the Portuguese Funds through FCT-“Fundação para a Ciência e a Tecnologia,” within the Project UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

Specification Tests of Parametric Dynamic Conditional Quantiles

This article proposes omnibus specification tests of parametric dynamic quantile regression models. Contrary to the existing procedures, we allow for a flexible and general specification framework where a possibly continuum of quantiles are simultaneously specified. This is the case for many econometric applications for both time series and cross section data which require a global diagnostic tool. We study the asymptotic distribution of the test statistics under fairly weak conditions on the serial dependence in the underlying data generating process. It turns out that the asymptotic null distribution depends on the data generating process and the hypothesized model. We propose a subsampling procedure for approximating the asymptotic critical values of the tests. An appealing property of the proposed tests is that they do not require estimation of the non-parametric (conditional) sparsity function. A Monte Carlo study compares the proposed tests and shows that the asymptotic results provide good approximations for small sample sizes. Finally, an application to some European stock indexes provides evidence that our methodology is a powerful and flexible alternative to standard backtesting procedures in evaluating market risk by using information from a range of quantiles in the lower tail of returns.

Nonparametric approaches for estimating risk maps

Assessment of environmental contamination is increasingly a concern in nowadays soci- ety. The maximum levels for pollutants are heavily regulated, being necessary to ensure compliance. Consequently, it becomes important to construct probability maps of the observation region, showing the complementary value of the distribution function of the variable involved at regulatory thresholds. These are usually called risk maps in the environmental setting. In this work, two kernel-type estimators of the spatial distribution function are constructed, which de- part from approximating the distribution at the sampled sites and then obtaining a weighted average of the resulting values, to derive a valid estimator at any random location. Consistency of both ap- proaches is proved under rather general conditions, such as local stationarity and the existence of a number of derivatives of the distribution function. Unlike other alternatives, the new proposals pro- vide non-decreasing functions and do not require a previous estimation of the indicator variogram or the trend function. However, appropriate bandwidths parameters are needed and selection of them in practice needs to be addressed. Numerical studies are carried out, aiming at comparing the current proposal with more usual methods, such as those based on the sill estimation or the indicator kriging, described in Journel (1983) or Goovaerts (1997), respectively, and redesigned in García-Soidán and Menezes (2012). Finally, the new proposal is applied to arsenic data from Portugal, so that pollution risk maps of the referred region are constructed. Moreover, accuracy maps of the probability estimates might be constructed based on bootstrap replicas, as described in García-Soidán, Menezes and Rubiños (2014).info:eu-repo/semantics/publishedVersio

When do jumps matter for portfolio optimization? : [Version 29 April 2013]

We consider the continuous-time portfolio optimization problem of an investor with constant relative risk aversion who maximizes expected utility of terminal wealth. The risky asset follows a jump-diffusion model with a diffusion state variable. We propose an approximation method that replaces the jumps by a diffusion and solve the resulting problem analytically. Furthermore, we provide explicit bounds on the true optimal strategy and the relative wealth equivalent loss that do not rely on results from the true model. We apply our method to a calibrated affine model and fine that relative wealth equivalent losses are below 1.16% if the jump size is stochastic and below 1% if the jump size is constant and γ ≥ 5. We perform robustness checks for various levels of risk-aversion, expected jump size, and jump intensity

More general credibility models

summary:This communication gives some extensions of the original Bühlmann model. The paper is devoted to semi-linear credibility, where one examines functions of the random variables representing claim amounts, rather than the claim amounts themselves. The main purpose of semi-linear credibility theory is the estimation of $\mu _0 (\theta ) = E[f_0 (X_{t+1})| \theta ]$ (the net premium for a contract with risk parameter $\theta$) by a linear combination of given functions of the observable variables: $\underline X' = (X_1, X_2, \ldots , X_t)$. So the estimators mainly considered here are linear combinations of several functions $f_1, f_2, \ldots , f_n$ of the observable random variables. The approximation to $\mu _0 (\theta )$ based on prescribed approximating functions $f_1, f_2, \ldots , f_n$ leads to the optimal non-homogeneous linearized estimator for the semi-linear credibility model. Also we discuss the case when taking $f_p = f$ for all $p$ to find the optimal function $f$. It should be noted that the approximation to $\mu _0 (\theta )$ based on a unique optimal approximating function $f$ is always better than the one in the semi-linear credibility model based on prescribed approximating functions: $f_1, f_2, \ldots , f_n$. The usefulness of the latter approximation is that it is easy to apply, since it is sufficient to know estimates for the structure parameters appearing in the credibility factors. Therefore we give some unbiased estimators for the structure parameters. For this purpose we embed the contract in a collective of contracts, all providing independent information on the structure distribution. We close this paper by giving the semi-linear hierarchical model used in the applications chapter