Sieve-based confidence intervals and bands for L\'{e}vy densities

The estimation of the L\'{e}vy density, the infinite-dimensional parameter controlling the jump dynamics of a L\'{e}vy process, is considered here under a discrete-sampling scheme. In this setting, the jumps are latent variables, the statistical properties of which can be assessed when the frequency and time horizon of observations increase to infinity at suitable rates. Nonparametric estimators for the L\'{e}vy density based on Grenander's method of sieves was proposed in Figueroa-L\'{o}pez [IMS Lecture Notes 57 (2009) 117--146]. In this paper, central limit theorems for these sieve estimators, both pointwise and uniform on an interval away from the origin, are obtained, leading to pointwise confidence intervals and bands for the L\'{e}vy density. In the pointwise case, our estimators converge to the L\'{e}vy density at a rate that is arbitrarily close to the rate of the minimax risk of estimation on smooth L\'{e}vy densities. In the case of uniform bands and discrete regular sampling, our results are consistent with the case of density estimation, achieving a rate of order arbitrarily close to $\log^{-1/2}(n)\cdot n^{-1/3}$, where $n$ is the number of observations. The convergence rates are valid, provided that $s$ is smooth enough and that the time horizon $T_n$ and the dimension of the sieve are appropriately chosen in terms of $n$.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ286 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

CONSUMER PREFERENCES FOR NON-CONVENTIONALLY GROWN PRODUCE

This study examines the potential for marketing fresh fruits and vegetables with labels indicating enhanced food and/or environmental safety attributes as compared to conventional produce. Four labels were investigated: Organic, Certified Organic, Certified Pesticide Residue-Free, and Grown with IPM. Results confirm findings of other surveys relating to concerns about pesticide residues. Seventy-one percent (71%) of respondents stated they believed that pesticide residues in food present a serious or moderate health hazard to consumers. In addition, 74% believed that pesticides pose a serious or moderate hazard to the environment, and 64% felt there was a serious or moderate hazard to farm workers. Results indicate there is a positive information effect for likelihood of purchasing for all of the labels, and this effect is statistically significant for all of the labels except for Certified Pesticide Residue-Free. The magnitude of the information effect for the Grown with IPM label was considerably higher than for the other labels, suggesting that there might be substantial payoffs for informing consumers about this label.Consumer/Household Economics,

Small-time asymptotics of stopped L\'evy bridges and simulation schemes with controlled bias

We characterize the small-time asymptotic behavior of the exit probability of a L\'evy process out of a two-sided interval and of the law of its overshoot, conditionally on the terminal value of the process. The asymptotic expansions are given in the form of a first-order term and a precise computable error bound. As an important application of these formulas, we develop a novel adaptive discretization scheme for the Monte Carlo computation of functionals of killed L\'evy processes with controlled bias. The considered functionals appear in several domains of mathematical finance (e.g., structural credit risk models, pricing of barrier options, and contingent convertible bonds) as well as in natural sciences. The proposed algorithm works by adding discretization points sampled from the L\'evy bridge density to the skeleton of the process until the overall error for a given trajectory becomes smaller than the maximum tolerance given by the user.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ517 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm