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Non-Parametric Threshold Estimation for the Wiener–Poisson Risk Model

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  • Honglong You

    (School of Statistics, Qufu Normal University, Qufu 273165, China)

  • Yuan Gao

    (School of Mathematics, Qufu Normal University, Qufu 273165, China)

Abstract

In this paper, we consider the Wiener–Poisson risk model, which consists of a Wiener process and a compound Poisson process. Given the discrete record of observations, we use a threshold method and a regularized Laplace inversion technique to estimate the survival probability. In addition, we also construct an estimator for the distribution function of jump size and study its consistency and asymptotic normality. Finally, we give some simulations to verify our results.

Suggested Citation

  • Honglong You & Yuan Gao, 2019. "Non-Parametric Threshold Estimation for the Wiener–Poisson Risk Model," Mathematics, MDPI, vol. 7(6), pages 1-11, June.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:6:p:506-:d:236681
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    References listed on IDEAS

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    1. Shimizu, Yasutaka, 2009. "Functional estimation for Lvy measures of semimartingales with Poissonian jumps," Journal of Multivariate Analysis, Elsevier, vol. 100(6), pages 1073-1092, July.
    2. S. Pitts, 1994. "Nonparametric estimation of compound distributions with applications in insurance," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 46(3), pages 537-555, September.
    3. Zhang, Zhimin & Yang, Hailiang, 2013. "Nonparametric estimate of the ruin probability in a pure-jump Lévy risk model," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 24-35.
    4. Morales, Manuel, 2007. "On the expected discounted penalty function for a perturbed risk process driven by a subordinator," Insurance: Mathematics and Economics, Elsevier, vol. 40(2), pages 293-301, March.
    5. Cecilia Mancini, 2009. "Non‐parametric Threshold Estimation for Models with Stochastic Diffusion Coefficient and Jumps," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(2), pages 270-296, June.
    6. Wang, Yinfeng & Yin, Chuancun, 2010. "Approximation for the ruin probabilities in a discrete time risk model with dependent risks," Statistics & Probability Letters, Elsevier, vol. 80(17-18), pages 1335-1342, September.
    7. Frees, Edward W., 1986. "Nonparametric Estimation of the Probability of Ruin," ASTIN Bulletin, Cambridge University Press, vol. 16(S1), pages 81-90, April.
    8. Shimizu, Yasutaka, 2009. "A new aspect of a risk process and its statistical inference," Insurance: Mathematics and Economics, Elsevier, vol. 44(1), pages 70-77, February.
    9. Dufresne, Francois & Gerber, Hans U., 1991. "Risk theory for the compound Poisson process that is perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 10(1), pages 51-59, March.
    10. Veraverbeke, Noel, 1993. "Asymptotic estimates for the probability of ruin in a Poisson model with diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 13(1), pages 57-62, September.
    11. Croux, Kristof & Veraverbeke, Noel, 1990. "Nonparametric estimators for the probability of ruin," Insurance: Mathematics and Economics, Elsevier, vol. 9(2-3), pages 127-130, September.
    12. Hipp, Christian, 1989. "Estimators and Bootstrap Confidence Intervals for Ruin Probabilities," ASTIN Bulletin, Cambridge University Press, vol. 19(1), pages 57-70, April.
    13. Furrer, H. J. & Schmidli, H., 1994. "Exponential inequalities for ruin probabilities of risk processes perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 15(1), pages 23-36, October.
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