Asymptotically Normal Estimators of the Ruin Probability for Lévy Insurance Surplus from Discrete Samples
Abstract
:1. Introduction
1.1. Ruin Probability Under Lévy Surplus
1.2. Earlier Works on Estimating Ruin Probability
1.3. Statistical Setting and General Notation
- and and .
- For a matrix , . Moreover, ⊤ stands for the transpose .
- For each , is the zero vector in . Moreover, and are the -zero matrix and identity matrix, respectively.
- For functions f and g, means that there exists a constant such that for all x.
- For , .
- For and , stands for the Laplace transform operator
- For functions ,
- stands for the convolution of f and g:
- and .
- For a -integrable function ,In particular, as for , we write
- for .
- Denote by the tail function of the exponential distribution with mean : for andMoreover, is its density function: .
2. Some Representations for the Ruin Probability
2.1. The Laguerre Expansion of
2.2. Coefficients and
3. Statistical Inference
3.1. Estimating the Lévy Characteristics
- (i)
- There exists some such that .
- (ii)
- For each , .
- (iii)
- For each ,
3.2. Joint Convergence and Asymptotic Normality
4. Main Theorems
5. Simulations
- (CP)
- Compound Poisson model: for
- (GS)
- Gamma subordinator model: for
6. Concluding Remarks
7. Preliminary Lemmas
Author Contributions
Funding
Conflicts of Interest
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Shimizu, Y.; Zhang, Z. Asymptotically Normal Estimators of the Ruin Probability for Lévy Insurance Surplus from Discrete Samples. Risks 2019, 7, 37. https://doi.org/10.3390/risks7020037
Shimizu Y, Zhang Z. Asymptotically Normal Estimators of the Ruin Probability for Lévy Insurance Surplus from Discrete Samples. Risks. 2019; 7(2):37. https://doi.org/10.3390/risks7020037
Chicago/Turabian StyleShimizu, Yasutaka, and Zhimin Zhang. 2019. "Asymptotically Normal Estimators of the Ruin Probability for Lévy Insurance Surplus from Discrete Samples" Risks 7, no. 2: 37. https://doi.org/10.3390/risks7020037