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Empirical Likelihood for Compound Poisson Processes

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  • Acknowledgements. The authors would like to thank two reviewers for their helpful comments. Li's research was supported by the National Natural Science Foundation of China (11201207). Zhou's work was partially supported by a grant R-155-000-095-112 at the National University of Singapore.

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Summary

Let {N(t), t > 0} be a Poisson process with rate λ > 0, independent of the independent and identically distributed random variables inline image with mean μ and variance inline image. The stochastic process inline image is then called a compound Poisson process and has a wide range of applications in, for example, physics, mining, finance and risk management. Among these applications, the average number of objects, which is defined to be λμ, is an important quantity. Although many papers have been devoted to the estimation of λμ in the literature, in this paper, we use the well-known empirical likelihood method to construct confidence intervals. The simulation results show that the empirical likelihood method often outperforms the normal approximation and Edgeworth expansion approaches in terms of coverage probabilities. A real data set concerning coal-mining disasters is analyzed using these methods.

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