Coupon collector's problem and generalized Pareto distributions

Jocković, Jelena; Mladenović, Pavle

doi:10.1016/j.jspi.2011.01.020

Cited by 8 publications

(5 citation statements)

References 18 publications

(15 reference statements)

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“…Algorithm 1: Algorithm for comparing the performance of the GPD estimators. 6 Journal of Applied Mathematics IPB estimators appear to be the best as their MSEs approach 0. This is in conformity with what have been reported in [18,19] and the results in Section 2.5.…”

Section: Simulation Results and Discussionmentioning

confidence: 99%

“…The generalized Pareto distribution (GPD) is one of the fundamental distributions used in extreme value theory. As reported in [1,2], the GPD is the limiting distribution of excesses over a large threshold regardless of the underlying distribution of data, F The significance of the GPD in extreme value theory cannot be overemphasized because of its extensive applicability: hydroelectric dam management [3], flood levels of river, insurance and finance [4,5], waiting time problems [6], ecology [7], and climatology [8], among others.…”

Section: Introductionmentioning

confidence: 99%

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Shrinkage Methods for Estimating the Shape Parameter of the Generalized Pareto Distribution

Pels,

Adebanji,

Twumasi-Ankrah

et al. 2023

Journal of Applied Mathematics

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The generalized Pareto distribution is one of the most important distributions in statistics of extremes as it has wide applications in fields such as finance, insurance, and hydrology. This study proposes two new methods for estimating the shape parameter of the generalized Pareto distribution (GPD). The proposed methods use the shrinkage principle to adapt the existing empirical Bayesian with data-based prior and the likelihood moment method to obtain two estimators. The performance of the proposed estimators is compared with the existing estimators (i.e., maximum likelihood, likelihood moment estimators, etc.) for the shape parameter of the generalized Pareto distribution in a simulation study. The results show that the proposed estimators perform better for small to moderate number of exceedances in estimating shape parameter of the light-tailed distributions and competitive when estimating heavy-tailed distributions. The proposed estimators are illustrated with practical datasets from climate and insurance studies.

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Section: Simulation Results and Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Shrinkage Methods for Estimating the Shape Parameter of the Generalized Pareto Distribution

Pels,

Adebanji,

Twumasi-Ankrah

et al. 2023

Journal of Applied Mathematics

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“…In the same way, we can use the second inequality of Lemma 2. Together with γ n,r → 1, log nR n,r → 0, and α + n,r → 0, where γ n,r , R n,r , and α + n,r are defined in ( 25), (22), and ( 26), respectively, we obtain the reverse inequality in (27), with each lim sup replaced by lim inf. We will now show that…”

Section: Lemma 2 We Havementioning

confidence: 90%

Two poisson limit theorems for the coupon collector’s problem with group drawings

Schilling

Henze

2021

J. Appl. Probab.

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In the collector’s problem with group drawings, s out of n different types of coupon are sampled with replacement. In the uniform case, each s-subset of the types has the same probability of being sampled. For this case, we derive a Poisson limit theorem for the number of types that are sampled at most $c-1$ times, where $c \ge 1$ is fixed. In a specified approximate nonuniform setting, we prove a Poisson limit theorem for the special case $c=1$ . As corollaries, we obtain limit distributions for the waiting time for c complete series of types in the uniform case and a single complete series in the approximate nonuniform case.

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“…In this paper we obtain the limiting distributions of the random variable M n in the coupon collector's problem, where M n is defined as the waiting time until all triples of elements jjj, j ∈ N n , are sampled. Corresponding results for the waiting time until all pairs of elements jj, j ∈ N n , are sampled are given in the papers Mladenović (2008) and Jocković, Mladenović (2011).…”

Section: Discussionmentioning

confidence: 99%