A hypothesis test for the domain of attraction of a random variable
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arXiv
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| Format: | Preprint |
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2022
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| author | Olivero, Héctor Talay, Denis |
| author_facet | Olivero, Héctor Talay, Denis |
| contents | In this work we address the problem of detecting whether a sampled probability distribution of a random variable $V$ has infinite first moment. This issue is notably important when the sample results from complex numerical simulation methods. For example, such a situation occurs when one simulates stochastic particle systems with complex and singular McKean-Vlasov interaction kernels. As stated, the detection problem is ill-posed. We thus propose and analyze an asymptotic hypothesis test for independent copies of a given random variable which is supposed to belong to an unknown domain of attraction of a stable law. The null hypothesis $\mathbf{H_0}$ is: `$X=\sqrt{V}$ is in the domain of attraction of the Normal law' and the alternative hypothesis is $\mathbf{H_1}$: `$X$ is in the domain of attraction of a stable law with index smaller than 2'. Our key observation is that~$X$ cannot have a finite second moment when $\mathbf{H_0}$ is rejected (and therefore $\mathbf{H_1}$ is accepted).
Surprisingly, we find it useful to derive our test from the statistics of random processes. More precisely, our hypothesis test is based on a statistic which is inspired by methodologies to determine whether a semimartingale has jumps from the observation of one single path at discrete times.
We justify our test by proving asymptotic properties of discrete time functionals of Brownian bridges. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_07885 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A hypothesis test for the domain of attraction of a random variable Olivero, Héctor Talay, Denis Statistics Theory Primary 62F05, 62G10, secondary 60J65 In this work we address the problem of detecting whether a sampled probability distribution of a random variable $V$ has infinite first moment. This issue is notably important when the sample results from complex numerical simulation methods. For example, such a situation occurs when one simulates stochastic particle systems with complex and singular McKean-Vlasov interaction kernels. As stated, the detection problem is ill-posed. We thus propose and analyze an asymptotic hypothesis test for independent copies of a given random variable which is supposed to belong to an unknown domain of attraction of a stable law. The null hypothesis $\mathbf{H_0}$ is: `$X=\sqrt{V}$ is in the domain of attraction of the Normal law' and the alternative hypothesis is $\mathbf{H_1}$: `$X$ is in the domain of attraction of a stable law with index smaller than 2'. Our key observation is that~$X$ cannot have a finite second moment when $\mathbf{H_0}$ is rejected (and therefore $\mathbf{H_1}$ is accepted). Surprisingly, we find it useful to derive our test from the statistics of random processes. More precisely, our hypothesis test is based on a statistic which is inspired by methodologies to determine whether a semimartingale has jumps from the observation of one single path at discrete times. We justify our test by proving asymptotic properties of discrete time functionals of Brownian bridges. |
| title | A hypothesis test for the domain of attraction of a random variable |
| topic | Statistics Theory Primary 62F05, 62G10, secondary 60J65 |
| url | https://arxiv.org/abs/2210.07885 |