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題 名 | Meeting Hausdorff in Monte Carlo: A Surprising Tour with Antihype Fractals |
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作 者 | Craiu, Radu V.; Meng, Xiao-li; | 書刊名 | Statistica Sinica |
卷 期 | 16:1 民95.01 |
頁 次 | 頁77-91 |
分類號 | 319 |
關鍵詞 | Antithetic variates; Extreme antithesis; Fractals; Hausdorff dimension; Koch's curve; Latin hypercube sampling; Markov chain Monte Carlo; Self-similar fractals; |
語 文 | 英文(English) |
英文摘要 | To many statistical researchers, fractals are aesthetically pleasing mathematical objects or ingredients of complex theoretical studies. This article documents and exception: during recent research on improving effectiveness of Markov chain mote Carlo (MCMC), we unexpectedly encountered a class of intriguing fractals in the simple context of generating negatively correlated random variates that achieve extreme antithesis. The calss of antihype fractals enticed us to tour the world of fractals, because it has intrinsic connections with classical fractals such as Koch’s snowflake and it illustrates theoretical concepts such as Hausdorff dimension in a very intuitive way. It also provides a practical example where a sequence of uniform variables converges exponentially in the Kolmogorov-Smirnov distance and Hellinger distance. We also show that this non-convergence result actually holds for any sequence of (proper) uniform distributions and supports formed by the generating process of a self-similar fractal. These negative results remind us that the choice of metrics, e.g., for diagnosing convergence of MCMC algorithms, do matter sometimes in practice. |
本系統中英文摘要資訊取自各篇刊載內容。