遍历扩散的经验$L^2$ -距离检验统计量
Empirical $L^2$-distance test statistics for ergodic diffusions
摘要：
本文的目的是介绍一种新的检验统计量的简单原假设的一维遍历扩散过程在离散时间采样。我们处理随机微分方程的准似然方法（即局部高斯近似的转移函数），并定义一个检验统计量的经验$L^2$之间的距离准似然。我们证明了引入的检验统计量是渐近分布自由的，即它弱收敛到一个$χ^2$随机变量。此外，我们还研究了参数检验在局部选择下的功效。我们通过Monte Carlo分析表明，在小样本的情况下，引入的测试似乎比文献中提出的其他测试更好地执行。
英文摘要：
The aim of this paper is to introduce a new type of test statistic for simple null hypothesis on one-dimensional ergodic diffusion processes sampled at discrete times. We deal with a quasi-likelihood approach for stochastic differential equations (i.e. local gaussian approximation of the transition functions) and define a test statistic by means of the empirical $L^2$-distance between quasi-likelihoods. We prove that the introduced test statistic is asymptotically distribution free; namely it weakly converges to a $χ^2$ random variable. Furthermore, we study the power under local alternatives of the parametric test. We show by the Monte Carlo analysis that, in the small sample case, the introduced test seems to perform better than other tests proposed in literature.
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