Study on Euler characteristic heuristics in distribution theory of random field

点击翻译按钮获取中文摘要

The tube method and the Euler characteristic heuristic are well known as integral-geometric methods for approximating the distribution of the maximum Pr (max X (t) > a) of a random field X (t), where t is an element of an index set I. Some characteristics of these methods and their applications to multiple comparisons (mainly genomic data analysis) were studied.(i)Approximating formulas for the maximum of a Gaussian random field with inhomogeneous mean and variance were derived. In addition, the validity of this approximation was proved.(ii)The expressions for the order of errors in the tube method and the Euler characteristic heuristic were given explicitly in the case where the random field is Gaussian with homogeneous mean and variance, and the critical radius is attained globally.(iii)Let A be an orthogonally invariant random matrix. By applying the Euler characteristic heuristic to a random field X(h) = h' A h, an approximating formulas as well as its error for the distribution of the largest eigenvalue were given. The errors of the Euler characteristic methods for many random matrices including the Wishart matrix, the Beta matrix, and the inverse Wishart matrix, were shown to be small.(iv)Methods for adjusting the multiplicity were studied in the problem of finding the fatal genes or the QTL analysis. The LOD score (test statistic) can be regarded as a chi-squared field with complicated correlation structure. A simple method for adjusting the multiplicity of p-values were studied.(v)In the problem of finding genes mentioned above, we have to treat the chi-squared random field with two indices in order to detect the interaction (epistasis) between two loci. When the spaces between maker loci are small, the chi-squared random field has the Ornstein-Uhlenbeck correlation structure. Under the approximation that the makers are located at even intervals, the distribution of the maximum of the random field (that is, the adjusted p-value) was given.