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The Minkowski functionals, including the Euler characteristic statistics, are standard tools for morphological analysis in cosmology. Motivated by cosmic research, we examine the Minkowski functional of the excursion set for an isotropic central limit random field, whose k-point correlation functions (kth-order cumulants) have the same structure as that assumed in cosmic research. Using 3- and 4-point correlation functions, we derive the asymptotic expansions of the Euler characteristic density, which is the building block of the Minkowski functional. The resulting formula reveals the types of non-Gaussianity that cannot be captured by the Minkowski functionals. As an example, we consider an isotropic chi-squared random field and confirm that the asymptotic expansion accurately approximates the true Euler characteristic density.

In matrix-valued datasets the sampled matrices often exhibit correlations among both their rows and their columns. A useful and parsimonious model of such dependence is the matrix normal model, in which the covariances among the elements of a random matrix are parameterized in terms of the Kronecker product of two covariance matrices, one representing row covariances and one representing column covariance. An appealing feature of such a matrix normal model is that the Kronecker covariance structure allows for standard likelihood inference even when only a very small number of data matrices is available. For instance, in some cases a likelihood ratio test of dependence may be performed with a sample size of one. However, more generally the sample size required to ensure boundedness of the matrix normal likelihood or the existence of a unique maximizer depends in a complicated way on the matrix dimensions. This motivates the study of how large a sample size is needed to ensure that maximum likelihood estimators exist, and exist uniquely with probability one. Our main result gives precise sample size thresholds in the paradigm where the number of rows and the number of columns of the data matrices differ by at most a factor of two. Our proof uses invariance properties that allow us to consider data matrices in canonical form, as obtained from the Kronecker canonical form for matrix pencils.

Reproductive isolation that initiates speciation is likely caused by incompatibility among multiple loci in organisms belonging to genetically diverging populations. Laboratory C57BL/6J mice, which predominantly originated from Mus musculus domesticus, and a MSM/Ms strain derived from Japanese wild mice (M. m. molossinus, genetically close to M. m. musculus) are reproductively isolated. Their F1 hybrids are fertile, but successive intercrosses result in sterility. A consomic strain, C57BL/6J-ChrXMSM, which carries the X chromosome of MSM/Ms in the C57BL/6J background, shows male sterility, suggesting a genetic incompatibility of the MSM/Ms X chromosome and other C57BL/6J chromosome(s). In this study, we conducted genomewide linkage analysis and subsequent QTL analysis using the sperm shape anomaly that is the major cause of the sterility of the C57BL/6J-ChrXMSM males. These analyses successfully detected significant QTL on chromosomes 1 and 11 that interact with the X chromosome. The introduction of MSM/Ms chromosomes 1 and 11 into the C57BL/6J-ChrXMSM background failed to restore the sperm-head shape, but did partially restore fertility. This result suggests that this genetic interaction may play a crucial role in the reproductive isolation between the two strains. A detailed analysis of the male sterility by intracytoplasmic sperm injection and zona-free in vitro fertilization demonstrated that the C57BL/6J-ChrXMSM spermatozoa have a defect in penetration through the zona pellucida of eggs.

Consider a Gaussian random field with a finite Karhunen-Loève expansion of the form Z(u) = ∑n i=1uizi, where zi, i = 1,..., n, are independent standard normal variables and u = (u1,..., un)' ranges over an index set M, which is a subset of the unit sphere Sn-1in Rn. Under a very general assumption that M is a manifold with a piecewise smooth boundary, we prove the validity and the equivalence of two currently available methods for obtaining the asymptotic expansion of the tail probability of the maximum of Z(u). One is the tube method, where the volume of the tube around the index set M is evaluated. The other is the Euler characteristic method, where the expectation for the Euler characteristic of the excursion set is evaluated. General discussion on this equivalence was given in a recent paper by R. J. Adler. In order to show the equivalence we prove a version of the Morse theorem for a manifold with a piecewise smooth boundary.

We consider approximate Bayesian model choice for model selection problems that involve models whose Fisher information matrices may fail to be invertible along other competing submodels. Such singular models do not obey the regularity conditions underlying the derivation of Schwarz's Bayesian information criterion BIC and the penalty structure in BIC generally does not reflect the frequentist large sample behaviour of the marginal likelihood. Although large sample theory for the marginal likelihood of singular models has been developed recently, the resulting approximations depend on the true parameter value and lead to a paradox of circular reasoning. Guided by examples such as determining the number of components in mixture models, the number of factors in latent factor models or the rank in reduced rank regression, we propose a resolution to this paradox and give a practical extension of BIC for singular model selection problems.

We consider the real and complex noncentral Wishart distributions. The moments of these distributions are shown to be expressed as weighted generating functions of graphs associated with the Wishart distributions. We give some bijections between sets of graphs related to moments of the real Wishart distribution and the complex noncentral Wishart distribution. By means of the bijections, we see that calculating these moments of a certain class the real Wishart distribution boils down to calculations for the case of complex Wishart distributions. Nous considérons les lois Wishart non-centrale réel et complexe. Les moments sont décrits comme fonctions génératrices de graphes associées avec les lois Wishart. Nous donnons bijections entre ensembles de graphes relatifs aux moments des lois Wishart non-centrale réel et complexe. Au moyen de la bijection, nous voyons que le calcul des moments d'une certaine classe la loi Wishart réel deviennent le calcul de moments de loi Wishart complexes.

We treat all the bivariate lack-of-memory (BLM) distributions in a unified approach and develop some new general properties of the BLM distributions, including joint moment generating function, product moments, and dependence structure. Necessary and sufficient conditions for the survival functions of BLM distributions to be totally positive of order two are given. Some previous results about specific BLM distributions are improved. In particular, we show that both the Marshall–Olkin survival copula and survival function are totally positive of all orders, regardless of parameters. Besides, we point out that Slepian’s inequality also holds true for BLM distributions.

We construct by using B-spline functions a class of copulas that includes the Bernstein copulas arising in Baker’s distributions. The range of correlation of the B-spline copulas is examined, and the Fréchet–Hoeffding upper bound is proved to be attained when the number of B-spline functions goes to infinity. As the B-spline functions are well-known to be an order-complete weak Tchebycheff system from which the property of total positivity of any order follows for the maximum correlation case, the results given here extend classical results for the Bernstein copulas. In addition, we derive in terms of the Stirling numbers of the second kind an explicit formula for the moments of the related B-spline functions on the right half-line.

It has recently been established that the tube formula and the Euler characteristic method give an identical and valid asymptotic expansion of the tail probability of the maximum of a Gaussian random field when the random field has finite Karhunen-Loève expansion and the index set has positive critical radius. We show that the positiveness of the critical radius is an essential condition. When the critical radius is zero, we prove that only the main term is valid and that other higher-order terms are generally not valid in the formal asymptotic expansion based on the tube formula. This is done by first establishing an exact tube formula and comparing the formal tube formula with the exact formula. Furthermore, we show that the equivalence of the formal tube formula and the Euler characteristic method no longer holds when the critical radius is zero. We conclude by applying our results to some specific examples.

Constructing a bivariate distribution with specific marginals and correlation has been a challenging problem since 1930s. In this survey we shall focus on the recent developments on the FGM-related distributions, including Sarmanov and Lee’s distributions, Baker’s distributions and Bayramoglu’s distributions. This complements the most recent works of (i) the review by Sarabia and Gómez-Déniz ( 2008 , SORT) and (ii) the monograph by Balakrishnan and Lai ( 2009 , Springer). Some new results are provided. Mathematics Subject Classification (2000) 62H20; 62H86; 62G30; 60E05; 62E10