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For a bivariate probability distribution, local dependence around a single point on the support is often formulated as the second derivative of the logarithm of the probability density function. However, this definition lacks the invariance under marginal distribution transformations, which is often required as a criterion for dependence measures. In this study, we examine the \textit{relative local dependence}, which we define as the ratio of the local dependence to the probability density function, for copulas. By using this notion, we point out that typical copulas can be characterised as the solutions to the corresponding partial differential equations, particularly highlighting that the relative local dependence of the Frank copula remains constant. The estimation and visualization of the relative local dependence are demonstrated using simulation data. Furthermore, we propose a class of copulas where local dependence is proportional to the $k$-th power of the probability density function, and as an example, we demonstrate a newly discovered relationship derived from the density functions of two representative copulas, the Frank copula and the Farlie-Gumbel-Morgenstern (FGM) copula., Comment: 25 pages,6 figures

In dependence modeling, various copulas have been utilized. Among them, the Frank copula has been one of the most typical choices due to its simplicity. In this work, we demonstrate that the Frank copula is the minimum information copula under fixed Kendall's $\tau$ (MICK), both theoretically and numerically. First, we explain that both MICK and the Frank density follow the hyperbolic Liouville equation. Moreover, we show that the copula density satisfying the Liouville equation is uniquely the Frank copula. Our result asserts that selecting the Frank copula as an appropriate copula model is equivalent to using Kendall's $\tau$ as the sole available information about the true distribution, based on the entropy maximization principle., Comment: 10 pages, 3 figures

Copulas have gained widespread popularity as statistical models to represent dependence structures between multiple variables in various applications. The minimum information copula, given a finite number of constraints in advance, emerges as the copula closest to the uniform copula when measured in Kullback-Leibler divergence. In prior research, the focus has predominantly been on constraints related to expectations on moments, including Spearman's $\rho$. This approach allows for obtaining the copula through convex programming. However, the existing framework for minimum information copulas does not encompass non-linear constraints such as Kendall's $\tau$. To address this limitation, we introduce MICK, a novel minimum information copula under fixed Kendall's $\tau$. We first characterize MICK by its local dependence property. Despite being defined as the solution to a non-convex optimization problem, we demonstrate that the uniqueness of this copula is guaranteed when the correlation is sufficiently small. Additionally, we provide numerical insights into applying MICK to real financial data., Comment: 35 pages, 10 figures, 4 tables

Phylogenetic trees are key data objects in biology, and the method of phylogenetic reconstruction has been highly developed. The space of phylogenetic trees is a nonpositively curved metric space. Recently, statistical methods to analyze the set of trees on this space are being developed utilizing this property. Meanwhile, in Euclidean space, the log-concave maximum likelihood method has emerged as a new nonparametric method for probability density estimation. In this paper, we derive a sufficient condition for the existence and uniqueness of the log-concave maximum likelihood estimator on tree space. We also propose an estimation algorithm for one and two dimensions. Since various factors affect the inferred trees, it is difficult to specify the distribution of sample trees. The class of log-concave densities is nonparametric, and yet the estimation can be conducted by the maximum likelihood method without selecting hyperparameters. We compare the estimation performance with a previously developed kernel density estimator numerically. In our examples where the true density is log-concave, we demonstrate that our estimator has a smaller integrated squared error when the sample size is large. We also conduct numerical experiments of clustering using the Expectation-Maximization (EM) algorithm and compare the results with k-means++ clustering using Fr\'echet mean., Comment: 41 pages, 10 figures

We propose a method to construct a joint statistical model for mixed-domain data to analyze their dependence. Multivariate Gaussian and log-linear models are particular examples of the proposed model. It is shown that the functional equation defining the model has a unique solution under fairly weak conditions. The model is characterized by two orthogonal parameters: the dependence parameter and the marginal parameter. To estimate the dependence parameter, a conditional inference together with a sampling procedure is proposed and is shown to provide a consistent estimator. Illustrative examples of data analyses involving penguins and earthquakes are presented., Comment: 41 pages, 3 figures

Multi-dimensional distributions whose marginal distributions are uniform are called copulas. Among them, the one that satisfies given constraints on expectation and is closest to the independent distribution in the sense of Kullback-Leibler divergence is called the minimum information copula. The density function of the minimum information copula contains a set of functions called the normalizing functions, which are often difficult to compute. Although a number of proper scoring rules for probability distributions having normalizing constants such as exponential families are proposed, these scores are not applicable to the minimum information copulas due to the normalizing functions. In this paper, we propose the conditional Kullback-Leibler score, which avoids computation of the normalizing functions. The main idea of its construction is to use pairs of observations. We show that the proposed score is strictly proper in the space of copula density functions and therefore the estimator derived from it has asymptotic consistency. Furthermore, the score is convex with respect to the parameters and can be easily optimized by the gradient methods., Comment: 21 pages, 4 figures; Figures are fixed (V2)

In causal inference, we can consider a situation in which treatment on one unit affects others, i.e., interference exists. In the presence of interference, we cannot perform a classical randomization test directly because a null hypothesis is not sharp. Instead, we need to perform the randomization test restricted to a subset of units and assignments that makes the null hypothesis sharp. A previous study constructed a useful testing method, a biclique test, by reducing the selection of the appropriate subsets to searching for bicliques in a bipartite graph. However, since the power depends on the features of selected subsets, there is still room to improve the power by refining the selection procedure. In this paper, we propose a method to improve the biclique test based on a power evaluation of the randomization test. We explicitly derived an expression for the power of the randomization test under several assumptions and found that a certain quantity calculated from a given assignment set characterizes the power. Based on this fact, we propose a method to improve the power of the biclique test by modifying the selection rule for subsets of units and assignments. Through a simulation with a spatial interference setting, we confirm that the proposed method has higher power than the existing method., Comment: 24 pages, 5 figures

We study properties of Fisher distribution (von Mises-Fisher distribution, matrix Langevin distribution) on the rotation group SO(3). In particular we apply the holonomic gradient descent, introduced by Nakayama et al. (2011), and a method of series expansion for evaluating the normalizing constant of the distribution and for computing the maximum likelihood estimate. The rotation group can be identified with the Stiefel manifold of two orthonormal vectors. Therefore from the viewpoint of statistical modeling, it is of interest to compare Fisher distributions on these manifolds. We illustrate the difference with an example of near-earth objects data., Comment: 25 pages, 1 figures

We propose a flexible statistical model for high-dimensional quantitative data on a hypercube. Our model, called the structural gradient model (SGM), is based on a one-to-one map on the hypercube that is a solution for an optimal transport problem. As we show with many examples, SGM can describe various dependence structures including correlation and heteroscedasticity. The maximum likelihood estimation of SGM is effectively solved by the determinant-maximization programming. In particular, a lasso-type estimation is available by adding constraints. SGM is compared with graphical Gaussian models and mixture models., Comment: 28pages, 6figures