Selection of variables in two-group discriminant analysis by error rate and Akaike's information criteria

This paper deals with two criteria for selection of variables for the discriminant analysis in the case of two multivariate normal populations with different means and a common covariance matrix. One is based on the estimated error rate of misclassification. The other uses Akaike's information criterion. The asymptotic distributions and error rate risks of the criteria are obtained. The result will prove that the two criteria are asymptotically equivalent in the sense of their asymptotic distributions and error rate risks being identical.     

Conjugacy relationship between M-convex and L-convex functions in continuous variables

By extracting combinatorial structures in well-solved nonlinear combinatorial optimization problems, Murota (1996,1998) introduced the concepts of M-convexity and L-convexity to functions defined over the integer lattice. Recently, Murota–Shioura (2000, 2001) extended these concepts to polyhedral convex functions and quadratic functions in continuous variables. In this paper, we consider a further extension to more general convex functions defined over the real space, and provide a proof for the conjugacy relationship between general M-convex and L-convex functions.Mathematics Subject Classification (1991): 90C10, 90C25, 90C27, 90C35This work is supported by Grant-in-Aid of the Ministry of Education, Culture, Sports, Science and Technology of Japan     

Relationship of causal effects in a causal chain and related inference

This paper discusses the relationship among the total causal effect and local causal effects in a causal chain and identifiability of causal effects. We show a transmission relationship of causal effects in a causal chain. According to the relationship, we give an approach to eliminating confounding bias through controlling for intermediate variables in a causal chain.     

Minimization of a functional over the set of causal operators of causal Hilbert space

The minimization problem for a quadratic functional defined on the set of nonwarning (causal) operators acting in a causal Hilbert space can be regarded as an abstrat analog of the Wiener problem on constructing the optimal nonwarning filter. A similar problem also arises in the linear control problem with the quadratic performance criterion (in this case the transfer operators of a closed control system serve as causal ones). The introduction of causal operators in filtering theory and control theory is a mathematical expression of the causality principle, which must be taken into account for a number of problems. In the present paper we attempt to systematize the mathematical foundations of the abstract linear filtering theory, for which its basic results are expressed in terms of operators describing the filtering problem. We introduce and study a class of finite operators, a natural generalization of the class of causal operators, and give a solution of the minimization problem for a quadratic positive functional defined on the set of causal operators acting in a “discrete” causal space. Bibliography: 54 titles. Translated fromProblemy Matematicheskogo Analiza, No. 14, 1995. pp. 143–187.     

A relationship between the maximum modulus and Taylor coefficients of entire functions of several complex variables

We present results on the relationship between the growth of the maximum modulus and the decay of Taylor coefficients of entire functions of several variables. The results are obtained by two different methods, the first of which had been proposed earlier by Oskolkov for the one-dimensional case, and the second is based on the use of the Legendre-Jung-Fenchel conjugates of the weight functions. Attention is mainly paid to the characterization of the growth of entire functions with respect to the conjunction of variables; however, some results are obtained for the case in which there is different growth with respect to different variables. Translated fromMatematicheskie Zametki, Vol. 62, No. 2, pp. 238–258, August, 1997. Translated by N. K. Kulman     

重尾随机变量加权和一致尾等价关系成立的条件

已有文献指出在次指数族重尾损失的情形下,独立同分布随机变量加权和的一致尾等价关系成立.文中主要在随机变量独立但不同分布的情况下,得到了上述尾等价关系一致成立时分布函数所应满足的一些条件.     

Nonparametric estimation of interval-censored failure time data in the presence of informative censoring

Nonparametric estimation of a survival function is one of the most commonly asked questions in the analysis of failure time data and for this, a number of procedures have been developed under various types of censoring structures (Kalbfleisch and Prentice, 2002). In particular, several algorithms are available for interval-censored failure time data with independent censoring mechanism (Sun, 2006; Turnbull, 1976). In this paper, we consider the interval-censored data where the censoring mechanism may be related to the failure time of interest, for which there does not seem to exist a nonparametric estimation procedure. It is well-known that with informative censoring, the estimation is possible only under some assumptions. To attack the problem, we take a copula model approach to model the relationship between the failure time of interest and censoring variables and present a simple nonparametric estimation procedure. The method allows one to conduct a sensitivity analysis among others.     

On a characterization of the essential spectra of some matrix operators and application to two-group transport operators

In this paper, we investigate the essential approximate point spectrum and the essential defect spectrum of a 2 × 2 block operator matrix on a Banach space. Furthermore, we apply the obtained results to two-group transport operators in the Banach space L _p ([−a, a] × [−1, 1]) × L _p ([−a, a] × [−1, 1]), a > 0, p ≥ 1.     

Inferencing the graphs of causal Markov fields

A statistical approach is presented for assessing the qualitative dependence structure of a causal or directed Markov field. A notion of a probability distance is established and a construction based hereon as well as an entropy related construction are investigated. Among others, the investigation goes along a particular notion of definiteness of directed acyclic graphs.     

Representations of ordered random variables in the terms of the sums or products of independent variables

This is a survey of the available representations capable of expressing dependent ordered variables like order statistics and records as sums of independent summands or as products of independent random multipliers. Some new similar representations are obtained that in a similar manner express the common distributions of representatives of these two types of ordered random variables.     

On the suppression of variables in Boolean equations

The resultant of suppression of variables from a Boolean equation is a Boolean equation, derived from the parent equation, whose solutions are exactly those of the parent equation that do not involve the suppressed variables. Two examples in the literature are discussed, in which it is necessary to solve a Boolean equation while excluding solutions involving certain variables. In such cases it would be advantageous to solve the resultant of suppression of those variables rather than solving the original equation and filtering the desired solutions from the results.     

Bilocal variables in the physics of condensed media

Institute of General Physics, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 81, No. 3, pp. 385–404, December, 1989.     

On the Mahler measure in several variables

If the total degree of a polynomial in n 2 variables of dimensionn is bounded by a double exponential function in n, then weshow that its Mahler measure is bounded from below by an absoluteconstant greater than 1.     

On the orthogonal polynomials in several variables

We study the connection between orthogonal polynomials in several variables and families of commuting symmetric operators of a special form.