scholarly journals Complete and sufficient statistics and perfect families in orthogonal and error orthogonal normal models

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Aníbal Areia ◽  
Francisco Carvalho ◽  
João T. Mexia
Keyword(s):  
Jordan Algebra ◽  
Algebraic Structure ◽  
Minimum Variance ◽  
Jordan Algebras ◽  
Symmetric Matrices ◽  
Sufficient Statistics ◽  
Unbiased Estimators

AbstractWe will discuss orthogonal models and error orthogonal models and their algebraic structure, using as background, commutative Jordan algebras. The role of perfect families of symmetric matrices will be emphasized, since they will play an important part in the construction of the estimators for the relevant parameters. Perfect families of symmetric matrices form a basis for the commutative Jordan algebra they generate. When normality is assumed, these perfect families of symmetric matrices will ensure that the models have complete and sufficient statistics. This will lead to uniformly minimum variance unbiased estimators for the relevant parameters.

Inference for the diffusion branching process

1975 ◽  
Vol 12 (3) ◽  
pp. 588-594 ◽  
Author(s):  
B. M. Brown ◽  
J. I. Hewitt
Keyword(s):  
Brownian Motion ◽  
Branching Process ◽  
Exponential Family ◽  
Minimum Variance ◽  
Sufficient Statistics ◽  
Sufficient Statistic ◽  
Unbiased Estimators ◽  
Standard Problem

For the diffusion branching process, we consider a method of inference that is essentially sequential in nature. The method allows us to simplify the natural sufficient statistics involved, and we are able to get their distributions quite easily by translating our problem into a standard problem in Brownian motion. Under certain circumstances, we are left with a complete sufficient statistic whose distribution belongs to an exponential family, and can therefore derive minimum variance unbiased estimators, etc.

Inference for the diffusion branching process

1975 ◽  
Vol 12 (03) ◽  
pp. 588-594 ◽  
Author(s):  
B. M. Brown ◽  
J. I. Hewitt
Keyword(s):  
Brownian Motion ◽  
Branching Process ◽  
Exponential Family ◽  
Minimum Variance ◽  
Sufficient Statistics ◽  
Sufficient Statistic ◽  
Unbiased Estimators ◽  
Standard Problem

For the diffusion branching process, we consider a method of inference that is essentially sequential in nature. The method allows us to simplify the natural sufficient statistics involved, and we are able to get their distributions quite easily by translating our problem into a standard problem in Brownian motion. Under certain circumstances, we are left with a complete sufficient statistic whose distribution belongs to an exponential family, and can therefore derive minimum variance unbiased estimators, etc.

SYMMETRIES OF CERTAIN PHYSICAL THEORIES AND THE JORDAN ALGEBRAS

1992 ◽  
Vol 07 (15) ◽  
pp. 3623-3637 ◽  
Author(s):  
R. FOOT ◽  
G. C. JOSHI
Keyword(s):  
Jordan Algebra ◽  
Space Time ◽  
Jordan Algebras ◽  
Bosonic String ◽  
The Other ◽  
Structure Group ◽  
Division Algebras ◽  
Hermitian Matrices ◽  
Higher Dimensional ◽  
Algebraic Framework

It is shown that the sequence of Jordan algebras [Formula: see text], whose elements are the 3 × 3 Hermitian matrices over the division algebras ℝ, [Formula: see text], ℚ and [Formula: see text], can be associated with the bosonic string as well as the superstring. The construction reveals that the space–time symmetries of the first-quantized bosonic string and superstring actions can be related. The bosonic string and the superstring are associated with the exceptional Jordan algebra while the other Jordan algebras in the [Formula: see text] sequence can be related to parastring theories. We then proceed to further investigate a connection between the symmetries of supersymmetric Lagrangians and the transformations associated with the structure group of [Formula: see text]. The N = 1 on-shell supersymmetric Lagrangians in 3, 4 and 6-dimensions with a spin 0 field and a spin 1/2 field are incorporated within the Jordan-algebraic framework. We also make some remarks concerning a possible role for the division algebras in the construction of higher-dimensional extended objects.

scholarly journals Eisenstein Series Arising from Jordan Algebras

2019 ◽  
Vol 72 (1) ◽  
pp. 183-201 ◽  
Author(s):  
Marcela Hanzer ◽  
Gordan Savin
Keyword(s):  
Eisenstein Series ◽  
Jordan Algebra ◽  
Jordan Algebras ◽  
Algebra Structure ◽  
Parabolic Subgroups ◽  
Automorphic Representations ◽  
Unipotent Radicals ◽  
Maximal Parabolic Subgroups

AbstractWe describe poles and the corresponding residual automorphic representations of Eisenstein series attached to maximal parabolic subgroups whose unipotent radicals admit Jordan algebra structure.

Equivalent norms on Banach Jordan algebras

1979 ◽  
Vol 86 (2) ◽  
pp. 261-270 ◽  
Author(s):  
M. A. Youngson
Keyword(s):  
Jordan Algebra ◽  
Sufficient Conditions ◽  
Equivalent Norm ◽  
Jordan Algebras ◽  
The Self ◽  
Equivalent Norms ◽  
Positive Elements ◽  
Definition Of ◽  
Necessary And Sufficient

1. Introduction. Recently Kaplansky suggested the definition of a suitable Jordan analogue of B*-algebras, which we call J B*-algebras (see (10) and (11)). In this article, we give a characterization of those complex unital Banach Jordan algebras which are J B*-algebras in an equivalent norm. This is done by generalizing results of Bonsall ((3) and (4)) to give necessary and sufficient conditions on a real unital Banach Jordan algebra under which it is the self-adjoint part of a J B*-algebra in an equivalent norm. As a corollary we also obtain a characterization of the cones in a Banach Jordan algebra which are the set of positive elements of a J B*-algebra.

Malcev–Poisson–Jordan algebras

2016 ◽  
Vol 15 (09) ◽  
pp. 1650159
Author(s):  
Malika Ait Ben Haddou ◽  
Saïd Benayadi ◽  
Said Boulmane
Keyword(s):  
Vector Space ◽  
Lie Algebras ◽  
Jordan Algebra ◽  
Jordan Algebras ◽  
Leibniz Rule ◽  
Malcev Algebra ◽  
Jordan Structure ◽  
Alternative Algebras

Malcev–Poisson–Jordan algebra (MPJ-algebra) is defined to be a vector space endowed with a Malcev bracket and a Jordan structure which are satisfying the Leibniz rule. We describe such algebras in terms of a single bilinear operation, this class strictly contains alternative algebras. For a given Malcev algebra [Formula: see text], it is interesting to classify the Jordan structure ∘ on the underlying vector space of [Formula: see text] such that [Formula: see text] is an MPJ-algebra (∘ is called an MPJ-structure on Malcev algebra [Formula: see text]. In this paper we explicitly give all MPJ-structures on some interesting classes of Malcev algebras. Further, we introduce the concept of pseudo-Euclidean MPJ-algebras (PEMPJ-algebras) and we show how one can construct new interesting quadratic Lie algebras and pseudo-Euclidean Malcev (non-Lie) algebras from PEMPJ-algebras. Finally, we give inductive descriptions of nilpotent PEMPJ-algebras.

scholarly journals Jordan Subalgebras of Banach algebras

1978 ◽  
Vol 21 (2) ◽  
pp. 103-110 ◽  
Author(s):  
F. F. Bonsall
Keyword(s):  
Hilbert Space ◽  
Jordan Algebra ◽  
Banach Algebras ◽  
Jordan Algebras ◽  
Adjoint Operators

We recall that a JC-algebra (Størmer (3)) is a norm closed Jordan algebra of self-adjoint operators on a Hilbert space. Recently, Alfsen, Shultz, and Størmer (1) have introduced a class of abstract normed Jordan algebras called JB-algebras, and have proved that every special JB-algebra is isometrically isomorphic to a JC-algebra. We show that this result brings to a satisfactory conclusion the discussion in (2) of certain wedges W in Banach algebras and their related Jordan algebras W–W, and leads to two characterisations of the bicontinuously isomorphic images of JC-algebras.

scholarly journals Cartan Subalgebras of Jordan Algebras

1966 ◽  
Vol 27 (2) ◽  
pp. 591-609 ◽  
Author(s):  
N. Jacobson
Keyword(s):  
Lie Algebra ◽  
Jordan Algebra ◽  
Cartan Subalgebra ◽  
Jordan Algebras ◽  
Classical Notion ◽  
Cartan Subalgebras ◽  
Definition Of

In this paper we shall give a definition of an analogue for Jordan algebras of the classical notion of a Cartan subalgebra of a Lie algebra. This is based on a notion of associator nilpotency of a Jordan algebra. A Jordan algebra is called associator nilpotent if there exists a positive (odd) integer M such that every associator of order M formed of elements of is 0 (§2).

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