Cumulants and partition lattices VI. variances and covariances of mean squares

Speed, T. P.; Silcock, Howard L.

doi:10.1017/s1446788700032158

Cited by 8 publications

(3 citation statements)

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“…The whole subject is later described by Stuart and Ord [75] in great detail. In the Eighties, tensor notation was taken by Speed [62] - [67] and extended to polykays and single k-statistics. This extension reveals the coefficients defining polykays to be values of the Moebius function over the lattice of set partitions.…”

Section: K-statisticsmentioning

confidence: 99%

From Combinatorics to Philosophy

Damiani¹,

DʼAntona²,

Marra³

et al. 2009

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Section: K-statisticsmentioning

confidence: 99%

From Combinatorics to Philosophy

Damiani¹,

DʼAntona²,

Marra³

et al. 2009

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“…In a paper summarizing a long study of the analysis of variance [17][18][19][20][21][22][23][24], T. P. Speed formulated the Orbit Problem.…”

Section: Introductionmentioning

confidence: 99%

Speed’s Orbit Problem: Variations on Themes of Burnside

Smith

1999

European Journal of Combinatorics

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“…It led to the development of an extended theory of symmetric functions for structured and nested arrays associated with a certain subgroup [20,21,22,23]. Elegant though they are, these papers are not for the faint of heart.…”

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confidence: 99%

Cumulants and Partition Lattices

McCullagh

2012

Selected Works of Terry Speed

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This is the first paper to appear in the statistical literature pointing out the importance of the partition lattice in the theory of statistical moments and their close cousins, the cumulants. The paper was first brought to my attention by Susan Wilson, shortly after I had given a talk at Imperial College on the Leonov-Shiryaev result expressed in graph-theoretic terms. Speed's paper was hot off the press, arriving a day or two after I had first become acquainted with the partition lattice from conversations with Oliver Pretzel. Naturally, I read the paper with more than usual attention to detail because I was still unfamiliar with Rota [18], and because it was immediately clear that Möbius inversion on the partition lattice E n , partially ordered by sub-partition, led to clear proofs and great simplification. It was a short paper packing a big punch, and for me it could not have arrived at a more opportune moment.The basic notion is a partition σ of the finite set [n] = {1,...,n}, a collection of disjoint non-empty subsets whose union is [n]. Occasionally, the more emphatic term set-partition is used to distinguish a partition of [n] from a partition of the integer n. For example 135|2|4 and 245|1|3 are distinct partitions of [5] corresponding to the same partition 3 + 1 + 1 of the integer 5. Altogether, there are two partitions of [2], five partitions of [3], 15 partitions of [4], 52 partitions of [5], and so on. These are the Bell numbers #E n , whose exponential generating function is exp(e t − 1). The symmetric group acting on E n preserves block sizes, and each integer partition is a group orbit. There are two partitions of the integer 2, three partitions of 3, five partitions of 4, seven partitions of 5, and so on.It turns out that, although set partitions are much larger, the additional structure they provide is essential for at least two purposes that are fundamental in modern probability and statistics. It is the partial order and the lattice property of E n that simplifies the description of moments and generalized cumulants in terms of cumulants. This is the subject matter of Speed's paper. At around the same time, from the late 1970s until the mid 1980s, Kingman was developing the theory of partition structures, or partition processes. These were initially described in terms of inte- [3,10], but subsequent workers including Kingman and Aldous have found it simpler and more natural to work with set partitions. In this setting, the simplification comes not from the lattice property, but from the fact that the family E = {E 1 , E 2 ,...} of set partitions is a projective system, closed under permutation and deletion of elements. The projective property makes it possible to define a process on E , and the mutual consistency of the Ewens formulae for different n implies an infinitely exchangeable partition process.In his 1964 paper, Rota pointed out that the inclusion-exclusion principle and much of combinatorics could be unified in the following manner. To any function f defined on a finite partially...

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Cumulants and partition lattices VI. variances and covariances of mean squares

Cited by 8 publications

References 24 publications

From Combinatorics to Philosophy

From Combinatorics to Philosophy

Speed’s Orbit Problem: Variations on Themes of Burnside

Cumulants and Partition Lattices

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