On linear combinations of weighted quasi-arithmetic means

Daróczy, Zoltán; Hajdu, Gabriella

doi:10.1007/s00010-004-2746-6

Cited by 11 publications

(11 citation statements)

References 6 publications

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“…In the special case when p = q = r in Eq. (3.18) the assertion of Proposition 3.32 follows immediately from [42,Theorem 4] where the extension theorem was proved for the equation…”

Section: Lemma 326 Let ϕ ψ ∈ Cm(i) Be Differentiable Functions Witmentioning

confidence: 86%

Invariance of means

Jarczyk

2018

Aequat. Math.

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We give a survey of results dealing with the problem of invariance of means which, for means of two variables, is expressed by the equality K • (M, N) = K. At the very beginning the Gauss composition of means and its strict connection with the invariance problem is presented. Most of the reported research was done during the last two decades, when means theory became one of the most engaging and influential topics of the theory of functional equations. The main attention has been focused on quasi-arithmetic and weighted quasi-arithmetic means, also on some of their surroundings. Among other means of great importance Bajraktarević means and Cauchy means are discussed.

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“…In the special case when p = q = r in Eq. (3.18) the assertion of Proposition 3.32 follows immediately from [42,Theorem 4] where the extension theorem was proved for the equation…”

Section: Lemma 326 Let ϕ ψ ∈ Cm(i) Be Differentiable Functions Witmentioning

confidence: 86%

Invariance of means

Jarczyk

2018

Aequat. Math.

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“…Observe that (16) is the generator of the Clayton family of copulas; refer to [17] for mathematical details about this class.…”

Section: Example 1 (The Clayton Family) Consider the Completely Monomentioning

confidence: 99%

“…More recent contributions are the works of Wimp (1986), Hutník (2006), Matkowski (1999; Jarczyk and Matkowski (2000), J. Marichal (2000), Daróczy and Hajdu (2005), and Abrahamovic et al (2006). Quasi-arithmetic means, in particular, have been applied in several disciplines.…”

Section: Introductionmentioning

confidence: 99%

Quasi-arithmetic means of covariance functions with potential applications to space–time data

Porcu

Mateu

Christakos

2009

Journal of Multivariate Analysis

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a b s t r a c tThe theory of quasi-arithmetic means represents a powerful tool in the study of covariance functions across space-time. In the present study we use quasi-arithmetic functionals to make inferences about the permissibility of averages of functions that are not, in general, permissible covariance functions. This is the case, e.g., of the geometric and harmonic averages, for which we obtain permissibility criteria. Also, some important inequalities involving covariance functions and preference relations as well as algebraic properties can be derived by means of the proposed approach. In particular, quasi-arithmetic covariances allow for ordering and preference relations, for a Jensen-type inequality and for a minimal and maximal element of their class. The general results shown in this paper are then applied to the study of spatial and spatio-temporal random fields. In particular, we discuss the representation and smoothness properties of a weakly stationary random field with a quasi-arithmetic covariance function. Also, we show that the generator of the quasiarithmetic means can be used as a link function in order to build a space-time nonseparable structure starting from the spatial and temporal margins, a procedure that is technically sound for those working with copulas. Several examples of new families of stationary covariances obtainable with this procedure are shown. Finally, we use quasi-arithmetic functionals to generalise existing results concerning the construction of nonstationary spatial covariances, and discuss the applicability and limits of this generalisation.

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“…Concretely, P. Burai considered the invariance of the arithmetic mean with the weighted quasi-arithmetic means in [2]. Z. Daróczy, G. Hajdu, J. Jarczyk and J. Matkowski studied the invariance equation involving three weighted quasi-arithmetic means [3,8,9]. J. Matkowski solved the invariance equation involving the arithmetic mean in class of Lagrangian mean-type mappings [13].…”

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confidence: 99%