scholarly journals On a Vector-Valued Measure of Multivariate Skewness

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1817
Author(s):  
Nicola Loperfido
Keyword(s):  
Multivariate Normality ◽  
Analysis Model ◽  
Third Order ◽  
Model Based Clustering ◽  
Semiparametric Modeling ◽  
Multivariate Skewness ◽  
Multivariate Kurtosis ◽  
Scalar Measure ◽  
Tensor Contraction ◽  
Vector Valued

The canonical skewness vector is an analytically simple function of the third-order, standardized moments of a random vector. Statistical applications of this skewness measure include semiparametric modeling, independent component analysis, model-based clustering, and multivariate normality testing. This paper investigates some properties of the canonical skewness vector with respect to representations, transformations, and norm. In particular, the paper shows its connections with tensor contraction, scalar measures of multivariate kurtosis and Mardia’s skewness, the best-known scalar measure of multivariate skewness. A simulation study empirically compares the powers of tests for multivariate normality based on the squared norm of the canonical skewness vector and on Mardia’s skewness. An example with financial data illustrates the statistical applications of the canonical skewness vector.

Well-posedness of Third Order Differential Equations in Hölder Continuous Function Spaces

2018 ◽  
Vol 62 (4) ◽  
pp. 715-726
Author(s):  
Shangquan Bu ◽  
Gang Cai
Keyword(s):  
Continuous Function ◽  
Differential Equations ◽  
Function Spaces ◽  
Linear Operators ◽  
Well Posedness ◽  
Third Order ◽  
Hölder Continuous ◽  
Vector Valued ◽  
Closed Linear Operators

AbstractIn this paper, by using operator-valued ${\dot{C}}^{\unicode[STIX]{x1D6FC}}$-Fourier multiplier results on vector-valued Hölder continuous function spaces, we give a characterization of the $C^{\unicode[STIX]{x1D6FC}}$-well-posedness for the third order differential equations $au^{\prime \prime \prime }(t)+u^{\prime \prime }(t)=Au(t)+Bu^{\prime }(t)+f(t)$, ($t\in \mathbb{R}$), where $A,B$ are closed linear operators on a Banach space $X$ such that $D(A)\subset D(B)$, $a\in \mathbb{C}$ and $0<\unicode[STIX]{x1D6FC}<1$.

Bilateral series and Ramanujan radial limits of mock (false) theta functions

2019 ◽  
Vol 30 (04) ◽  
pp. 1950023
Author(s):  
Bin Chen
Keyword(s):  
Modular Forms ◽  
Theta Functions ◽  
Mock Theta Functions ◽  
Third Order ◽  
Radial Limits ◽  
The Third ◽  
False Theta Functions ◽  
Even Order ◽  
Fifth Order ◽  
Vector Valued

Ramanujan gave a list of seventeen functions which he called mock theta functions. For one of the third-order mock theta functions [Formula: see text], he claimed that as [Formula: see text] approaches an even order [Formula: see text] root of unity [Formula: see text], then [Formula: see text] He also pointed at the existence of similar properties for other mock theta functions. Recently, [J. Bajpai, S. Kimport, J. Liang, D. Ma and J. Ricci, Bilateral series and Ramanujan’s radial limits, Proc. Amer. Math. Soc. 143(2) (2014) 479–492] presented some similar Ramanujan radial limits of the fifth-order mock theta functions and their associated bilateral series are modular forms. In this paper, by using the substitution [Formula: see text] in the Ramanujan’s mock theta functions, some associated false theta functions in the sense of Rogers are obtained. Such functions can be regarded as Eichler integral of the vector-valued modular forms of weight [Formula: see text]. We find two associated bilateral series of the false theta functions with respect to the fifth-order mock theta functions are special modular forms. Furthermore, we explore that the other two associated bilateral series of the false theta functions with respect to the third-order mock theta functions are mock modular forms. As an application, the associated Ramanujan radial limits of the false theta functions are constructed.

scholarly journals The product of vector-valued measures

1973 ◽  
Vol 8 (3) ◽  
pp. 359-366 ◽  
Author(s):  
Charles Swartz
Keyword(s):  
Scalar Field ◽  
Banach Spaces ◽  
Vector Measure ◽  
Sufficient Condition ◽  
Pettis Integral ◽  
Necessary And Sufficient Condition ◽  
Bilinear Map ◽  
Scalar Measure ◽  
Necessary And Sufficient ◽  
Vector Valued

Let M (N) be a σ–algebra of subsets of a set S (T) and let X, Y be Banach spaces with (,) a continuous bilinear map from X × Y into the scalar field. If μ: M → X (v: N → Y) is a vector measure and λ is the scalar measure defined on the measurable rectangles A × B, A ∈ M, B ∈ N, by λ(A×B) = 〈μ(A), v(B)〉, it is known that λ is generally not countably additive on the algebra generated by the measurable rectangles and therefore has no countably additive extension to the σ-algebra generated by the measurable rectangles. If μ (v) is an indefinite Pettis integral it is shown that a necessary and sufficient condition that λ have a countable additive extension to the σ-algebra generated by the measurable rectangles is that the function F: (s, t) → 〈f(s), g(t)〉 is integrable with respect to α × β.

scholarly journals MaxSkew and MultiSkew: Two R Packages for Detecting, Measuring and Removing Multivariate Skewness

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 970 ◽  
Author(s):  
Cinzia Franceschini ◽  
Nicola Loperfido
Keyword(s):  
Statistical Methods ◽  
Multivariate Data ◽  
Moment Matrix ◽  
Third Order ◽  
The Third ◽  
Multivariate Skewness ◽  
Sample Test ◽  
R Packages

The R packages MaxSkew and MultiSkew measure, test and remove skewness from multivariate data using their third-order standardized moments. Skewness is measured by scalar functions of the third standardized moment matrix. Skewness is tested with either the bootstrap or under normality. Skewness is removed by appropriate linear projections. The packages might be used to recover data features, as for example clusters and outliers. They are also helpful in improving the performances of statistical methods, as for example the Hotelling’s one-sample test. The Iris dataset illustrates the usages of MaxSkew and MultiSkew.

Sign in / Sign up

Export Citation Format

Share Document