A derivation of the multivariate singular skew-normal density function

2016 ◽  
Vol 117 ◽  
pp. 40-45 ◽  
Author(s):  
Phil D. Young ◽  
Jane L. Harvill ◽  
Dean M. Young
Keyword(s):  
Density Function ◽  
Normal Density ◽  
Normal Density Function ◽  
Skew Normal

scholarly journals Point-Symmetric Multivariate Density Function and Its Decomposition

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Kiyotaka Iki ◽  
Sadao Tomizawa
Keyword(s):  
Density Function ◽  
Point Symmetry ◽  
Normal Density ◽  
Multivariate Normal ◽  
Normal Density Function ◽  
Multivariate Density

For aT-variate density function, the present paper defines the point-symmetry, quasi-point-symmetry of orderk(<T), and the marginal point-symmetry of orderkand gives the theorem that the density function isT-variate point-symmetric if and only if it is quasi-point-symmetric and marginal point-symmetric of orderk. The theorem is illustrated for the multivariate normal density function.

Distribution of the branching-process population among generations

1971 ◽  
Vol 8 (4) ◽  
pp. 655-667 ◽  
Author(s):  
M. L. Samuels
Keyword(s):  
Normal Form ◽  
Density Function ◽  
Branching Process ◽  
Random Variables ◽  
Normal Density ◽  
Asymptotically Linear ◽  
Random Functions ◽  
Normal Density Function ◽  
Age Dependent ◽  
The Mean

SummaryIn a standard age-dependent branching process, let Rn(t) denote the proportion of the population belonging to the nth generation at time t. It is shown that in the supercritical case the distribution {Rn(t); n = 0, 1, …} has asymptotically, for large t, a (non-random) normal form, and that the mean ΣnRn(t) is asymptotically linear in t. Further, it is found that, for large n, Rn(t) has the shape of a normal density function (of t).Two other random functions are also considered: (a) the proportion of the nth generation which is alive at time t, and (b) the proportion of the nth generation which has been born by time t. These functions are also found to have asymptotically a normal form, but with parameters different from those relevant for {Rn(t)}.For the critical and subcritical processes, analogous results hold with the random variables replaced by their expectations.

Distribution of the branching-process population among generations

1971 ◽  
Vol 8 (04) ◽  
pp. 655-667 ◽  
Author(s):  
M. L. Samuels
Keyword(s):  
Normal Form ◽  
Density Function ◽  
Branching Process ◽  
Random Variables ◽  
Normal Density ◽  
Asymptotically Linear ◽  
Random Functions ◽  
Normal Density Function ◽  
Age Dependent ◽  
The Mean

Summary In a standard age-dependent branching process, let Rn (t) denote the proportion of the population belonging to the nth generation at time t. It is shown that in the supercritical case the distribution {Rn (t); n = 0, 1, …} has asymptotically, for large t, a (non-random) normal form, and that the mean ΣnRn (t) is asymptotically linear in t. Further, it is found that, for large n, Rn (t) has the shape of a normal density function (of t). Two other random functions are also considered: (a) the proportion of the nth generation which is alive at time t, and (b) the proportion of the nth generation which has been born by time t. These functions are also found to have asymptotically a normal form, but with parameters different from those relevant for {Rn (t)}. For the critical and subcritical processes, analogous results hold with the random variables replaced by their expectations.

On the normal moments distributions and its characterizations

2017 ◽  
Vol 6 (1-2) ◽  
pp. 16
Author(s):  
A. A. Olosunde
Keyword(s):  
Density Function ◽  
Shape Parameter ◽  
Normal Density ◽  
Normal Density Function ◽  
Moment Distribution

Normal moment distribution is a family of elliptical density, the density which depends on the shape parameter \(\alpha\), such that when \(\alpha=0\), it corresponds to the normal density function. Several properties of this class of density function and characterizations are established.

The Integration of the Multivariate Normal Density Function for the Triangular Method

1987 ◽  
pp. 321-328 ◽  
Author(s):  
John A. Kapenga ◽  
Kenneth Mullen ◽  
Elise Doncker ◽  
Daniel M. Ennis
Keyword(s):  
Density Function ◽  
Normal Density ◽  
Multivariate Normal ◽  
Normal Density Function ◽  
Triangular Method

Accumulated (integrated) interest and the square root process

2019 ◽  
Vol 32 (4) ◽  
pp. 678-691
Author(s):  
Diandian Ma ◽  
Xiaojing Song ◽  
Mark Tippett ◽  
Thu Phuong Truong
Keyword(s):  
Density Function ◽  
Strong Mixing ◽  
Normal Density ◽  
Square Root ◽  
Convergence In Distribution ◽  
Mixing Conditions ◽  
Instantaneous Rate ◽  
Content Type ◽  
Normal Density Function ◽  
Distributional Properties

Purpose The purpose of this study is to determine distributional properties of the accumulated rate of interest when the instantaneous rate of interest evolves in terms of the Cox et al. (1985) square root process. Design/methodology/approach The law of iterated (or double) expectations is used to determine the mean and variance of the accumulated rate of interest on a cash management (or loan) account when interest accumulates at the instantaneous rates of interest implied by the square root process. Findings This study demonstrates how the accumulated rate of interest does not satisfy the strong mixing conditions necessary for convergence in distribution to the normal density function. Originality/value This study has strong educational value in determining distributional properties of the accumulated rate of interest when the instantaneous rate of interest evolves in terms of the Cox et al. (1985) square root process and demonstrating how the accumulated rate of interest does not satisfy the strong mixing conditions necessary for convergence in distribution to the normal density function.

On generalized measures of entropy and dependence

2008 ◽  
Vol 58 (3) ◽  
Author(s):  
Parmil Kumar ◽  
D. Hooda
Keyword(s):  
Applied Sciences ◽  
Density Function ◽  
Shannon’S Entropy ◽  
Normal Density ◽  
Multivariate Normal ◽  
Shannon's Entropy ◽  
Measures Of Dependence ◽  
Normal Density Function ◽  
Generalized Entropies

AbstractVarious authors have studied extensions of Shannon’s entropy but their inferential properties and applications in applied sciences have not invited proper attention from researchers. In the present paper we explore the motivation and implication of using various classes of the generalized entropies and conditional entropies. We evaluate β-class and (α, β)-class entropies for multivariate normal density function. We also obtain the measures of dependence in terms of the classes of generalized entropies.

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