scholarly journals The general theory of canonical correlation and its relation to functional analysis

1961 ◽  
Vol 2 (2) ◽  
pp. 229-242 ◽  
Author(s):  
E. J. Hannan
Keyword(s):  
Linear Combination ◽  
Canonical Correlation ◽  
Random Variables ◽  
Random Variable ◽  
Finite Variance ◽  
Finite Sample ◽  
Standard Description ◽  
Joint Distributions ◽  
Gaussian Case ◽  
The Relationship

The classical theory of canonical correlation is concerned with a standard description of the relationship between any linear combination of ρ random variablesxs, and any linear combination ofqrandom variablesytinsofar as this relation can be described in terms of correlation. Lancaster [1] has extended this theory, forp=q= 1, to include a description of the correlation of any function of a random variablexand any function of a random variabley(both functions having finite variance) for a class of joint distributions ofxandywhich is very general. It is the purpose of this paper to derive Lancaster's results from general theorems concerning the spectral decomposition of operators on a Hilbert space. These theorems lend themselves easily to the generalisation of the theory to situations wherepandqare not finite. In the case of Gaussian, stationary, processes this generalisation is equivalent to the classical spectral theory and corresponds to a canonical reduction of a (finite) sample of data which is basic. The theory also then extends to any number of processes. In the Gaussian case, also, the present discussion-is connected with the results of Gelfand and Yaglom [2] relating to the amount of information in one random process about another.

A mixed autoregressive-moving average exponential sequence and point process (EARMA 1,1)

1977 ◽  
Vol 9 (1) ◽  
pp. 87-104 ◽  
Author(s):  
P. A. Jacobs ◽  
P. A. W. Lewis
Keyword(s):  
Linear Combination ◽  
Point Process ◽  
Moving Average ◽  
Random Variables ◽  
Stationary Sequence ◽  
Autoregressive Moving Average ◽  
Marginal Distributions ◽  
Joint Distributions ◽  
Exponential Random Variables ◽  
Exponential Sequence

A stationary sequence of random variables with exponential marginal distributions and the correlation structure of an ARMA (1, 1) process is defined. The process is formed as a random linear combination of i.i.d. exponential random variables and is very simple to generate on a computer. Moments and joint distributions for the sequence are obtained, as well as limiting properties of sums of the random variables and of the point process whose intervals have the EARMA (1, 1) structure.

A Poisson limit theorem for weakly exchangeable events

1979 ◽  
Vol 16 (04) ◽  
pp. 794-802 ◽  
Author(s):  
G. K. Eagleson
Keyword(s):  
Limit Theorem ◽  
Spatial Data ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Random Variable ◽  
Limit Distributions ◽  
Joint Distributions ◽  
Poisson Limit ◽  
Image Position ◽  
Independent Identically Distributed

Let Y 1, Y2 , · ·· be a sequence of independent, identically distributed random variables, g some symmetric 0–1 function of m variables and set Silverman and Brown (1978) have shown that under certain conditions the statistic is asymptotically distributed as a Poisson random variable. They then use this result to derive limit distributions for various statistics, useful in the analysis of spatial data. In this paper, it is shown that Silverman and Brown's theorem holds under much weaker assumptions; assumptions which involve only the symmetry of the joint distributions of the X il…i m .

Computation of Density for A Linear Combination of –Generalized Normal Random Variables

1975 ◽  
Vol 24 (1-4) ◽  
pp. 101-116
Author(s):  
Ru-Ying Lee ◽  
I. R. Goodman
Keyword(s):  
Linear Combination ◽  
Power Series Expansion ◽  
Random Variables ◽  
Random Variable ◽  
Computational Procedure ◽  
Normal Random Variable ◽  
Characteristic Functions ◽  
Accuracy Control ◽  
Truncated Form ◽  
Fourier Inversion

A computational procedure is presented for the approximation of the density of a linear combination of univariate -generalized normal random variables. (The -generalized normal random variable generalizes the ordinary normal one by replacing the power two in the exponent of the density by an arbitrary positive number.) The procedure applies a truncated form of the Fourier Inversion Theorem to the power series expansion of the characteristic function of a -generalized normal random variable. Because of the unimodal nature of -generalized normal characteristic functions for ⩽ 2 and the oscillatory nature for > 2, much of the computational procedure divides into two corresponding parts. Complete error analysis and accuracy control in all computations are also presented.

scholarly journals Exponential and Hypoexponential Distributions: Some Characterizations

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2207
Author(s):  
George P. Yanev
Keyword(s):  
Linear Combination ◽  
Exponential Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Unknown Distribution ◽  
Converse Result ◽  
Hypoexponential Distribution ◽  
Exponential Random Variables

The (general) hypoexponential distribution is the distribution of a sum of independent exponential random variables. We consider the particular case when the involved exponential variables have distinct rate parameters. We prove that the following converse result is true. If for some n≥2, X1,X2,…,Xn are independent copies of a random variable X with unknown distribution F and a specific linear combination of Xj’s has hypoexponential distribution, then F is exponential. Thus, we obtain new characterizations of the exponential distribution. As corollaries of the main results, we extend some previous characterizations established recently by Arnold and Villaseñor (2013) for a particular convolution of two random variables.

Asymptotic properties for the loglog laws under positive association

2012 ◽  
Vol 62 (5) ◽  
Author(s):  
Xiao-Rong Yang ◽  
Ke-Ang Fu
Keyword(s):  
Gamma Function ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Positive Association ◽  
Random Variable ◽  
Stationary Sequence ◽  
Normal Random Variable ◽  
Finite Variance ◽  
Associated Random Variables ◽  
Standard Normal

AbstractLet {X n: n ≥ 1} be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set $$S_n = \sum\limits_{k = 1}^n {X_k }$$, $$Mn = \mathop {\max }\limits_{k \leqslant n} \left| {S_k } \right|$$, n ≥ 1. Suppose that $$0 < \sigma ^2 = EX_1^2 + 2\sum\limits_{k = 2}^\infty {EX_1 X_k < \infty }$$. In this paper, we prove that if E|X 1|2+δ < for some δ ∈ (0, 1], and $$\sum\limits_{j = n + 1}^\infty {Cov\left( {X_1 ,X_j } \right) = O\left( {n^{ - \alpha } } \right)}$$ for some α > 1, then for any b > −1/2 $$\mathop {\lim }\limits_{\varepsilon \searrow 0} \varepsilon ^{2b + 1} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^{b - 1/2} }} {{n^{3/2} \log n}}} E\left\{ {M_n - \sigma \varepsilon \sqrt {2n\log \log n} } \right\}_ + = \frac{{2^{ - 1/2 - b} E\left| N \right|^{2(b + 1)} }} {{(b + 1)(2b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2(b + 1)} }}}$$ and $$\mathop {\lim }\limits_{\varepsilon \nearrow \infty } \varepsilon ^{ - 2(b + 1)} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^b }} {{n^{3/2} \log n}}E\left\{ {\sigma \varepsilon \sqrt {\frac{{\pi ^2 n}} {{8\log \log n}}} - M_n } \right\}} _ + = \frac{{\Gamma (b + 1/2)}} {{\sqrt 2 (b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2b + 2} }}} ,$$ where x + = max{x, 0}, N is a standard normal random variable, and Γ(·) is a Gamma function.

Convex majorization with an application to the length of critical paths

1979 ◽  
Vol 16 (03) ◽  
pp. 671-677 ◽  
Author(s):  
Isaac Meilijson ◽  
Arthur Nádas
Keyword(s):  
Maximal Function ◽  
Completion Time ◽  
Random Variables ◽  
Random Variable ◽  
List Type ◽  
Type Number ◽  
Joint Distributions ◽  
Littlewood Maximal Function ◽  
Delay Times ◽  
Critical Paths

1. (Y) for all non-negative, non-decreasing convex functions φ (X is convexly smaller than Y) if and only if, for all . 2. Let H be the Hardy–Littlewood maximal function HY (x) = E(Y – X | Y &gt; x). Then HY (Y) is the smallest random variable exceeding stochastically all random variables convexly smaller than Y. 3. Let X 1 X 2 · ·· Xn be random variables with given marginal distributions, let I 1, I 2, ···, Ik be arbitrary non-empty subsets of {1,2, ···, n} and let M = max (M is the completion time of a PERT network with paths Ij , and delay times Xi .) The paper introduces a computation of the convex supremum of M in the class of all joint distributions of the Xi 's with specified marginals, and of the ‘bottleneck probability' of each path.

A mixed autoregressive-moving average exponential sequence and point process (EARMA 1,1)

1977 ◽  
Vol 9 (01) ◽  
pp. 87-104
Author(s):  
P. A. Jacobs ◽  
P. A. W. Lewis
Keyword(s):  
Linear Combination ◽  
Point Process ◽  
Moving Average ◽  
Random Variables ◽  
Stationary Sequence ◽  
Autoregressive Moving Average ◽  
Marginal Distributions ◽  
Joint Distributions ◽  
Exponential Random Variables ◽  
Exponential Sequence

A stationary sequence of random variables with exponential marginal distributions and the correlation structure of an ARMA (1, 1) process is defined. The process is formed as a random linear combination of i.i.d. exponential random variables and is very simple to generate on a computer. Moments and joint distributions for the sequence are obtained, as well as limiting properties of sums of the random variables and of the point process whose intervals have the EARMA (1, 1) structure.

A Poisson limit theorem for weakly exchangeable events

1979 ◽  
Vol 16 (4) ◽  
pp. 794-802 ◽  
Author(s):  
G. K. Eagleson
Keyword(s):  
Limit Theorem ◽  
Spatial Data ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Random Variable ◽  
Limit Distributions ◽  
Joint Distributions ◽  
Poisson Limit ◽  
Image Position ◽  
Independent Identically Distributed

Let Y1, Y2, · ·· be a sequence of independent, identically distributed random variables, g some symmetric 0–1 function of m variables and set Silverman and Brown (1978) have shown that under certain conditions the statistic is asymptotically distributed as a Poisson random variable. They then use this result to derive limit distributions for various statistics, useful in the analysis of spatial data. In this paper, it is shown that Silverman and Brown's theorem holds under much weaker assumptions; assumptions which involve only the symmetry of the joint distributions of the Xil…im.

Convex majorization with an application to the length of critical paths

1979 ◽  
Vol 16 (3) ◽  
pp. 671-677 ◽  
Author(s):  
Isaac Meilijson ◽  
Arthur Nádas
Keyword(s):  
Maximal Function ◽  
Completion Time ◽  
Random Variables ◽  
Random Variable ◽  
List Type ◽  
Type Number ◽  
Joint Distributions ◽  
Littlewood Maximal Function ◽  
Delay Times ◽  
Critical Paths

1. (Y) for all non-negative, non-decreasing convex functions φ (X is convexly smaller than Y) if and only if, for all .2.Let H be the Hardy–Littlewood maximal function HY(x) = E(Y – X | Y > x). Then HY(Y) is the smallest random variable exceeding stochastically all random variables convexly smaller than Y.3.Let X1X2 · ·· Xn be random variables with given marginal distributions, let I1,I2, ···, Ik be arbitrary non-empty subsets of {1,2, ···, n} and let M = max (M is the completion time of a PERT network with paths Ij, and delay times Xi.) The paper introduces a computation of the convex supremum of M in the class of all joint distributions of the Xi's with specified marginals, and of the ‘bottleneck probability' of each path.

Analysis of Maximum Expiratory Flow Volume Curves Using Canonical Correlation Analysis

1985 ◽  
Vol 24 (02) ◽  
pp. 91-100 ◽  
Author(s):  
W. van Pelt ◽  
Ph. H. Quanjer ◽  
M. E. Wise ◽  
E. van der Burg ◽  
R. van der Lende
Keyword(s):  
Correlation Analysis ◽  
Canonical Correlation Analysis ◽  
Canonical Correlation ◽  
Flow Volume ◽  
Non Linear ◽  
Maximum Expiratory Flow ◽  
Expiratory Flow ◽  
Relationship Of ◽  
The Relationship ◽  
Age And Sex

SummaryAs part of a population study on chronic lung disease in the Netherlands, an investigation is made of the relationship of both age and sex with indices describing the maximum expiratory flow-volume (MEFV) curve. To determine the relationship, non-linear canonical correlation was used as realized in the computer program CANALS, a combination of ordinary canonical correlation analysis (CCA) and non-linear transformations of the variables. This method enhances the generality of the relationship to be found and has the advantage of showing the relative importance of categories or ranges within a variable with respect to that relationship. The above is exemplified by describing the relationship of age and sex with variables concerning respiratory symptoms and smoking habits. The analysis of age and sex with MEFV curve indices shows that non-linear canonical correlation analysis is an efficient tool in analysing size and shape of the MEFV curve and can be used to derive parameters concerning the whole curve.

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