On the ith Latent Root of a Complex Matrix(1)

1972 ◽  
Vol 15 (3) ◽  
pp. 323-327
Author(s):  
Sabri Al-Ani
Keyword(s):  
Statistical Analysis ◽  
Covariance Matrix ◽  
Joint Distribution ◽  
Latent Root ◽  
Multivariate Normal ◽  
Complex Matrix ◽  
Image Position ◽  
Latent Roots

Goodman [1] has pointed out the applications of the distributional results of the complex multivariate normal statistical analysis. Khatri [4], has suggested the maximum latent root statistic for testing the reality of a covariance matrix. The joint distribution of the latent roots under certain null hypotheses can be written as, [2], [3],1whereand

scholarly journals On n/p-Asymptotic Distribution Of Vector Of Weighted Traces Of Powers Of Wishart Matrices

2018 ◽  
Vol 33 ◽  
pp. 24-40 ◽  
Author(s):  
Jolanta Pielaszkiewicz ◽  
Dietrich Von Rosen ◽  
Martin Singull
Keyword(s):  
Normal Distribution ◽  
Covariance Matrix ◽  
Asymptotic Distribution ◽  
Joint Distribution ◽  
Recursive Formula ◽  
Multivariate Normal ◽  
Wishart Matrices ◽  
The Matrix ◽  
Matrix Normal ◽  
Matrix Normal Distribution

The joint distribution of standardized traces of $\frac{1}{n}XX'$ and of $\Big(\frac{1}{n}XX'\Big)^2$, where the matrix $X:p\times n$ follows a matrix normal distribution is proved asymptotically to be multivariate normal under condition $\frac{{n}}{p}\overset{n,p\rightarrow\infty}{\rightarrow}c>0$. Proof relies on calculations of asymptotic moments and cumulants obtained using a recursive formula derived in Pielaszkiewicz et al. (2015). The covariance matrix of the underlying vector is explicitely given as a function of $n$ and $p$.

scholarly journals Multivariate Statistical Analysis of Glacier Annual Balances

1974 ◽  
Vol 13 (69) ◽  
pp. 371-392 ◽  
Author(s):  
L. Lliboutry
Keyword(s):  
Statistical Analysis ◽  
Linear Model ◽  
Covariance Matrix ◽  
Random Errors ◽  
Multivariate Statistical ◽  
Routine Work ◽  
Image Position ◽  
Annual Balance ◽  
Comprehensive Study ◽  
Variance Covariance Matrix

A statistical analysis has been made of the annual balances collected during 16 consecutive years at 32 sites on the ablation area of the Glacier de Saint-Sorlin (French Alps). Only 38% of the 32 × 16 balances are known; moreover in 8 cases only the total balance for 2 consecutive years is known, and in one case the balance for 4 consecutive years. A comprehensive study of the errors leads us to assume the following linear model for the annual balance xjt at site j for year t:where αj and βt are parameters depending upon the site and the year respectively, ηjt and η’jt are random errors with a Gaussian distribution and standard errors σ and σ’ respectively. Assuming some known value for σ’2/σ2 = ρ, the parameters αj and βt, their variance–covariance matrix, and the variance covariance matrix of the residuals are estimated in the most general case. The estimators being stable against variations in ρ, the value ρ = o may be assumed; this value docs not conflict with the behaviour of the estimates of the residuals. A test of the linear model derived from Tukey’s non-additivity test is positive. Although a much more general, non-linear model gives a better representation of 13 × 6 balances forming a complete table of data, the linear model with σ ≈ 0.20 m is good enough to be used in theoretical studies or in routine work.

scholarly journals Multivariate Statistical Analysis of Glacier Annual Balances

1974 ◽  
Vol 13 (69) ◽  
pp. 371-392 ◽  
Author(s):  
L. Lliboutry
Keyword(s):  
Statistical Analysis ◽  
Linear Model ◽  
Covariance Matrix ◽  
Random Errors ◽  
Multivariate Statistical ◽  
Routine Work ◽  
Image Position ◽  
Annual Balance ◽  
Comprehensive Study ◽  
Variance Covariance Matrix

A statistical analysis has been made of the annual balances collected during 16 consecutive years at 32 sites on the ablation area of the Glacier de Saint-Sorlin (French Alps). Only 38% of the 32 × 16 balances are known; moreover in 8 cases only the total balance for 2 consecutive years is known, and in one case the balance for 4 consecutive years. A comprehensive study of the errors leads us to assume the following linear model for the annual balancexjtat site j for yeart:whereαjandβtare parameters depending upon the site and the year respectively,ηjtandη’jtare random errors with a Gaussian distribution and standard errorsσandσ’respectively. Assuming some known value forσ’2/σ2=ρ, the parametersαjandβt,their variance–covariance matrix, and the variance covariance matrix of the residuals are estimated in the most general case. The estimators being stable against variations inρ,the valueρ= o may be assumed; this value docs not conflict with the behaviour of the estimates of the residuals. A test of the linear model derived from Tukey’s non-additivity test is positive. Although a much more general, non-linear modelgives a better representation of 13 × 6 balances forming a complete table of data, the linear model withσ≈ 0.20 m is good enough to be used in theoretical studies or in routine work.

The asymptotic distribution of serial correlation coefficients for autoregressive processes with dependent residuals

1954 ◽  
Vol 50 (1) ◽  
pp. 60-64 ◽  
Author(s):  
A. M. Walker
Keyword(s):  
Finite Number ◽  
Joint Distribution ◽  
Goodness Of Fit ◽  
Autoregressive Process ◽  
Linear Functions ◽  
Fourth Moment ◽  
Autoregressive Processes ◽  
Multivariate Normal ◽  
Goodness Of Fit Test ◽  
Image Position

In a recent paper (3) Diananda has given an extension of the central limit theorem to stationary sequences of random variables {Yt} (t = 1,2,3,…), which are m–dependent, i.e. are such that m + 1 is the smallest integer r having the property that two sets of the Y's are independent whenever the suffix of any member of one set differs from that of any member of the other set by at least r. As Diananda points out, this can be used to show that for linear autoregressive processes with m–dependent residuals, the joint distribution of any finite number of the forms which occur in Bartlett and Diananda's goodness of fit test (see (1)) is asymptotically multivariate normal, provided also that the fourth moment of the residuals is finite. These forms are linear functions of the sample serial correlations rt, and if the autoregressive process is defined bythey may be denoted by , whereEt being the usual shift operator such that for any function ft defined for integral t, In particular, when , as for a moving average process, the joint distribution of any finite number of the rt, is asymptotically multivariate normal.

scholarly journals An asymptotic expansion for a class of multivariate normal integrals

1962 ◽  
Vol 2 (3) ◽  
pp. 253-264 ◽  
Author(s):  
Harold Ruben
Keyword(s):  
Asymptotic Expansion ◽  
Covariance Matrix ◽  
Random Vector ◽  
Multivariate Normal ◽  
Image Position ◽  
Variance Covariance Matrix

Let x = (x1, x2,…, xn) be a normal random vector with zero expectation vector and with a variance-covariance matrix which has 1 for its diagonal elements and ρ for its off-diagonal elements. Consider the quantity where.

The focusing of radiation by a random surface when the source is at a finite distance

1960 ◽  
Vol 56 (1) ◽  
pp. 27-40 ◽  
Author(s):  
M. S. Longuet-Higgins
Keyword(s):  
Mean Curvature ◽  
Joint Distribution ◽  
Total Curvature ◽  
Statistical Distribution ◽  
Constant Parameter ◽  
Random Surface ◽  
Second Derivatives ◽  
Finite Distance ◽  
The Mean ◽  
Image Position

Imagine a nearly horizontal, statistically uniform, random surface ζ(x, y), Gaussian in the sense that the second derivatives , , have a normal joint distribution. The problem considered is the statistical distribution of the quantitywhere J and Ω denote the mean curvature and total curvature of the surface, respectively, and ν is a constant parameter.

Central Limit Theorems for Interchangeable Processes

1958 ◽  
Vol 10 ◽  
pp. 222-229 ◽  
Author(s):  
J. R. Blum ◽  
H. Chernoff ◽  
M. Rosenblatt ◽  
H. Teicher
Keyword(s):  
Stochastic Process ◽  
Probability Measure ◽  
Central Limit ◽  
Joint Distribution ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Probability Measures ◽  
Image Position ◽  
Positive Integers

Let {Xn} (n = 1, 2 , …) be a stochastic process. The random variables comprising it or the process itself will be said to be interchangeable if, for any choice of distinct positive integers i 1, i 2, H 3 … , ik, the joint distribution of depends merely on k and is independent of the integers i 1, i 2, … , i k. It was shown by De Finetti (3) that the probability measure for any interchangeable process is a mixture of probability measures of processes each consisting of independent and identically distributed random variables.

scholarly journals Genz and Mendell-Elston Estimation of the High-Dimensional Multivariate Normal Distribution

Algorithms ◽  
2021 ◽  
Vol 14 (10) ◽  
pp. 296
Author(s):  
Lucy Blondell ◽  
Mark Z. Kos ◽  
John Blangero ◽  
Harald H. H. Göring
Keyword(s):  
Monte Carlo ◽  
Statistical Analysis ◽  
Execution Time ◽  
Absolute Error ◽  
High Dimensional ◽  
Multivariate Normal ◽  
Multinomial Data ◽  
Efficient Performance ◽  
Simulation Based ◽  
Scale Characteristics

Statistical analysis of multinomial data in complex datasets often requires estimation of the multivariate normal (mvn) distribution for models in which the dimensionality can easily reach 10–1000 and higher. Few algorithms for estimating the mvn distribution can offer robust and efficient performance over such a range of dimensions. We report a simulation-based comparison of two algorithms for the mvn that are widely used in statistical genetic applications. The venerable Mendell-Elston approximation is fast but execution time increases rapidly with the number of dimensions, estimates are generally biased, and an error bound is lacking. The correlation between variables significantly affects absolute error but not overall execution time. The Monte Carlo-based approach described by Genz returns unbiased and error-bounded estimates, but execution time is more sensitive to the correlation between variables. For ultra-high-dimensional problems, however, the Genz algorithm exhibits better scale characteristics and greater time-weighted efficiency of estimation.

Chronological Development of Bones and Teeth A twin study

1970 ◽  
Vol 19 (1-2) ◽  
pp. 75-75 ◽  
Author(s):  
L. Gedda ◽  
G. Brenci
Keyword(s):  
Statistical Analysis ◽  
Correlation Coefficient ◽  
Twin Study ◽  
Bone Age ◽  
Dental Age ◽  
Hereditary Component ◽  
Image Position ◽  
Two Parameters

The hereditary component of the chronological development of bones and teeth has been studied, in 40 twin pairs aged 5-7 years, through dental age (defined on account of the mineralization of the permanent dentition's dental buds) and bone age (denned on account of the presence and form of the hand ossification nuclei).The statistical analysis shows a correlation coefficient of 0.95 in MZ and 0.84 in DZ twins for dental age; and of 0.94 in MZ and 0.81 in DZ twins for bone age.The following are therefore the estimates of the hereditary component (based on Holzinger's formula) for the two parameters studied:

Sign in / Sign up

Export Citation Format

Share Document