On Sum of 0–1 Random Variables II. Multivariate Case

2020 ◽  
pp. 381-399
Author(s):  
Kei Takeuchi
Keyword(s):  
Random Variables ◽  
Multivariate Case

Identically distributed linear forms and the normal distribution

1973 ◽  
Vol 5 (1) ◽  
pp. 138-152 ◽  
Author(s):  
S. G. Ghurye ◽  
I. Olkin
Keyword(s):  
Normal Distribution ◽  
Random Variables ◽  
Linear Forms ◽  
Multivariate Case ◽  
Identical Distribution

A general discussion and survey is provided of the characterization of the normal distribution by the identical distribution of linear forms. The first result dates to 1923 when Pólya showed that if X and Y are i.i.d. random variables satisfying certain conditions, and if aX + bY is distributed as (a2 + b2)1/2X, then X has a normal distribution. This result has been generalized in several directions. In addition to a recasting of some of the results, an extension in the multivariate case is provided.

Identically distributed linear forms and the normal distribution

1973 ◽  
Vol 5 (01) ◽  
pp. 138-152
Author(s):  
S. G. Ghurye ◽  
I. Olkin
Keyword(s):  
Normal Distribution ◽  
Random Variables ◽  
Linear Forms ◽  
Multivariate Case ◽  
Identical Distribution

A general discussion and survey is provided of the characterization of the normal distribution by the identical distribution of linear forms. The first result dates to 1923 when Pólya showed that if X and Y are i.i.d. random variables satisfying certain conditions, and if aX + bY is distributed as (a 2 + b 2)1/2 X, then X has a normal distribution. This result has been generalized in several directions. In addition to a recasting of some of the results, an extension in the multivariate case is provided.

On sum of 0–1 random variables II. Multivariate case

1987 ◽  
Vol 39 (2) ◽  
pp. 307-324 ◽  
Author(s):  
Kei Takeuchi ◽  
Akimichi Takemura
Keyword(s):  
Random Variables ◽  
Multivariate Case

A Test for Mean with Additional Observations

1979 ◽  
Vol 28 (1-4) ◽  
pp. 47-56 ◽  
Author(s):  
S. K. Sarkar
Keyword(s):  
Normal Distribution ◽  
Random Variables ◽  
Bivariate Normal Distribution ◽  
Bivariate Normal ◽  
Multivariate Case ◽  
Independent Observations ◽  
Available Information

X1 and X2 are two random variables following a bivariate normal distribution, [Formula: see text], with all parameters unknown. Given n1 indepen­ dent observations on (X1 , X2) and n2 additional independent observations on X1 , we propose a procedure for inferring about µ2 utilizing all available information. The proposed procedure is compared with that based on student's t­statistic utilizing only the observations on X2. The generalization to the multivariate case is also done.

Normal approximations to discrete unimodal distributions

1986 ◽  
Vol 23 (04) ◽  
pp. 1013-1018
Author(s):  
B. G. Quinn ◽  
H. L. MacGillivray
Keyword(s):  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Hypergeometric Distribution ◽  
Death Process ◽  
Normal Approximations ◽  
Discrete Random Variables ◽  
Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

On the Mathematical Basis of Zelen’s Prerandomized Designs

1985 ◽  
Vol 24 (03) ◽  
pp. 120-130 ◽  
Author(s):  
E. Brunner ◽  
N. Neumann
Keyword(s):  
Clinical Trial ◽  
Normal Distribution ◽  
Random Variables ◽  
Small Samples ◽  
Selection Effects ◽  
Sample Sizes ◽  
Mathematical Basis ◽  
Additional Assumptions

SummaryThe mathematical basis of Zelen’s suggestion [4] of pre randomizing patients in a clinical trial and then asking them for their consent is investigated. The first problem is to estimate the therapy and selection effects. In the simple prerandomized design (PRD) this is possible without any problems. Similar observations have been made by Anbar [1] and McHugh [3]. However, for the double PRD additional assumptions are needed in order to render therapy and selection effects estimable. The second problem is to determine the distribution of the statistics. It has to be taken into consideration that the sample sizes are random variables in the PRDs. This is why the distribution of the statistics can only be determined asymptotically, even under the assumption of normal distribution. The behaviour of the statistics for small samples is investigated by means of simulations, where the statistics considered in the present paper are compared with the statistics suggested by Ihm [2]. It turns out that the statistics suggested in [2] may lead to anticonservative decisions, whereas the “canonical statistics” suggested by Zelen [4] and considered in the present paper keep the level quite well or may lead to slightly conservative decisions, if there are considerable selection effects.

Modified algorithm for fast bandwidth selection for kernel estimates of multidimensional probability densities

2020 ◽  
pp. 9-13
Author(s):  
A. V. Lapko ◽  
V. A. Lapko
Keyword(s):  
Probability Density ◽  
Optimal Parameter ◽  
Random Variables ◽  
Kernel Functions ◽  
Independent Random Variables ◽  
Functional Dependencies ◽  
Approximation Properties ◽  
Probability Density Estimation ◽  
Multidimensional Probability ◽  
Selection Of

An original technique has been justified for the fast bandwidths selection of kernel functions in a nonparametric estimate of the multidimensional probability density of the Rosenblatt–Parzen type. The proposed method makes it possible to significantly increase the computational efficiency of the optimization procedure for kernel probability density estimates in the conditions of large-volume statistical data in comparison with traditional approaches. The basis of the proposed approach is the analysis of the optimal parameter formula for the bandwidths of a multidimensional kernel probability density estimate. Dependencies between the nonlinear functional on the probability density and its derivatives up to the second order inclusive of the antikurtosis coefficients of random variables are found. The bandwidths for each random variable are represented as the product of an undefined parameter and their mean square deviation. The influence of the error in restoring the established functional dependencies on the approximation properties of the kernel probability density estimation is determined. The obtained results are implemented as a method of synthesis and analysis of a fast bandwidths selection of the kernel estimation of the two-dimensional probability density of independent random variables. This method uses data on the quantitative characteristics of a family of lognormal distribution laws.

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