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A note on characterization of some laws by the symmetrical distribution of one linear function given another
Author(s) -
Khatri C.G.
Publication year - 1974
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3314696
Subject(s) - mathematics , linear form , distribution (mathematics) , function (biology) , linear predictor function , random variable , mathematical analysis , linear regression , statistics , proper linear model , polynomial regression , evolutionary biology , biology
Results corresponding to the symmetrical distribution of a linear vector function of vector variables given another linear vector function of vector variables are obtained by reducing the problem to one of identical distribution of two linear functions. To have normality for each vector variable, one requires stronger conditions on the coefficients of the linear functions than those required in the Darmois‐Skitovich theorem. It is pointed out that Heyde's result (1970) on the symmetrical distribution of a linear function given another linear function, using the same conditions on the coefficients as those of the Darmois‐Skitovich theorem, is not true. For some cases, there do exist some non‐normal random variables among the components of the linear functions.