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03.4.1. Normal's Deconvolution and the Independence of Sample Mean and Variance

Published online by Cambridge University Press:  01 August 2003

Karim M. Abadir  and
Jan R. Magnus
Karim M. Abadir
Affiliation:
University of York, UK
Jan R. Magnus
Affiliation:
Tilburg University, The Netherlands
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Extract

Normal's deconvolution and the independence of sample mean and variance.

Information

Type
PROBLEMS AND SOLUTIONS
Information
Econometric Theory , Volume 19 , Issue 4 , August 2003 , pp. 691
Copyright
© 2003 Cambridge University Press

(a) Let x1 and x2 be independent variates having m.g.f.s m1(t1) and m2(t2), respectively, and define y := x1 + x2. Prove that y is normal if and only if x1 and x2 are both normal. Is the existence of m.g.f.s necessary for this result?

(b) Let x := (x1,…,xn)′ be a vector of independent (but not necessarily identically distributed) components, where 2 ≤ n < ∞. Define

. It is well known that if

independently from z/σ2 ∼ χ2(n − 1). For n ≥ 3, prove that if

and z/σ2 ∼ χ2(n − 1), then

.

(c) Why is the last statement in (b) not necessarily true for n = 2? What additional conditions are needed to make it hold for n = 2?

ACKNOWLEDGMENT

We thank Stéphane Gregoir, Paolo Paruolo, and two referees for their help and acknowledge support from the British ESRC grant R000239538.