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13) From (13) it can be noted that the estimator rI for ρ1 is somewhat more complicated than the sample analog r˜ I of ρ1 , which has the form r˜ I = = n k=1 (d − 1) {s2a l=l (ykl − y)(ykl − y) n d 2 k=1 l=1 (ykl − y) {s2a − s2w } − {s2w /(n − 1)} . + (d − 1)s2w } + {(d − 1)s2w /(n − 1)} (14) Thus rI is not an intraclass correlation coefficient for the sample; and r˜ I is not the natural ratio estimator for ρI . Otherwise, . when d/n = 0, then rI = r˜ I . If N is very large for the population so .

If φ(x) with values in Y is a maximal invariant on X , φ · h is a maximal invariant on X with values in Z. This fact is often utilized in writing the maximal invariant in a simpler form. Defining, for any x in X , the totality of points g(x) with g in G as the orbit of x, it follows from above that a function φ(x) is invariant under G if and only if it is constant on each orbit, and it is a maximal invariant under G if it is constant on each orbit and takes different values on different orbits.

REFERENCES 1. Kannemann, K. (1980). Biom. , 22, 377–390. 2. Kannemann, K. (1982). Biom. , 24, 679–684. BIBLIOGRAPHY Gibbons, J. D. (1971). Nonparametric Statistical Inference. McGraw-Hill, Toronto. ) Kannemann, K. (1980). Biom. , 22, 229–239. ) Kendall, M. G. and Stuart, A. (1979). The Advanced Theory of Statistics, Vol. 2, 3rd. Edn. Charles Griffin, London. ) K. KANNEMANN INVARIANCE INTRODUCTION In statistics the term ‘‘invariance’’ is generally used in its mathematical sense to denote a property that remains unchanged under a particular transformation, and, in practice, many statistical problems possess such a property.

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