Statistical Seminvariants and Their Setimates with Particular Emphasis on Their Relation to Algebraic Invariants

Application of saddlepoint approximation in reliability analysis of dynamic systems

“…The unbiased estimators for the cumulants of a population are generally provided by k-statistics (Fisher, 1928;Dressel, 1940). Based on the sample values {Y 1 , ,Y N } of the random variable Y, the fi rst four cumulants are estimated by …”

Computation with Multiplek-Statistics

“…Dressel [7] defined statistics whose expected values are products of cumulants. Tukey [16], [17] showed that these same statistics are unbiased estimates of the corresponding finite population parameters.…”

“…Now one would hardly expect that an unbiased estimate of a quantity which is not a seminvariant would be a seminvariant and indeed it can be shown that any k . .. with one or more unit subscripts is not independent of the origin.s This is the reason that Dressel [7] did not develop formulae for estimates of products of cumulants containing K1. For the same reason we may, for practical purposes, limit the discussion to the calculation of those k .…”

“…The treatment through weight 8, with special consideration to the calculation of the seminvariant k's, is given below. 7. CALCULATION OF SEMINVARIANT k ...…”

“…T H E use of the estimates of the cumulants by Fisher [9] led to a remarkable simplification of the moment formulae for infinite sampling, and the introduction of the estimates of the products of the cumulants by Dressel [7], followed by the work of Tukey [16], [17], [18] and Wishart [20], [21], has led to improvements in the simplicity of the finite moment formulae. But this simplicity of theory may be, from the standpoint of practical computation, something of an illusion since the multiple k-statistics featured in the new formulae are more involved and more difficult to compute, except in the most trivial cases, than are the moment statistics featured in the old formulae.…”