k-Statistic

The th -statistic is the unique symmetric unbiased estimator of the cumulant of a given statistical distribution, i.e., is defined so that

(1)

where denotes the expectation value of (Kenney and Keeping 1951, p. 189; Rose and Smith 2002, p. 256). In addition, the variance

(2)

is a minimum compared to all other unbiased estimators (Halmos 1946; Rose and Smith 2002, p. 256). Most authors (e.g., Kenney and Keeping 1951, 1962) use the notation for -statistics, while Rose and Smith (2002) prefer .

The -statistics can be given in terms of the sums of the th powers of the data points as

(3)

then

			(4)
			(5)
			(6)
			(7)

(Fisher 1928; Rose and Smith 2002, p. 256). These can be given by KStatistic[r] in the Mathematica application package mathStatica.

For a sample size , the first few -statistics are given by

			(8)
			(9)
			(10)
			(11)

where is the sample mean, is the sample variance, and is the th sample central moment (Kenney and Keeping 1951, pp. 109-110, 163-165, and 189; Kenney and Keeping 1962).

The variances of the first few -statistics are given by

			(12)
			(13)
			(14)
			(15)

An unbiased estimator for is given by

(16)

(Kenney and Keeping 1951, p. 189). In the special case of a normal parent population, an unbiased estimator for is given by

(17)

(Kenney and Keeping 1951, pp. 189-190).

For a finite population, let a sample size be taken from a population size . Then unbiased estimators for the population mean , for the population variance , for the population skewness , and for the population kurtosis excess are

			(18)
			(19)
			(20)
			(21)

(Church 1926, p. 357; Carver 1930; Irwin and Kendall 1944; Kenney and Keeping 1951, p. 143), where is the sample skewness and is the sample kurtosis excess.