K-Statistic -- From Wolfram MathWorld

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

<k_n>=kappa_n, (1)

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

var(k_r)=<(k_r-kappa_r)^2> (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 k_n for k-statistics, while Rose and Smith (2002) prefer k_n.

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

S_r=sum_(i=1)^nX_i^r, (3)

then

k_1=(S_1)/n (4)
k_2=(nS_2-S_1^2)/(n(n-1)) (5)
k_3=(2S_1^3-3nS_1S_2+n^2S_3)/(n(n-1)(n-2)) (6)
k_4=(-6S_1^4+12nS_1^2S_2-3n(n-1)S_2^2-4n(n+1)S_1S_3+n^2(n+1)S_4)/(n(n-1)(n-2)(n-3)) (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 n, the first few k-statistics are given by

k_1=mu (8)
k_2=n/(n-1)m_2 (9)
k_3=(n^2)/((n-1)(n-2))m_3 (10)
k_4=(n^2[(n+1)m_4-3(n-1)m_2^2])/((n-1)(n-2)(n-3)), (11)

where mu is the sample mean, m_2 is the sample variance, and m_i is the ith sample central moment (Kenney and Keeping 1951, pp. 109-110, 163-165, and 189; Kenney and Keeping 1962).

The variances of the first few k-statistics are given by

var(k_1)=(kappa_2)/n (12)
var(k_2)=(kappa_4)/n+(2kappa_2^2)/(n-1) (13)
var(k_3)=(kappa_6)/n+(9kappa_2kappa_4)/(n-1)+(9kappa_3^2)/(n-1)+(6nkappa_2^3)/((n-1)(n-2)) (14)
var(k_4)=(kappa_8)/n+(16kappa_2kappa_6)/(n-1)+(48kappa_3kappa_5)/(n-1)+(34kappa_4^2)/(n-1)+(72nkappa_2^2kappa_4)/((n-1)(n-2))+(144nkappa_2kappa_3^2)/((n-1)(n-2))+(24(n+1)nkappa_2^4)/((n-1)(n-2)(n-3)). (15)

An unbiased estimator for var(k_2) is given by

var(k_2)^^=(2k_2^2n+(n-1)k_4)/(n(n+1)) (16)

(Kenney and Keeping 1951, p. 189). In the special case of a normal parent population, an unbiased estimator for var(k_3) is given by

var(k_3)^^=(6k_2^3n(n-1))/((n-2)(n+1)(n+3)) (17)

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

For a finite population, let a sample size n be taken from a population size N. Then unbiased estimators M_1 for the population mean mu, M_2 for the population variance mu_2, G_1 for the population skewness gamma_1, and G_2 for the population kurtosis excess gamma_2 are

M_1=mu (18)
M_2=(N-n)/(n(N-1))mu_2 (19)
G_1=(N-2n)/(N-2)sqrt((N-1)/(n(N-n)))gamma_1 (20)
G_2=((N-1)(N^2-6Nn+N+6n^2)gamma_2)/(n(N-2)(N-3)(N-n))-(6N(Nn+N-n^2-1))/(n(N-2)(N-3)(N-n)) (21)

(Church 1926, p. 357; Carver 1930; Irwin and Kendall 1944; Kenney and Keeping 1951, p. 143), where gamma_1 is the sample skewness and gamma_2 is the sample kurtosis excess.

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