## k-Statistic

An Unbiased Estimator of the Cumulants of a Distribution. The expectation values of the -statistics are therefore given by the corresponding Cumulants

 (1) (2) (3) (4)

(Kenney and Keeping 1951, p. 189). For a sample of size, , the first few -statistics are given by
 (5) (6) (7) (8)

where is the sample Mean, is the sample Variance, and is the sample th Moment about the Mean (Kenney and Keeping 1951, pp. 109-110, 163-165, and 189; Kenney and Keeping 1962). These statistics are obtained from inverting the relationships

 (9) (10) (11) (12) (13)

The first moment (sample Mean) is

 (14)

and the expectation value is
 (15)

The second Moment (sample Standard Deviation) is
 (16)

and the expectation value is
 (17)

since there are terms , using
 (18)

and where is the Moment about 0. Using the identity
 (19)

to convert to the Moment about the Mean and simplifying then gives
 (20)

The factor is known as Bessel's Correction.

The third Moment is

 (21)

Now use the identities
 (22) (23)
where it is understood that sums over products of variables exclude equal indices. Plugging in

 (24)

The expectation value is then given by

 (25)

where is the Moment about 0. Plugging in the identities
 (26) (27)

and simplifying then gives
 (28)

(Kenney and Keeping 1951, p. 189).

The fourth Moment is

 (29)

Now use the identities

 (30) (31) (32)
Plugging in,

 (33)

The expectation value is then given by
 (34)

where are Moments about 0. Using the identities
 (35) (36) (37)

and simplifying gives
 (38)

(Kenney and Keeping 1951, p. 189).

The square of the second moment is

 (39)

Now use the identities

 (40) (41) (42)
Plugging in,

 (43)

The expectation value is then given by
 (44)

where are Moments about 0. Using the identities
 (45) (46) (47)

and simplifying gives
 (48)

(Kenney and Keeping 1951, p. 189).

The Variance of is given by

 (49)

so an unbiased estimator of is given by
 (50)

(Kenney and Keeping 1951, p. 189). The Variance of can be expressed in terms of Cumulants by
 (51)

An Unbiased Estimator for is
 (52)

(Kenney and Keeping 1951, p. 190).

Now consider a finite population. Let a sample of be taken from a population of . Then Unbiased Estimators for the population Mean , for the population Variance , for the population Skewness , and for the population Kurtosis are

 (53) (54) (55) (56)

(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.

References

Carver, H. C. (Ed.). Fundamentals of the Theory of Sampling.'' Ann. Math. Stat. 1, 101-121, 1930.

Church, A. E. R. On the Means and Squared Standard-Deviations of Small Samples from Any Population.'' Biometrika 18, 321-394, 1926.

Irwin, J. O. and Kendall, M. G. Sampling Moments of Moments for a Finite Population.'' Ann. Eugenics 12, 138-142, 1944.

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951.

Kenney, J. F. and Keeping, E. S. The -Statistics.'' §7.9 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 99-100, 1962.