On the extreme values and range of samples from non-normal populations

Moments of the Range of Samples from Nonnormal Populations

“…The coefficient of X4 and in pl' are found to be in agreement with those given by the author [7, Table 1 (a)]. As a further check on the values in Table 1, we find that the coefficients of X d and in p2' for n = 2(2)12, in p3' for n = 5, and in p4' for n = 4 and ) (2.2) 6, are in agreement with those calculated directly from (1.1) by use of the values given by the author [7] in his Tables 1. These checks establish the accuracy of (2.6)…”

“…It may, however, be noted that for certain populations of Burr's system our approximate values of E(R/ud,) are slightly lower for n = 5 and higher for n = 10. However, when such mean values are compared [7] with the true values available for certain other populations, such as xZ, log x2, logistic, exponential, and rectangular, the error (approximate -true) does not change sign up to n = 12. As regards cv, the present approximation leads to values consistently higher for the Burr system and so also for the xaa distribution, but with comparatively negligible error in the It may be noted that for most populations considered, AS and A4 are close to the effectively positive definite and unimodal region [l, 91. It seems that for the same first four cumulants, the Burr system represents somewhat different populations compared to Edgeworth expansion or Type I11 distributions.…”

“…But, using the method of David and Johnson [4], Bhattacharjee [Z] approximates E(zlz,) and whence E'(Ra) for symmetrical populations ( A 3 = 0). Incidentally, Subrahmaniam's [lo] results for E ( x l z n ) , where the coefficient of h3 # 0, seem to contradict those obtained in [7].…”

“…4 and of below 0.2. But, Singh [7] shows that the series with haa = 0.5 can be regarded as effectively positive definite and unimodal and that in some cases the errors in numerical terms are not serious even if ha2 exceeds 0.5. Thus the representation may cover a wider class of nonnormal populations.…”

The Method of Least Squares and Some Alternatives.

“…(1) The trimmed mean, scaled by the Winscrized e.s.d., has both excellent small sample and excellent large sample properties; since it is also easy to compute, it can be strongly recamended for practical use; (2) 7he trimad mean and a maxinan likelihood type estimate T devel~,ped by Huber behave well even if th, unierlying distributti, fails to have a density, but the corresponding estimates Cox (1954), David (1954 and Singh (1967). He concludes that, for Edgeworth type populations, the effect of parental skewness on the probability integral and percentage points of the range is comparatively small for samples of size n<5, but as n becomes larger this effect becomes as prominent as that of kurtosis 22z…”