ABSTRACT
With estimates of f.l;, c::r, a 3 , and a 4 calculated as described in Section 11.4, the type of distribution can be established from the original Pearson criteria, or from the Carver-Craig criteria as presented by Craig (1936). Excellent descriptions of the original Pearson criteria are contained in volumes by Elderton (1938) and by Elderton and Johnson (1969). In the present instance, however, since estimates b0 , b1, and b2 must of necessity be computed before estimates of the population parameters can be obtained, it seems more appropriate to determine the type directly from the quadratic equation
(11.5.1) The general solution of the differential equation (11.2.1) can be written as
(11.5.2) where r 1 and r 2 are roots of (11.5.1) (cf., for example Craig (1936). The nature of these roots determines the type of the distribution. Let D designate the discriminant, D = hi - 4b0 b2 , and the principal Pearson distributions may be classified as follows
A necessary condition for the odd central moments to equal zero [i.e., for f(x) to be symmetrical about the mean] is that
(11.5.3)
we can translate the origin to the truncation point on the right, and again drop the last equation from ( ll.4. l)
In practical applications, finding either J or H equal or nearly equal to zero in a sample that is represented as being doubly truncated suggests that perhaps the sample was in fact only singly truncated. If both J and H are zero or nearly zero, this suggests that the sample is in fact from a complete rather than from a truncated distribution. In this case, parameter estimates would be calculated from complete sample estimators.
