Suppose the pdf g(x) is PHLD with the form as in (5) and then substituting in (7) and further on integrating, we get
is the pdf of PHLD if, and only if, f(x) is the pdf of HLD.
One may derive further characterizations of PHLD by taking the (1/[alpha])th power of half-logistic variables in the results of Olapade , which described some characterizations of half-logistic distribution.
Suppose a sample of size n is taken from PHLD with density function (5).
Least square estimation method involves the least squares regression to estimate the two parameters based on the linearized PHLD distribution function.
Samples of sizes 100, 50, and 20 are generated from PHLD for different values of parameters.
If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where X ~ PHLD, the pdf of Y is given by
Immediately, we have the following result, which may be exploited for generating random variables from PHLD.
If X ~ U (0,1), then the random variable Z = [[(1/[beta]) log (a - X)/X].sup.1/[alpha]] has truncated PHLD (TPHLD).
Note that when [alpha] = 1, we have the Truncated Half-Logistic Distribution; when [alpha] = 2, [alpha] = 1, it gives HLD; and when a = 2, [alpha] > 0, it gives PHLD.
Families of Distributions Generated from PHLD. We have, in the literature, quite a few families of distributions generated from Beta and Gamma distributions (see [7,12,13]).
where H(x) is the distribution function of PHLD or its generalizations and [phi](x) takes different forms of [bar.G](x), the survival function of a random variable.