Normal-Based Methods for a Gamma Distribution

Abstract

In this article we propose inferential procedures for a gamma distribution using the Wilson-Hilferty (WH) normal approximation. Specifically, using the result that the cube root of a gamma random variable is approximately normally distributed, we propose normal-based approaches for a gamma distribution for (a) constructing prediction limits, one-sided tolerance limits, and tolerance intervals; (b) for obtaining upper prediction limits for at least l of m observations from a gamma distribution at each of r locations; and (c) assessing the reliability of a stress-strength model involving two independent gamma random variables. For each problem, a normal-based approximate procedure is outlined, and its applicability and validity for a gamma distribution are studied using Monte Carlo simulation. Our investigation shows that the approximate procedures are very satisfactory for all of these problems. For each problem considered, the results are illustrated using practical examples. Our overall conclusion is that the WH normal approximation provides a simple, easy-to-use unified approach for addressing various problems for the gamma distribution.

Tolerance limits for gamma distribution based on generalized fiducial method

“…As we can see, our proposed methods perform quite well in regardless of different settings. It is worth mentioning that the normal-based method for a gamma distribution proposed in [15] actually does not work very well for the one-sided tolerance limit case. In their study, the coverage probabilities for the upper tolerance limit are uniformly smaller than the nominal values.…”

“…In this section, we use the same data set in [15] for illustrative purpose. This data set in Table III represents Table I the alkalinity concentrations in groundwater.…”

“…Based on this approximation, the tolerance limits for the gamma distribution can be constructed by using the tolerance limits for a normal distribution, which has been well received in the literature. However, according to the simulation results in [15], the coverage probabilities of an upper (lower) tolerance limit based on this method are uniformly smaller (larger) than the nominal value. This is undesirable because one-sided tolerance limit, especially the upper tolerance limit, is very important in the monitoring problems as we introduced before.…”

“…However, the true parameter values are never known in practice and the estimators may have large biases when the sample size is small. On the other hand, the second method is based on the normal approximation to the gamma distribution [14], [15]. They assumed that the cube root of a gamma random variable is approximately normal distributed.…”

Demographic variation and habitat specialization of tree species in a diverse tropical forest of Cameroon

“…Hinkley 1977), and the option for negative numbers is a crucial advantage. The precedent for a power transformation of negative numbers is the cube root, which is defined for negatives and normalizes the gamma-distribution (Krishnamoorthy et al 2008). We explored powers between 0.3 and 0.5, maintaining negatives, and found that the exponent 0.45 was most effective at reducing skewness; the cube-root overtransformed, and produced skewness in the opposite direction.…”