A new method for interval estimation of the mean of the Gamma distribution

Abstract

The two parameter Gamma distribution is widely used for modeling lifetime distributions in reliability theory. There is much literature on the inference on the individual parameters of the Gamma distribution, namely the shape parameter k and the scale parameter theta when the other parameter is known. However, usually the reliability professionals have a major interest in making statistical inference about the mean lifetime mu, which equals the product thetak for the Gamma distribution. The problem of inference on the mean mu when both parameters theta and k are unknown has been less attended in the literature for the Gamma distribution. In this paper we review the existing methods for interval estimation of mu. A comparative study in this paper indicates that the existing methods are either too approximate and yield less reliable confidence intervals or are computationally quite complicated and need advanced computing facilities. We propose a new simple method for interval estimation of the Gamma mean and compare its performance with the existing methods. The comparative study showed that the newly proposed computationally simple optimum power normal approximation method works best even for small sample sizes.

Using equity premium survey data to estimate future wealth

“…This allows us to use the numerical OPNAM technique of Kulkarni and Powar (2010) to understand the probability density function of the future value. In addition, we derive closed-form solutions for unbiased and MLSE estimates of future value, and quantiles of future value, thus extending the work of Blume (1974), Indro and Lee (1997), Jacquier et al (2003Jacquier et al ( , 2005 and Kan and Zhou (2009) when forecasts are unbiased.…”

“…As we note below, there are a number of techniques available for this; we use the one proposed by Kulkarni and Powar (2010). While we assume that the population frequency distribution of expert opinions can be well characterized through ( i ; ; ) ; there is no assumption that g p ( p ) will also be gamma distributed; in fact, it will not be.…”

“…Hawkins and Wixley (1986) propose instead p = 1=4. In this paper, we follow the Optimal Power Normal Approximation Method (OPNAM) method of Kulkarni and Powar (2010) who propose using p = 0:246 when > 1:5 as is the case for Welch's data. When we make the transformation Y i = 0:246 i we …nd that the sample mean Y = 0:4712, the sample standard deviation Y = 4:40%, the skewness is -0.52 and the excess kurtosis is 1.67.…”

“…In this appendix, we present further details of the Kulkarni and Powar (2010) OPNAM method for estimating the density function of the mean of a gamma distribution conditional on a …nite sample, the Winkler (1981) method for combining correlated expert forecasts, and the application of these two methods to the problem at hand. We also, by simulation, demonstrate the accuracy of these two techniques.…”

“…The …rst four moments of Y i are known in closed form -we derive the …rst two here and note that the equivalent equations for skewness and kurtosis are given on page 437 of Kulkarni and Powar (2010). The pdf of a gamma distribution is given by:…”