In many application areas, such as demography, management science, hydrology, finance, etc., data are frequently positive and right-skewed. Recently, the inverse Gaussian (IG) distribution has drawn many attentions and the inferences concerned with the IG distribution have also grown rapidly because the IG distribution is an ideal candidate for modeling and analyzing the right-skewed and positive data. For instance, Wise (1971, 1975) and Wise et al. (1968) developed the IG population as a possible model to describe cycle time distribution for particles in the blood; Lancaster (1972) made use of the IG distribution in describing strike duration data, etc.
The history of the inverse Gaussian distribution can be traced back to 1915 when Schrödinger and Smoluchowski presented independent derivations of the density of the first passage time distribution of Bownian motion with positive drift. The modern day statistical community became aware of this distribution through the pioneering work of Tweedie (1941, 1945, 1946, 1947, 1956, 1957). The name “inverse Gaussian”
was also given by Tweedie based on his discovers that the cumulant function of IG distribution is the inverse of the cumulant function of the normal distribution. For further details about the IG distribution and for applications, the reader is referred to the books by Chhikara and Folks (1989) and Seshadri (1999), respectively.
The probability density function (pdf) of IG distribution, IG( , )μ λ , is defined as
1/ 2
Fig. 1.1 Density functions of IG( , )μ λ for fixed λ = and five values of 1 μ
Fig. 1.2 Density functions of IG( , )μ λ for fixed μ =1 and five values of λ
In Fig. 1.1 and Fig. 1.2, the density of IG distribution varies from the highly right-skewed to almost symmetric for different parameter configurations. Hence, the inverse Gaussian distribution is very flexible in describing various data.
The inference methods of the IG model are very analogous to those of the Gaussian model; for example, a very common problem in applied field is to compare the means of several Gaussian populations, i.e.
0 : 1 2 I
H μ = μ =L=μ vs. HA: not all ' are equalμi s . (1.2) If the variance of each population is homogeneous, the analysis of variance (ANOVA) can be used to perform the test. Similarly, the analysis of reciprocals (ANORE) can also be used to test the equality of means of several IG samples if all scale parameters among groups are assumed to be equal (Tweedie, 1956 ; Chhikara and Folks, 1989).
The testing procedure will be briefly introduced in Remark 1. But when the scale parameters are non-homogeneous, the ANORE fails to solve the problem as ANOVA fails to test (1.2) when these populations are not homogeneous. Tian (2005) proposed a method to test the equality of IG means under heterogeneity based on generalized test variable method. However, if the null hypothesis is not been rejected, the inferences for the common mean remain unsolved. Therefore, in this paper, we would like to estimate and construct the 100(1−α)% confidence interval for the common mean of several non-homogeneous IG populations. Our method is based on a higher order asymptotic likelihood based method. This method, in theory, has a higher order accuracy, , and is very accurate even when the sample size is small. Reid (1996) gave some review and annotation of the development. The method has also been applied to solve many practical problems involving interval estimation for a skewed distribution, e.g. Wu et al. (2002) applied this procedure to make the confidence interval estimation of the ratio of two independent lognormal distribution, Wu et al. (2003) presented a confidence interval for a log-normal mean based on this
( -3/2
O n )
method; Wu and Wong (2004) used the method to improve the interval estimation for the two-parameter Birnbaum-Saunders distribution; Tian and Wilding (2005) used the method to construct confidence interval for the ratio of means of two independent IG distributions, etc. In our case, the likelihood-based method gives a satisfactory result as well.
Remark 1. One of the resemblances between the IG distribution and the ordinary Gaussian distribution is the method for the analysis of residuals. We are familiar with the analysis of variance (ANOVA) in normal inference theory, Tweedie (1956) introduce the so called analysis of reciprocals (ANORE) for the inverse Gaussian distribution. The procedure is as follows:
Assume that there are components in the th populations and each population is distributed as
ni i
IG( , )μ λi , whereμi, i=1,..., and I λare unknown. Further assume the random samples Xij, 1,..., , 1,...,i= I j = ni
I
are independent. We are interested in the problem of testing (1.2)
The likelihood function, denoted by L(μ1,...,μ λI, ;Xij =xij, 1,..., , 1,...,i= I j= n )is
where . Differentiation with respect to
1
1 1
The likelihood ratio can be reduced to
Note that Qcan be decomposed into
0 1 1 It is easy to verify that λQ0 follows a chi-squared distribution with degrees of freedom N−Iwhile λQ1 is a chi-squared distribution with degrees of freedom of (I−1) and are independent. It follows that the likelihood ratio test statistic α level rejection region is given by solving the following inequality 1 1, ,1 For illustration examples of ANORE, see Chhikara and Folks (1989) and Seshadri (1999) for further details.
This article is organized as follows. In section 2, we will briefly introduce the properties of IG distribution and the concepts of the signed log-likelihood ratio statistic and a higher order asymptotic method. Then the method is applied to construct a confidence interval for the common mean of several independent IG populations in section 3. The classical procedure under the assumption of identical scale is also described in section 3. We will present two numerical examples and two
simulation studies in section 4 to illustrate the merits of our proposed method. Some concluding remarks are given in section 5.