Statistics and Probability Letters 82 (2012) 1755–1760

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Statistics and Probability Letters

journal homepage: www.elsevier.com/locate/stapro

Testing the homogeneity of inverse Gaussian scale-like parameters

Ming Chang a,∗, Xuqun You a, Muqing Wen b

a College of Psychology, Shaanxi Normal University, Xian, 710062, Chinab College of Finance and Economics, North-West University of Politics and Law, Xian, 710061, China

a r t i c l e i n f o

Article history:Received 28 September 2011Received in revised form 3 February 2012Accepted 17 May 2012Available online 4 June 2012

Keywords:Inverse Gaussian populationsScale-like parametersGeneralized p-value

a b s t r a c t

A test for the homogeneity of normal variances was proposed by Liu and Xu [Liu, X.H.,Xu, X.Z., 2010. A generalized p-value approach for testing the homogeneity of variances.Statistics and Probability Letters 80, 1486–1491]. For testing the homogeneity of inverseGaussian scale-like parameters, a parallel test is developed in this article. The proposedtest is proved to have exact frequent property. The merits of the proposed method arenumerically compared with the existing method with respect to their sizes and powersunder different scenarios. The simulation results show that the proposed approach canperform hypothesis testing with satisfactory sizes and powers.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

The density function of the two-parameter inverse Gaussian (IG) distribution IG(µ, λ) is defined as

f (x, µ, λ) =

λ

2πx3

1/2

exp−

λ

2µ2x(x − µ)2

, x > 0, µ, λ > 0, (1.1)

whereµ is the mean parameter and λ is the scale-like parameter. The IG distribution has already been applied in describingand analyzing right-skewed data more and more extensively. Durham and Padgett (1997) used the IG models to develop anew general method based on cumulative damage for describing the failure of a system, and Doksum and Hóyland (1992)developed a model for variable-stress-accelerated life testing experiments based on the IG distributions. Seshadri (1993,1999) discussed more applications of IG distribution in life tests, etc.

The IG distributions share many inference theories based on t , F and χ2 distributions as for the Gaussian distribution.We refer readers to Chhikara and Folks (1989), Seshadri (1993, 1999) and Mudholkar and Natarajan (2002) for more detailsof Gaussian and IG analogies. In many statistical applications, a test of the equality of IG scale-like parameters is of interest,that is

H0 : λ1 = λ2 = · · · = λk, vs. H1 : not all λ’is are equal. (1.2)

For example, when testing for homogeneity of inverse Gaussian means, homogeneity of the scale-like parameters λ is animportant assumption in the analysis of reciprocals (Chhikara and Folks, 1989). The sampling distributions of the rescaledmaximum likelihood estimators (MLEs) of λ−1 and σ 2 for the inverse Gaussian and normal populations, respectively, areboth chi-square with n − 1 degrees of freedom. Interestingly, inference procedures concerning λ are remarkably similar to

∗ Corresponding author.E-mail addresses: sms1226moon@yahoo.com.cn, changmll@163.com (M. Chang).

0167-7152/$ – see front matter© 2012 Elsevier B.V. All rights reserved.doi:10.1016/j.spl.2012.05.013

1756 M. Chang et al. / Statistics and Probability Letters 82 (2012) 1755–1760

those concerning the normal scale parameter. Let Xi1, Xi2, . . . , Xini be a random sample from an IG(µi, λi) population fori = 1, . . . , k. Denote

Vi =

nij=1

(X−1ij − X−1

i ), N =

ki=1

(ni − 1) (1.3)

and

V =

ki=1

Vi, (1.4)

where Xi is the ith sample mean. For testing the homogeneity of k IG scale-like parameters, Chhikara and Folks (1989)developed a modified likelihood ratio (MLR) test, which is analogous to the approximated test presented by Bartlett andKendall (1946) for normal theory. The test statistics isM/C , where

C = 1 +1

3(k − 1)

ki=1

1ni − 1

−1

ki=1

(ni − 1)

and

M = N ln

VN

−

ki=1

(ni − 1) ln

Vi

(ni − 1)

.

Under H0, M/C is distributed approximately as χ2k−1, and the level α rejection region for H0 is given by

M/C > χ2k−1,1−α, (1.5)

where χ2k−1,1−α is the 100(1 − α) percentage point of the χ2 distribution with k − 1 degrees of freedom.

The MLR test for the homogeneity of IG scale-like parameters is based on large-sample approximations whose reliabilitycan be quite poor, in particular, little is knownabout their validity in small samples. Therefore, it is of practical and theoreticalimportance to develop a test which does not depend on large-sample approximations. This paper will fill this gap bydeveloping an approach using the concepts of generalized test variables and generalized p-values. The generalized p-valuewas introducedby Tsui andWeerahandi (1989), and the generalized confidence interval byWeerahandi (1993). The conceptsof generalized test variable and generalized pivot quantities have been widely applied to a variety of practical settingswhere standard inference methods do not exist. For example, see Zhou and Mathew (1994), Tian (2006) and Li (2009), etc.A generalized test variable for the homogeneity of normal variances was proposed by Liu and Xu (2010). In this paper, aparallel generalized p-value procedure for testing the homogeneity of IG scale-like parameters is proposed. The proposedprocedure is based on an exact probability statement as required in the context of generalized inference, and it has the exactfrequent property. Simulation results also show that the proposed approach has satisfactory sizes and powers.

This article is organized as follows. Section 2 reviews the concept of the generalized p-value. In Section 3, a test basedon the generalized test variable for testing homogeneity of IG scale-like parameters is presented, and the exact frequencyproperty of the proposed test is proved. In Section 4, simulation results on the sizes and powers are presented. Concludingremarks are summarized in Section 5.

2. Generalized test variables and generalized p-values

The generalized p-value approach will be valuable whenever the standard test statistic and standard pivot quantity areeither non-existent or difficult to obtain based on the conventional method. To illustrate these concepts, let X be a randomvariable whose cumulative function is F(X; ξ), where ξ = (θ, η) is a vector of unknown parameters, assuming values in aparameter space Θ , where θ is the parameter of interest and η represents the nuisance parameter. Note that θ and η maybe more than one parameter of interest and more than one nuisance parameter. Let Ω be the sample space and x be theobserved value of X .

Consider testing H0 : θ ≤ θ0 versus H1 : θ > θ0, where θ is the parameter of interest, θ0 is a specified value. Then definethe generalized test variable T (X; x, θ, η) satisfying the following conditions:

(a) The distribution of T (X; x, θ, η) is free of the nuisance parameter η.

(b) The observed value of T (X; x, θ, η), i.e., T (x; x, θ, η) is free of η.

(c) PrT (X; x, θ, η) ≥ T (x; x, θ, η) is nondecreasing in θ, for fixed x and η. (2.1)If the distribution of T (X; x, θ, η) is stochastically increasing in θ , then the generalized p-value for testing the above

hypothesis is defined as Pr(T (X; x, θ, η) ≥ t), where t = T (x; x, θ, η). But if the distribution of T (X; x, θ, η) is stochasticallydecreasing in θ , then the generalized p-value for testing the above hypothesis is defined as Pr(T (X; x, θ, η) ≤ t).

For further details on the concept of the generalized p-value, we refer readers to the book by Weerahandi (1995).

M. Chang et al. / Statistics and Probability Letters 82 (2012) 1755–1760 1757

3. The generalized p-value approach

A generalized test variable for testing homogeneity of normal variances was proposed by Liu and Xu (2010). In thissection, a parallel test for IG scale-like parameters is developed as follows.

Let Xi1, Xi2, . . . , Xini be a random sample from an IG(µi, λi) population with Xi as the ith sample mean for i = 1, . . . , k.Let xi denote the ith observed sample mean for i = 1, . . . , k. It is well known that

Xi ∼ IG(µi, niλ), λiVi ∼ χ2ni−1. (3.1)

Denote λ = (λ1, λ2, . . . , λk)′ and

H =

1 0 · · · 0 −10 1 · · · 0 −1...

......

......

0 0 · · · 1 −1

(k−1)×k

. (3.2)

Then the testing problem (1.2) is equivalent to testing

H0 : Hλ = 0, vs. H1 : Hλ = 0. (3.3)

In this section, a new generalized p-value is developed for testing (3.3). According to Tian (2006), the the generalizedpivot quantity for λi based on the ith sample is given by

Tλi =λiVi

vi=

Ui

vi, (3.4)

wherevi is the observed value ofVi,Ui ∼ χ2ni−1, i = 1, 2, . . . , k, and theUi aremutually independent. Obviously, Tλi coincides

with the traditional pivot quantity for λi. Then the generalized pivot quantity for Hλ can be derived as

THλ = H(Tλ1 , Tλ2 , . . . , Tλk). (3.5)

Let V = (V1, V2, . . . , Vk)′ and X = (X1, X2, . . . , Xk)

′. From (3.5), we see that the conditional expectation of THλ given(X, V ) = (x, v) is

µT = E(THλ|(x, v)) = HE(Tλ|(x, v)) = H(E(Tλ1 |(x, v)), . . . , E(Tλk |(x, v)))′ (3.6)

and the conditional covariance matrix of THλ given (X, V ) = (x, v) is

ΣT = Cov(THλ|(x, v)) = HCov(Tλ|(x, v))H ′= Hdiag(Cov(Tλ1 |(x, v), . . . , Tλk |(x, v)))H ′, (3.7)

where E(Tλi |(x, v)) =ni−1vi

and Cov(Tλi |(x, v)) =2(ni−1)

v2ifor i = 1, 2, . . . , k, respectively.

Let T denote the standardized expression of THλ with T = Σ−1/2T (THλ − µT ), and t the observed value of T for (X, V ) =

(x, v), where µT and ΣT are given by (3.6) and (3.7), respectively. Given (x, v), obviously, the distribution of T is free of anyunknown parameter. Then ∥T∥

2= (THλ − µT )

′Σ−1T (THλ − µT ) does not depend on any unknown parameter too, and the

observed value of ∥t∥2, underH0 : Hλ = 0, is equal toµ′

TΣ−1T µT which is a known constant free of any parameter. Therefore,

∥T∥2 is a generalized test variable satisfying conditions similar to (2.1). Then the generalized p-value can be obtained as

p(x, v) = Pr(∥T∥2

≥ ∥t∥2|H0) = Pr((THλ − µT )

′Σ−1T (THθ − µT ) ≥ µ′

TΣ−1T µT ), (3.8)

andH0 will be rejectedwhenever p(x, v) is less than the level α. Furthermore, the rejection region can be derived as follows:

Cα(x11, . . . , xknk) = (x11, . . . , xknk) : p(x, v) ≤ α. (3.9)

The following theorem shows that the new generalized p-value method has exact frequent property; i.e., the p-valuestatistic p(X, V ) has a uniform distribution on (0, 1) under the null hypothesis H0.

Theorem 1. About the above problem,

Pr(Cα(x11, . . . , xknk)|H0) = PrH0(p(X, V ) ≤ α) = α

where p(X, V ) and Cα(X11, . . . , Xknk) are defined through (3.8) and (3.9), respectively.

1758 M. Chang et al. / Statistics and Probability Letters 82 (2012) 1755–1760

Proof. Assume that U∗

i and V ∗

i are the independent copies of Ui and Vi; then

PrH0 [p(X, V ) ≤ α] = PrH0 [Pr∗((THλ − µT )

′Σ−1T (THθ − µT ) ≥ µ′

TΣ−1T µT ) ≤ α]

= PrH0

Pr∗

k

i=1

(U∗

i − (ni − 1))2

2(ni − 1)−

1k

i=1

V∗i

2(ni−1)

k

i=1

V ∗

i (U∗

i − (ni − 1))2(ni − 1)

2

≥

ki=1

ni − 12

−1

ki=1

Vi2(ni−1)

k

i=1

Vi

2

2

≤ α

.

Under the null hypothesis H0, µ′

TΣ−1µT = (Hλ − µT )

′Σ−1(Hλ − µT ),

= PrH0

Pr∗

k

i=1

(U∗

i − (ni − 1))2

2(ni − 1)−

1k

i=1

V∗i

2(ni−1)

k

i=1

V ∗

i (U∗

i − (ni − 1))2(ni − 1)

2

≥

ki=1

(λiVi − (ni − 1))2

2(ni − 1)−

1k

i=1

Vi2(ni−1)

k

i=1

Vi(λiVi − (ni − 1))2(ni − 1)

2

≤ α

= PrH0

Pr∗

k

i=1

(U∗

i − (ni − 1))2

2(ni − 1)−

1k

i=1

V∗i

2(ni−1)

k

i=1

V ∗

i (U∗

i − (ni − 1))2(ni − 1)

2

≥

ki=1

(Ui − (ni − 1))2

2(ni − 1)−

1k

i=1

Vi2(ni−1)

k

i=1

Vi(Ui − (ni − 1))2(ni − 1)

2

≤ α

.

Obviously,

Pr∗

k

i=1

(U∗

i − (ni − 1))2

2(ni − 1)−

1k

i=1

V∗i

2(ni−1)

k

i=1

V ∗

i (U∗

i − (ni − 1))2(ni − 1)

2

≥

ki=1

(Ui − (ni − 1))2

2(ni − 1)−

1k

i=1

Vi2(ni−1)

k

i=1

Vi(Ui − (ni − 1))2(ni − 1)

2

∼ U(0, 1).

Then, PrH0(p(x, v) ≤ α) = α is proved.

In practice, the generalized p-value is computed through the following algorithm.

Algorithm 1. For a given (n1, . . . , nk), (x1, . . . , xk) and (v1, . . . , vk).For l = 1, . . . , L;Generate Ui ∼ χ2

ni−1, i = 1, . . . , k.(End l loop)Compute Tl = THλ.

Compute µT =1L

Ll=1 Tl and ΣT =

1L−1

Ll=1(Tl − µT )(Tl − µT )

′.

M. Chang et al. / Statistics and Probability Letters 82 (2012) 1755–1760 1759

Compute ∥T∥2l = (Tl − µT )

′Σ−1T (Tl − µT ), l = 1, . . . , L and µ′

T Σ−1T µT .

LetWl = 1 if ∥T∥2l ≥ µ′

T Σ−1T µT , elseWl = 0.

1L

Ll=1 Wl is a simulated estimate of generalized p-value in (3.8) for testing (3.3).

4. Numerical results

In this section, some simulated type I error probabilities and powers are given in Tables 1 and 2 for the test proposed inSection 3 and the modified likelihood ratio (MLR) test proposed by Chhikara and Folks (1989). In Tables 1 and 2, GP, MLRdenote the Monte Carlo estimators of the probabilities using the new proposed generalized p-value test and the modifiedlikelihood ratio test, respectively. The type I error probability and power of the test based on (3.8) were computed as follows.

Algorithm 2. Form = 1 toMFor a given (n1, n2, . . . , nk) and (µ, λ1, λ2, . . . , λk), generate (x1, x2, . . . , xk) and (v1, v2, . . . , vk) according to (3.1).Use Algorithm 1 to compute the generalized p-value pm;(Endm loop).

If the parametersµi are chosen such thatHθ = 0, (Number of pm < α)/M is a simulated estimate of type I error probability;otherwise (Number of pm < α)/M is a simulated estimate of the power. For each setting of sample size n = (n1, n2, . . . , nk),we take µi = 1, L = 2000 andM = 5000. The nominal levels are set at α = 0.05 and α = 0.1, respectively.

From the numerical results in Table 1, it appears that the test based on the generalized p-value in (3.8) has a type I errorprobability very close to the nominal level for all cases. However, theMLR test is more liberal even for large samples. Table 2presents the powers of the proposed generalized p-value test are better than the MLR test except when the ni are small. Butwhen the ni are small, as in an asymptotic test, the MLR test is not suitable in practice. The exact frequency property of theproposed test also is proved. Therefore, the new generalized p-value approach performs better than the MLR test.

5. Concluding remarks

For the problem of testing the homogeneity of inverse Gaussian scale-like parameters, this article derives a newgeneralized test variable and gives the generalized p-value based on the test variable (the test is analogous to the methodproposed by Liu and Xu (2010)). Although the literature abounds with examples of generalized p-value tests, no exactfrequent property has been proved. The only evidence of their acceptability for practical use is through simulation studieswhich suggest that almost all reported generalized p-value test methods appear to have type I error probabilities close tothe nominal significant level. But for the problem of testing the homogeneity of IG scale-like parameters, the proposed newgeneralized p-value approach has an exact frequency property. Therefore, except for the simulation results in Section 4, thisgood property also shows that the new method is worth using in practice.

Table 1Simulated type I error (based on 5000 simulations).

GP (α = 0.05) MLR GP (α = 0.1) MLR

λ = (0.3, 0.3, 0.3)n = (5, 5, 5) 0.054 0.051 0.102 0.103n = (5, 8, 10) 0.051 0.058 0.094 0.109n = (10, 10, 10) 0.046 0.068 0.099 0.117n = (10, 15, 20) 0.044 0.057 0.098 0.109n = (30, 30, 30) 0.047 0.069 0.088 0.122λ = (0.5, 0.5, 0.5, 0.5, 0.5)n = (5, 5, 5, 5, 5) 0.056 0.050 0.106 0.104n = (7, 8, 9, 10, 12) 0.050 0.067 0.101 0.106n = (10, 10, 10, 10, 10) 0.046 0.055 0.089 0.121n = (12, 15, 16, 18, 20) 0.051 0.068 0.092 0.115n = (20, 20, 30, 30, 30) 0.047 0.079 0.097 0.124λ = (1, 1, 1, 1, 1, 1, 1, 1)n = (5, 5, 5, 5, 5, 5, 5, 5) 0.059 0.051 0.112 0.103n = (7, 8, 10, 12, 12, 14, 14, 15) 0.051 0.049 0.102 0.125n = (10, 10, 10, 10, 10, 10, 10, 10) 0.046 0.062 0.091 0.118n = (10, 12, 15, 15, 18, 18, 20, 20) 0.048 0.074 0.100 0.119n = (20, 20, 30, 30, 30, 30, 40, 40) 0.049 0.070 0.087 0.136λ = (1.2, 1.2, 1.2, 1.2, 1.2, 1.2, 1.2, 1.2, 1.2, 1.2)n = (5, 5, 5, 5, 5, 5, 5, 5, 5, 5) 0.056 0.052 0.105 0.104n = (7, 8, 8, 10, 10, 12, 12, 12, 15, 15) 0.059 0.057 0.107 0.138n = (10, 10, 10, 10, 10, 10, 10, 10, 10, 10) 0.048 0.074 0.099 0.124n = (10, 10, 12, 12, 15, 15, 18, 18, 20, 20) 0.047 0.069 0.098 0.135n = (20, 20, 20, 30, 30, 30, 30, 40, 40, 40) 0.044 0.071 0.092 0.126

1760 M. Chang et al. / Statistics and Probability Letters 82 (2012) 1755–1760

Table 2Simulated powers (based on 5000 simulations).

GP (α = 0.05) MLR GP (α = 0.1) MLR

λ = (0.3, 0.5, 0.8)n = (5, 5, 5) 0.089 0.098 0.156 0.154n = (5, 8, 10) 0.112 0.109 0.186 0.191n = (10, 10, 10) 0.133 0.125 0.209 0.197n = (10, 15, 20) 0.284 0.271 0.398 0.372n = (30, 30, 30) 0.366 0.349 0.498 0.476λ = (0.3, 0.5, 0.7, 1)n = (5, 5, 5, 5) 0.079 0.081 0.162 0.163n = (5, 8, 10, 12) 0.091 0.089 0.182 0.175n = (10, 10, 10, 10) 0.122 0.117 0.231 0.218n = (10, 15, 18, 20) 0.208 0.204 0.298 0.292n = (30, 30, 30, 30) 0.448 0.434 0.587 0.566λ = (0.5, 0.5, 1, 1, 1, 1.2, 1.2)n = (5, 5, 5, 5, 5, 5, 5) 0.071 0.078 0.140 0.158n = (7, 8, 10, 12, 12, 14, 14) 0.109 0.098 0.172 0.174n = (10, 10, 10, 10, 10, 10, 10) 0.187 0.171 0.244 0.237n = (10, 12, 15, 15, 18, 18, 18) 0.214 0.202 0.278 0.270n = (20, 20, 30, 30, 30, 40, 40) 0.407 0.405 0.561 0.550λ = (0.3, 0.3, 0.5, 0.5, 1, 1, 1, 1.2, 1.2, 1.2)n = (5, 5, 5, 5, 5, 5, 5, 5, 5, 5) 0.082 0.086 0.144 0.152n = (7, 8, 8, 10, 10, 12, 12, 12, 15, 15) 0.102 0.096 0.188 0.169n = (10, 10, 10, 10, 10, 10, 10, 10, 10, 10) 0.184 0.175 0.225 0.213n = (10, 10, 12, 12, 15, 15, 18, 18, 20, 20) 0.279 0.274 0.362 0.363n = (20, 20, 20, 30, 30, 30, 30, 40, 40, 40) 0.387 0.362 0.499 0.481

Acknowledgments

The authors cordially thank the Editor and two referees for their kind help and comments which led to the improvementof this paper.

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