# Equivariance and Generalized Inference in Two-Sample Location-Scale Families

Authors: Sévérien Nkurunziza; Fuqi Chen
Journal: Journal of Probability and Statistics
 (2011)

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## Abstract

We are interested in-typical Behrens-Fisher problem in general location-scale families. We present a method of constructing generalized pivotal quantity (GPQ) and generalizedP value (GPV) for the difference between two location parameters. The suggested method is based on the minimum risk equivariant estimators (MREs), and thus, it is an extension of the methods based on maximum likelihood estimators and conditional inference, which have been, so far, applied to some specific distributions. The efficiency of the procedure is illustrated by Monte Carlo simulation studies. Finally, we apply the proposed method to two real datasets.


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## Body

## 1. Introduction

In statistical problems involving nuisance parameters, the small-sample optimal solution may not be available. For example, for the difference between means of two exponential distributions, or two normal distributions with different variances, small sample-optimal test and confidence intervals do not exist (see, [1]). To overcome this problem, Tsui and Weerahandi [2] introduced the concept of generalized Pvalue (GPV) and generalized test variable (GTV). Further, Weerahandi [1] developed the concept of generalized pivotal quantity (GPQ) and generalized confidence interval (GCI). The GCI and GPV have been revealed to perform well for some small-sample problems where classical procedures are not optimal. For example, Weerahandi [1] applied the GCIs to the difference in two exponential means and two normal means. In addition, Bebu and Mathew [3] developed a generalized pivotal quantity for comparing the means and variances of a bivariate log-normal distribution.In this paper, we present a method of constructing the GPQ and GTV in two-sample location-scale families. Also, we extend the method in Sprott [4] where the author applied conditional inference to some particular bivariate location-scale families. In the quoted book, the author uses the maximum likelihood estimator (MLE). However, it is well known that the MLE does not exist in some location and scale families. For more details, we refer to Pitman [5], and Gupta and Székely [6] among others.Our proposed method is based on Pitman estimator that is the minimum risk equivariant estimator (MRE). It is noticed that, when MLE of a location parameter (or scale parameter) exists, it is an equivariant estimator. Indeed, the suggested method is more general, and our simulation studies show that it provides a high coverage probability, high power and preserves the nominal level of the test.The rest of this paper is organized as follows. In Section1, we present some background about generalized inference in location and scale family. We also establish in Section 1 the proposed generalized pivotal quantity and generalized test variable in two-sample location-scale family. Section 2 gives the main result of this paper. Namely, in this section, we present the algorithm of the proposed method. In Section 3, we discuss the application of the method in some specific location-scale families. Section 4 presents some simulation studies as well as analysis results of two real datasets. Finally, Section 5 gives discussion and concluding remarks. Details and technical results are outlined in two appendices.


## 2. Background and Preliminary Results

In this section, we present some concepts of generalized inference for the convenience of the reader. Also, we set up notation which is used in this paper. For more details about the concepts of GPQ, GTV, and GPV, the reader is referred to Tsui and Weerahandi [2], Weerahandi [1], and Krishnamoorthy et al. [7] among others. Let X1,...,Xn be i.i.d. random variables from the population probability density function (pdf) fx(x∣η1). Also, let Y1,...,Ym be iid random variables from the population pdf fy(y∣η2). We assume that the two random samples (X1,...,Xn) and (Y1,...,Ym) are independent. Also, let η=(η1′,η2′)′ be a p-column vector of unknown parameters (with p≥2). Further, let τ(η) be a q-column vector function of η with q≤p, and to simplify the notation, let τ(η)=θ=(θ1,θ2′)′ where θ1 is the parameter of interest and θ2 is a vector of nuisance parameters. Let 𝒳 denote the sample space of possible values of (X,Y)′, where X=X1,...,Xn, Y=Y1,...,Ym, and let Θ denote the parameter space of θ. In addition, we denote (x,y)((x,y)∈𝒳) as an observation from (X,Y). Given this statistical model, two statistical problems about θ1 are considered.First, we are interested in deriving confidence interval estimation ofθ1. Second, for a given θ0, we consider the testing problem (2.1)H0:θ1≥θ0versusH1:θ1<θ0.Definition 2.1.
LetR=R(X,Y,x,y,θ) be a function of X, Y, x, y, θ, where θ=(θ1,θ2). Then, the function R is said to be a generalized pivotal quantity for θ1 if (1)
givenx,y, the distribution of R is free from unknown parameters;(2)
the observed value, defined asRobs=R(x,y,x,y,θ), does not depend on the nuisance parameter θ2.Definition 2.2.
The generalized test variable forθ1, is defined as a function of (X,Y,x,y,θ), say T(X,Y,x,y,θ), which satisfies the following requirements.  (1)
t=T(x,y,x,y,θ) is free of θ2.(2)
For fixedx, y, θ, the distribution of T(X,Y,x,y,θ) is free of the nuisance parameter θ2.(3)
For fixedx, y and θ2, P[T(X,Y,x,y,θ)≥T(x,y,x,y,θ)] is stochastically monotone in θ1.To make the connection between GPQ and GTV, it is noticed that the GTV can be derived from GPQR(X,Y,x,y,θ). In fact, if R(X,Y,x,y,θ) is a GPQ for θ1 then, (2.2)T1(X,Y,x,y,θ)=R(X,Y,x,y,θ)-R(x,y,x,y,θ)
is a GTV. For instance, if R(x,y,x,y,θ)=θ1, we have T1(X,Y,x,y,θ)=R(X,Y,x,y,θ)-θ1. For more details, see Krishnamoorthy et al. [7]. Also, the generalized P value for the testing problem (2.1) is defined as p=supH0P[T1(X,Y,x,y,θ)≥0]. More specifically, for the case where T1(X,Y,x,y,θ)=R(X,Y,x,y,θ)-θ1, the GPV for the testing problem (2.1) becomes
(2.3)p=sup⁡H0P[R(X,Y,x,y,θ)-θ0≥0]=P(R(X,Y,x,y,θ)≥θ0).
Thus, since the distribution of R(X,Y,x,y,θ) is free of any unknown parameters, the GPV for θ1 can be obtained from (2.3) by either analytical method or Monte Carlo simulation. We consider the case where η=(μ1,σ1,μ2,σ2)′, τ(η)=μ1-μ2. Thus, we present the GPQ and GTV for the difference between two location parameters δ=μ1-μ2. On one hand, we are interested in deriving GCIs for δ. On the other hand, we consider solving the following testing problem:
(2.4)H0:δ≥δ0versusH1:δ<δ0.
Let Rδ denote the GPQ for δ. For the testing problem (2.4), the generalized P value is
(2.5)pδ=P(Rδ≥δ0).


### 2.1. Equivariance and Minimum Risk Equivariant Estimators

In this subsection, we give a brief background about the concept of equivariance and minimum risk equivariant estimators in location-scale family. For more details about this concept, we refer to Lehmann and Casella [8, page 171–173], Schervish [9, chapter 6]  among others. To set up some notation, let X=(X1,X2,…,Xn)′ be a random sample whose joint pdf can be written as (1/σn)∏i=1ng((xi-μ)/σ) where g is a pdf which does not depend on μ and σ. Then, X is said to be from the location-scale family with location parameter μ and scale parameter σ.An estimatorδ1(X) for the scale parameter σ is said to be equivariant if it satisfies (2.6)δ1(bX+a)=bδ1(X),∀-∞<a<∞,b>0.
An estimator δ2(X) for the location parameter μ is said to be equivariant if it satisfies (2.7)δ2(bX+a)=bδ2(X)+a,∀-∞<a<∞,b>0.
Also, let δ(X) be equivariant estimator for the scale (or location) parameter θ and let R(δ(X),θ) be its risk function, that is, the expected value of a certain loss function which is invariant under the scale (or location) transformation. Then, the estimator δ(X) is said to be minimum risk equivariant estimator (MRE) if for any other equivariant estimator for θ, ϕ(X), we have
(2.8)R(δ(X),θ)≤R(ϕ(X),θ),∀θ∈Θ.In this paper, the loss function under consideration is the quadratic error loss function, and in this case, the minimum risk equivariant estimator is also known as Pitman estimator (see Lehmann and Casella [8, pages 154–174]).In particular, letμ̂lp and σ̂lp, l=1,2 denote the minimum risk equivariant estimator for μl and σl, l=1,2, respectively. In this notation, the subscript p refers to Pitman estimator. Further, let μ̂lobs, σ̂lobs denote the observed values of μ̂lp and σ̂lp, l=1,2, respectively. We close this section by recalling the result which is used in computing μ̂lp, and σ̂lp, l=1,2.Theorem 2.3.
LetX1,X2,...,Xn be iid random sample from scale-location family with pdf f(x∣μ,σ)=σ-1g((x-μ)/σ), where μ and σ are unknown. Also, under quadratic error loss function, suppose that there exists an equivariant estimator with finite risk. Then, under quadratic loss function the MRE of μ and σ are, respectively
(2.9)μ̂p(x)=∫0∞∫-∞∞uvn+1∏i=1ng((xi-u)v)dudv×(∫0∞∫-∞∞vn+1∏i=1ng((xi-u)v)dudv)-1,σ̂p(x)=∫0∞vn∫-∞∞∏i=1ng((xi-u)v)dudv×(∫0∞vn+1∫-∞∞∏i=1ng((xi-u)v)dudv)-1.For proof, the reader is referred to Lehmann and Casella [8, page 154], Schervish  [9, chapter 6] and references therein.


### 2.2. GPQ and GTV in Two-Sample Location-Scale Families

Let(X1,…,Xn) and (Y1,…,Ym) be two-sample iid from the population pdfs (2.10)fx(xi∣μ1,σ1)=σ1-1g1(xi-μ1σ1),fy(yj∣μ2,σ2)=σ2-1g2(yj-μ2σ2),
respectively with i=1,…,n; j=1,…,m; where μ1, μ2, σ1, σ2 unknown parameters, and g1 and g2 are pdfs. Then, the joint pdf of X1,…,Xn, Y1,…,Ym is given by
(2.11)fxy(x1,…,xn,y1,…,ym)=σ1-nσ2-m∏i=1ng1(xi-μ1σ1)∏j=1mg2(yj-μ2σ2).We are interested in inference problems concerning the difference between the location parametersδ=μ1-μ2, with ρ=σ2/σ1 unknown. To set up notation, let Z3=(Z31,Z32), where (2.12)Z3l=μ̂lp-μlσ̂lp,l=1,2.
By using equivariance property of μ̂l and σ̂l, l=1,2, we derive the GPQ, GTV of δ. Indeed, let (2.13)R31=μ̂1pobs-σ̂1pobsZ31,R32=μ̂2pobs-σ̂2pobsZ32,
where Z3l, l=1,2 are defined by (2.12). Then, R3l are GPQ for μl, l=1,2. Using R3l, l=1,2, we derive the GPQ and GTV for δ as given by the following proposition.Proposition 2.4.
If the two samples are from the pdf in (2.10), the GPQ for δ is
(2.14)Rδ=μ̂1pobs-μ̂2pobs-σ̂1pobsZ31+σ̂2pobsZ32.
Furthermore, the GTV is T3=Rδ-δ.Proof.
Obviously, the observed value ofRδ is δ. Further, since μ̂l and σ̂l, l=1,2 are equivariant for μl and σl, respectively, by using Lemma A.3, we conclude that the distributions of R3l, l=1,2 are not dependent on parameter. Therefore, the distribution of Rδ=R31-R32 does not depend on parameter, and this completes the proof.In the following section, we present an algorithm which is used in computing the GCI and GPV. The proposed algorithm extensively uses Proposition2.5 and Corollary 2.6 given below. To the best of our knowledge, these two results are not in the existing literature. To set up notation, let a=(a1,…,an)′, and let b=(b1,…,bm)′, where
(2.15)ai=Xi-μ̂1pσ̂1p,bj=Yj-μ̂2pσ̂2p,i=1,…,n,j=1,…,m.
Further, let a̲=(a1,a2,…,an-2)′, let b̲=(b1,b2,…,bn-2)′ and let Z4l=σ̂lp/σl, l=1,2.Proposition 2.5.
Assume that the two random samples are from the pdfs in (2.10). Then, conditionally to a̲, b̲, the joint pdf of (Z31,Z32,Z41,Z42) is
(2.16)f(x,y,t,s∣a̲,b̲)=C-1tn-1sm-1∏i=1ng1(t(x+ai))∏j=1mg2(s(y+bj)),
t,s>0, -∞<x,y<+∞ where
(2.17)C=∬-∞∞∬0∞z1n-1z2m-1∏i=1ng1(z1(u+ai))∏j=1mg2(z2(w+bj))dz1dz2dudw.Proposition2.5 extends Corollary A.2 that is established in Appendix A. The proof follows from similar arguments as for Corollary A.2. Further, from Proposition 2.5, we establish Corollary 2.6 that gives the joint pdf of (Z31,Z32) conditionally to a̲, b̲.Corollary 2.6.
If Proposition2.5 holds then, conditionally to a̲, b̲, the joint pdf of (Z31,Z32) is
(2.18)f3(x,y∣a̲,b̲)=C-1∬0∞tn-1sm-1∏i=1ng1(t(x+ai))∏j=1mg2(s(y+bj))dsdt,
where with C given in (2.17).Proof.
The proof follows directly from Proposition2.5.


## 2.1. Equivariance and Minimum Risk Equivariant Estimators

In this subsection, we give a brief background about the concept of equivariance and minimum risk equivariant estimators in location-scale family. For more details about this concept, we refer to Lehmann and Casella [8, page 171–173], Schervish [9, chapter 6]  among others. To set up some notation, let X=(X1,X2,…,Xn)′ be a random sample whose joint pdf can be written as (1/σn)∏i=1ng((xi-μ)/σ) where g is a pdf which does not depend on μ and σ. Then, X is said to be from the location-scale family with location parameter μ and scale parameter σ.An estimatorδ1(X) for the scale parameter σ is said to be equivariant if it satisfies (2.6)δ1(bX+a)=bδ1(X),∀-∞<a<∞,b>0.
An estimator δ2(X) for the location parameter μ is said to be equivariant if it satisfies (2.7)δ2(bX+a)=bδ2(X)+a,∀-∞<a<∞,b>0.
Also, let δ(X) be equivariant estimator for the scale (or location) parameter θ and let R(δ(X),θ) be its risk function, that is, the expected value of a certain loss function which is invariant under the scale (or location) transformation. Then, the estimator δ(X) is said to be minimum risk equivariant estimator (MRE) if for any other equivariant estimator for θ, ϕ(X), we have
(2.8)R(δ(X),θ)≤R(ϕ(X),θ),∀θ∈Θ.In this paper, the loss function under consideration is the quadratic error loss function, and in this case, the minimum risk equivariant estimator is also known as Pitman estimator (see Lehmann and Casella [8, pages 154–174]).In particular, letμ̂lp and σ̂lp, l=1,2 denote the minimum risk equivariant estimator for μl and σl, l=1,2, respectively. In this notation, the subscript p refers to Pitman estimator. Further, let μ̂lobs, σ̂lobs denote the observed values of μ̂lp and σ̂lp, l=1,2, respectively. We close this section by recalling the result which is used in computing μ̂lp, and σ̂lp, l=1,2.Theorem 2.3.
LetX1,X2,...,Xn be iid random sample from scale-location family with pdf f(x∣μ,σ)=σ-1g((x-μ)/σ), where μ and σ are unknown. Also, under quadratic error loss function, suppose that there exists an equivariant estimator with finite risk. Then, under quadratic loss function the MRE of μ and σ are, respectively
(2.9)μ̂p(x)=∫0∞∫-∞∞uvn+1∏i=1ng((xi-u)v)dudv×(∫0∞∫-∞∞vn+1∏i=1ng((xi-u)v)dudv)-1,σ̂p(x)=∫0∞vn∫-∞∞∏i=1ng((xi-u)v)dudv×(∫0∞vn+1∫-∞∞∏i=1ng((xi-u)v)dudv)-1.For proof, the reader is referred to Lehmann and Casella [8, page 154], Schervish  [9, chapter 6] and references therein.


## 2.2. GPQ and GTV in Two-Sample Location-Scale Families

Let(X1,…,Xn) and (Y1,…,Ym) be two-sample iid from the population pdfs (2.10)fx(xi∣μ1,σ1)=σ1-1g1(xi-μ1σ1),fy(yj∣μ2,σ2)=σ2-1g2(yj-μ2σ2),
respectively with i=1,…,n; j=1,…,m; where μ1, μ2, σ1, σ2 unknown parameters, and g1 and g2 are pdfs. Then, the joint pdf of X1,…,Xn, Y1,…,Ym is given by
(2.11)fxy(x1,…,xn,y1,…,ym)=σ1-nσ2-m∏i=1ng1(xi-μ1σ1)∏j=1mg2(yj-μ2σ2).We are interested in inference problems concerning the difference between the location parametersδ=μ1-μ2, with ρ=σ2/σ1 unknown. To set up notation, let Z3=(Z31,Z32), where (2.12)Z3l=μ̂lp-μlσ̂lp,l=1,2.
By using equivariance property of μ̂l and σ̂l, l=1,2, we derive the GPQ, GTV of δ. Indeed, let (2.13)R31=μ̂1pobs-σ̂1pobsZ31,R32=μ̂2pobs-σ̂2pobsZ32,
where Z3l, l=1,2 are defined by (2.12). Then, R3l are GPQ for μl, l=1,2. Using R3l, l=1,2, we derive the GPQ and GTV for δ as given by the following proposition.Proposition 2.4.
If the two samples are from the pdf in (2.10), the GPQ for δ is
(2.14)Rδ=μ̂1pobs-μ̂2pobs-σ̂1pobsZ31+σ̂2pobsZ32.
Furthermore, the GTV is T3=Rδ-δ.Proof.
Obviously, the observed value ofRδ is δ. Further, since μ̂l and σ̂l, l=1,2 are equivariant for μl and σl, respectively, by using Lemma A.3, we conclude that the distributions of R3l, l=1,2 are not dependent on parameter. Therefore, the distribution of Rδ=R31-R32 does not depend on parameter, and this completes the proof.In the following section, we present an algorithm which is used in computing the GCI and GPV. The proposed algorithm extensively uses Proposition2.5 and Corollary 2.6 given below. To the best of our knowledge, these two results are not in the existing literature. To set up notation, let a=(a1,…,an)′, and let b=(b1,…,bm)′, where
(2.15)ai=Xi-μ̂1pσ̂1p,bj=Yj-μ̂2pσ̂2p,i=1,…,n,j=1,…,m.
Further, let a̲=(a1,a2,…,an-2)′, let b̲=(b1,b2,…,bn-2)′ and let Z4l=σ̂lp/σl, l=1,2.Proposition 2.5.
Assume that the two random samples are from the pdfs in (2.10). Then, conditionally to a̲, b̲, the joint pdf of (Z31,Z32,Z41,Z42) is
(2.16)f(x,y,t,s∣a̲,b̲)=C-1tn-1sm-1∏i=1ng1(t(x+ai))∏j=1mg2(s(y+bj)),
t,s>0, -∞<x,y<+∞ where
(2.17)C=∬-∞∞∬0∞z1n-1z2m-1∏i=1ng1(z1(u+ai))∏j=1mg2(z2(w+bj))dz1dz2dudw.Proposition2.5 extends Corollary A.2 that is established in Appendix A. The proof follows from similar arguments as for Corollary A.2. Further, from Proposition 2.5, we establish Corollary 2.6 that gives the joint pdf of (Z31,Z32) conditionally to a̲, b̲.Corollary 2.6.
If Proposition2.5 holds then, conditionally to a̲, b̲, the joint pdf of (Z31,Z32) is
(2.18)f3(x,y∣a̲,b̲)=C-1∬0∞tn-1sm-1∏i=1ng1(t(x+ai))∏j=1mg2(s(y+bj))dsdt,
where with C given in (2.17).Proof.
The proof follows directly from Proposition2.5.


## 3. Framework

In general, the distributions of the GPQRδ in (2.14) do not have a closed form. Accordingly, Monte Carlo simulations are needed in order to compute numerically the distributions of Rδ. In this section, we present an algorithm which is used in computing the GCI and GPV for δ. The proposed algorithm is applicable to all members of location-scale families, and in particular, it is applicable to the normal family that is the most commonly discussed in the literature. To the best of our knowledge, there does not exist a similar algorithm in the literature.The proposed GCI and GPV are obtained by using the following algorithm.(1)
For a given datasetx, using Theorem 2.3, compute μ̂lobs(x), σ̂lobs(x), the observed values of μ̂l(X), σ̂l(X), l=1,2, respectively.(2)
By using (2.15), compute {ai}, i=1,2,…,n, and {bj}, j=1,2,…,m.(3)
GenerateU1l~U(0,1) for l=1,2.(4)
From the pdf of(Z31,Z32), f3(x,y∣a̲,b̲) given in (2.18), determine Z31 and Z32 such that ∫-∞Z31∫-∞∞f3(x,y∣a̲,b̲)dydx=U11,∫-∞Z32∫-∞∞f3(x,y∣a̲,b̲)dxdy=U12.(5)
By using (2.14), compute Rδ.(6)
Repeat from step (3) to (5),M times (with M large), and set Rδ,k the value of Rδ obtained at the kth replicate, k=1,2,…,M.(7)
Findℛδ,α/2(x) and ℛδ,1-α/2(x) as, respectively, 100α/2 and 100(1-α/2) percentiles of Rδ,1,Rδ,2,…,Rδ,M.(8)
Let𝕀A denote the indicator function of the event A. Using (2.3), estimate the GPV for δ by p̂δ=M-1∑k=1M𝕀{Rδ,k≥δ0}.Remark 3.1.
The equations in step (4) of the above algorithm do not generally give a closed-form solution. Thus, some numerical methods are needed in order to find the quantilesZ31 and Z32. In this paper, we applied Newton's method.Remark 3.2.
For the normal sample case, the proposed algorithm produces the same solution as in Weerahandi [1]. Indeed, at normal case, as established in Section 3, Z31 and Z32 have Student's distributions with, respectively, n-1 and m-1 degrees of freedom.


## 4. Some Cases of Two-Sample Location-Scale Families

In this section, we discuss the application of the proposed method to some specific two-sample location-scale families. More precisely, we discuss the application of the proposed method to the two-sample location-scale families for which MLEs do not exist. Also, in order to illustrate the fact that the proposed approach generalizes the method designed at normal case, we discuss briefly the two-sample normal families case.


### 4.1. Two-Sample Normal Case

Letx=(x1,…,xn)′ and let y=(y1,…,ym)′. From (2.3), we have
(4.1)fxy(x,y)=σ1-nσ2-m(2π)-(n+m)/2exp⁡[-(2σ12)-1∑i=1n(xi-μ1)2-(2σ22)-1∑j=1m(yj-μ2)2].
Under the model in (4.1), we illustrate the computation of GCI and GPV, based on the proposed GPQ. To set up notation, let
(4.2)a̅=1n∑i=1nai,S12=∑i=1n(ai-a̅)2,b̅=1m∑j=1mbi,S22=∑j=1m(bi-b̅)2.
If Xi~𝒩(μ1,σ1) and Yi~𝒩(μ2,σ2), using Theorem 2.3, we have(4.3)μ̂1p=n-1∑i=1nXi,μ̂2p=m-1∑j=1mYj,σ̂1p=(Γ(n/2)Γ((n+1)/2))12∑i=1n(Xi-X̅)2,
and σ̂2p=(Γ(m/2)/Γ((m+1)/2))∑j=1m(Yj-Y̅)2/2. Also, a̅=b̅=0,S12=(2Γ2((n+1)/2))/Γ2(n/2), S22=(2Γ2((m+1)/2))/Γ2(m/2).Then, using Proposition2.4 and some computations, we have (4.4)R31=μ̂1pobs-σ̂1pobs(S1obsn(n-1))×(n(n-1)((μ̂1p-μ1)/σ̂1p)S1)=μ̂1pobs-σ̂1pobs(S1obsTn-1n(n-1)),
where 𝒯n-1 stands for a Student't variate with n-1 degrees of freedom. Similarly (4.5)R32=μ̂2pobs-σ̂2pobs(S2obsn(n-1))(n(n-1)((μ̂2p-μ2)/σ̂2p)S2)=μ̂2pobs-σ̂2pobs(S2obsTm-1m(m-1)),
and taking R31-R32, we get, (4.6)Rδ=μ̂1pobs-μ̂2pobs-σ̂1pobsS1obsTn-1n(n-1)+σ̂2pobsS2obsTm-1m(m-1).


### 4.2. Two Location-Scale Families Case Where MLE Does Not Exist

Following Gupta and Székely [6], let the families σl-1gl((x-μl)/σl), l=1,2 where
(4.7)g1(x)=c(xlog⁡2x)-1,g2(y)=c(ylog⁡2y)-1,0<x, y≤k<1, k is any constant that satisfies 0<k<1 and c=-1/log(k) is a constant. Gupta and Székely [6] proved that MLEs for σl, μl, l=1,2 do not exist.The second illustrative example is based on the result in Pitman [5]. Namely, we consider families σl-1gl((x-μl)/σl), l=1,2, where (4.8)gl(xl)=(2(1+|xl|)(1+log⁡⁡(1+|xl|))2)-1,-∞<xl<∞,l=1,2.Pitman [5] proved that MLEs for σl, μl, l=1,2 do not exist. For the families in (4.7) and (4.8), the pdf of R3l and R4l, l=1,2 do not have a closed form and thus, the distribution of Rδ is obtained numerically by using the algorithm given in Section 2.


## 4.1. Two-Sample Normal Case

Letx=(x1,…,xn)′ and let y=(y1,…,ym)′. From (2.3), we have
(4.1)fxy(x,y)=σ1-nσ2-m(2π)-(n+m)/2exp⁡[-(2σ12)-1∑i=1n(xi-μ1)2-(2σ22)-1∑j=1m(yj-μ2)2].
Under the model in (4.1), we illustrate the computation of GCI and GPV, based on the proposed GPQ. To set up notation, let
(4.2)a̅=1n∑i=1nai,S12=∑i=1n(ai-a̅)2,b̅=1m∑j=1mbi,S22=∑j=1m(bi-b̅)2.
If Xi~𝒩(μ1,σ1) and Yi~𝒩(μ2,σ2), using Theorem 2.3, we have(4.3)μ̂1p=n-1∑i=1nXi,μ̂2p=m-1∑j=1mYj,σ̂1p=(Γ(n/2)Γ((n+1)/2))12∑i=1n(Xi-X̅)2,
and σ̂2p=(Γ(m/2)/Γ((m+1)/2))∑j=1m(Yj-Y̅)2/2. Also, a̅=b̅=0,S12=(2Γ2((n+1)/2))/Γ2(n/2), S22=(2Γ2((m+1)/2))/Γ2(m/2).Then, using Proposition2.4 and some computations, we have (4.4)R31=μ̂1pobs-σ̂1pobs(S1obsn(n-1))×(n(n-1)((μ̂1p-μ1)/σ̂1p)S1)=μ̂1pobs-σ̂1pobs(S1obsTn-1n(n-1)),
where 𝒯n-1 stands for a Student't variate with n-1 degrees of freedom. Similarly (4.5)R32=μ̂2pobs-σ̂2pobs(S2obsn(n-1))(n(n-1)((μ̂2p-μ2)/σ̂2p)S2)=μ̂2pobs-σ̂2pobs(S2obsTm-1m(m-1)),
and taking R31-R32, we get, (4.6)Rδ=μ̂1pobs-μ̂2pobs-σ̂1pobsS1obsTn-1n(n-1)+σ̂2pobsS2obsTm-1m(m-1).


## 4.2. Two Location-Scale Families Case Where MLE Does Not Exist

Following Gupta and Székely [6], let the families σl-1gl((x-μl)/σl), l=1,2 where
(4.7)g1(x)=c(xlog⁡2x)-1,g2(y)=c(ylog⁡2y)-1,0<x, y≤k<1, k is any constant that satisfies 0<k<1 and c=-1/log(k) is a constant. Gupta and Székely [6] proved that MLEs for σl, μl, l=1,2 do not exist.The second illustrative example is based on the result in Pitman [5]. Namely, we consider families σl-1gl((x-μl)/σl), l=1,2, where (4.8)gl(xl)=(2(1+|xl|)(1+log⁡⁡(1+|xl|))2)-1,-∞<xl<∞,l=1,2.Pitman [5] proved that MLEs for σl, μl, l=1,2 do not exist. For the families in (4.7) and (4.8), the pdf of R3l and R4l, l=1,2 do not have a closed form and thus, the distribution of Rδ is obtained numerically by using the algorithm given in Section 2.


## 5. Simulation Study and Data Analysis

### 5.1. Simulation Study

In this section, we carry out intensive simulation studies in order to evaluate the performances of the suggested approach in small and moderate sample sizes. To this end, we generate 10000 two-samples from logistic distribution, from the distribution in (4.7), and from the distribution in (4.8). In order to save space, we report below the empirical coverage probability and the empirical power for the location-scale family given in (4.8). Namely, the simulated coverage probabilities of the 95% GCI are presented in Table 1, and the empirical powers of the proposed test are given in Table 2, at significance level α=.05.Table 1
The coverage probabilities (CPR) of the 95% GCI forδ (family in (4.8)).
Sizes(n,m)CPR:(μ1,μ2,σ1,σ2)=(2,2,2,2)CPR:(μ1,μ2,σ1,σ2)=(2,1,2,1)(5, 5)0.8470.839(10, 10)0.9250.920(20, 20)0.9420.938(50, 50)0.9460.942(60, 60)0.9460.945(10, 20)0.9670.957(10, 50)0.9700.972(30, 50)0.9620.969Table 2
The simulated powers forδ (the location-scale family in (4.8)).
Sizes(n,m)(δ,σ1,σ2)PowerSizes(n,m)(δ,σ1,σ2)Power(5,5)(-2,2,2)0.194(20,20)(-2,2,2)0.816(-1,2,2)0.102(-1,2,2)0.453(0,2,2)0.037(0,2,2)0.046(10,10)(-2,2,2)0.527(50,50)(-2,2,2)0.946(-1,2,2)0.251(-1,2,2)0.752(0,2,2)0.044(0,2,2)0.054(10,20)(-4,2,2)0.831(10,50)(-4,2,2)0.819(-2,2,2)0.510(-2,2,2)0.465(0,2,2)0.042(0,2,2)0.038In particular, concerning the GCI ofδ, Table 1 shows that, for n≥20, the coverage probabilities are also relatively close to the nominal confidence level of 95%. Interestingly, the case of equal scale parameters and that of unequal scale parameters seem to provide similar results. Further, it is noticed that as the sample size increases, the coverage probability gets closer to the nominal confidence level (95%). Concerning the performance of the solution to the testing problem (2.4), Table 2 shows that the power function varies with different values of m, n, μ1, σ1, σ2, and δ=μ1-μ2. In fact, from Figure 1, it can  be seen that when δ=δ0=0, the powers are all approximately equal to 0.05. But on the left-hand side of 0, the power continually increases to 1 when the distance between δ and 0 increases. Also, in the right hand side, the power decreases to 0 as the distance increases. Furthermore, in the left hand side of 0, for each exact value of δ, the power increases as the sample size increases.The simulated power versusδ (the family case in (4.8)).
(a)
Ifσ1=1,σ2=2 (the family in (4.8))(b)
Ifσ1=1,σ2=200 (the family in (4.8))


### 5.2. Illustrative Examples and Data Analysis

#### 5.2.1. Normal Body Temperature Dataset

This dataset is found in Mackowiak et al. [10]. In this dataset, a total number of 130 patients have been assigned, with 65 males and 65 females. Their body temperatures have been tested and recorded. Furthermore, it is already confirmed that the temperatures in these 2 gender groups are normally distributed. In particular, for the male group, one can consider X~𝒩(μ1,σ1), and for the female group, one can consider Y~𝒩(μ2,σ2). From Table 3, a 95% GCI for δ is (-0.54288915,-0.03725161) and thus, since the interval does not contain 0, there is a significant difference between the two location parameters. By applying (2.5) to the testing problem H0:δ≥0 versus H0:δ<0, the GPV is found to be 0.0133, and this result indicates that the null hypothesis should be rejected at 2% significant level, that is, this confirms that μ1<μ2.Table 3
Numerical results for the normal body temperature dataset.
Parameters of interestPoint estimate95% GCIμ198.1(97.92873, 98.28112)σ1.696(0.5283083, 0.8186700)μ298.39(98.20655, 98.57497)σ20.74(0.5623908, 0.8679169)δ−.2892308(−0.54288915, −0.03725161)


#### 5.2.2. Cloud Seeding Dataset

The cloud seeding dataset consists of the amount of rainfall (in acre-feet) which have been recorded. The dataset is given in Krishnamoorthy, and Mathew [11]. For this dataset, 26 clouds were randomly seeded with silver nitrate, and 26 others were unseeded. In the above quoted paper, the authors showed that lognormal model fits the dataset very well. Thus, we assume unseeded cloud group X1~Lognormal(μ1,σ1) and seeded cloud group Y1~Lognormal(μ2,σ2). We set X=log(X1)~𝒩(μ1,σ1) and Y=log(Y1)~𝒩(μ2,σ2).From Table4, the GCI for δ indicates that the difference between the two location parameters is statistically significant. Also, for the testing problem H0:δ≥0 versus H1:δ<0, the GPV is 0.007 which indicates that μ1<μ2. Note that this finding corroborates the result given in Krishnamoorthy and Mathew [11], where the authors concluded that μ1 is statistically different from μ2.Table 4
Numerical results for the cloud seeding dataset.
Parameters of interestPoint estimate95% GCIμ13.990406(3.325968, 4.641512)σ11.625515(1.071410, 2.148553)μ25.134187(4.496115, 5.775415)σ21.583602(1.019519, 2.084743)δ−1.143781(−2.073196, −0.2211248)


## 5.1. Simulation Study

In this section, we carry out intensive simulation studies in order to evaluate the performances of the suggested approach in small and moderate sample sizes. To this end, we generate 10000 two-samples from logistic distribution, from the distribution in (4.7), and from the distribution in (4.8). In order to save space, we report below the empirical coverage probability and the empirical power for the location-scale family given in (4.8). Namely, the simulated coverage probabilities of the 95% GCI are presented in Table 1, and the empirical powers of the proposed test are given in Table 2, at significance level α=.05.Table 1
The coverage probabilities (CPR) of the 95% GCI forδ (family in (4.8)).
Sizes(n,m)CPR:(μ1,μ2,σ1,σ2)=(2,2,2,2)CPR:(μ1,μ2,σ1,σ2)=(2,1,2,1)(5, 5)0.8470.839(10, 10)0.9250.920(20, 20)0.9420.938(50, 50)0.9460.942(60, 60)0.9460.945(10, 20)0.9670.957(10, 50)0.9700.972(30, 50)0.9620.969Table 2
The simulated powers forδ (the location-scale family in (4.8)).
Sizes(n,m)(δ,σ1,σ2)PowerSizes(n,m)(δ,σ1,σ2)Power(5,5)(-2,2,2)0.194(20,20)(-2,2,2)0.816(-1,2,2)0.102(-1,2,2)0.453(0,2,2)0.037(0,2,2)0.046(10,10)(-2,2,2)0.527(50,50)(-2,2,2)0.946(-1,2,2)0.251(-1,2,2)0.752(0,2,2)0.044(0,2,2)0.054(10,20)(-4,2,2)0.831(10,50)(-4,2,2)0.819(-2,2,2)0.510(-2,2,2)0.465(0,2,2)0.042(0,2,2)0.038In particular, concerning the GCI ofδ, Table 1 shows that, for n≥20, the coverage probabilities are also relatively close to the nominal confidence level of 95%. Interestingly, the case of equal scale parameters and that of unequal scale parameters seem to provide similar results. Further, it is noticed that as the sample size increases, the coverage probability gets closer to the nominal confidence level (95%). Concerning the performance of the solution to the testing problem (2.4), Table 2 shows that the power function varies with different values of m, n, μ1, σ1, σ2, and δ=μ1-μ2. In fact, from Figure 1, it can  be seen that when δ=δ0=0, the powers are all approximately equal to 0.05. But on the left-hand side of 0, the power continually increases to 1 when the distance between δ and 0 increases. Also, in the right hand side, the power decreases to 0 as the distance increases. Furthermore, in the left hand side of 0, for each exact value of δ, the power increases as the sample size increases.The simulated power versusδ (the family case in (4.8)).
(a)
Ifσ1=1,σ2=2 (the family in (4.8))(b)
Ifσ1=1,σ2=200 (the family in (4.8))


## 5.2. Illustrative Examples and Data Analysis

### 5.2.1. Normal Body Temperature Dataset

This dataset is found in Mackowiak et al. [10]. In this dataset, a total number of 130 patients have been assigned, with 65 males and 65 females. Their body temperatures have been tested and recorded. Furthermore, it is already confirmed that the temperatures in these 2 gender groups are normally distributed. In particular, for the male group, one can consider X~𝒩(μ1,σ1), and for the female group, one can consider Y~𝒩(μ2,σ2). From Table 3, a 95% GCI for δ is (-0.54288915,-0.03725161) and thus, since the interval does not contain 0, there is a significant difference between the two location parameters. By applying (2.5) to the testing problem H0:δ≥0 versus H0:δ<0, the GPV is found to be 0.0133, and this result indicates that the null hypothesis should be rejected at 2% significant level, that is, this confirms that μ1<μ2.Table 3
Numerical results for the normal body temperature dataset.
Parameters of interestPoint estimate95% GCIμ198.1(97.92873, 98.28112)σ1.696(0.5283083, 0.8186700)μ298.39(98.20655, 98.57497)σ20.74(0.5623908, 0.8679169)δ−.2892308(−0.54288915, −0.03725161)


### 5.2.2. Cloud Seeding Dataset

The cloud seeding dataset consists of the amount of rainfall (in acre-feet) which have been recorded. The dataset is given in Krishnamoorthy, and Mathew [11]. For this dataset, 26 clouds were randomly seeded with silver nitrate, and 26 others were unseeded. In the above quoted paper, the authors showed that lognormal model fits the dataset very well. Thus, we assume unseeded cloud group X1~Lognormal(μ1,σ1) and seeded cloud group Y1~Lognormal(μ2,σ2). We set X=log(X1)~𝒩(μ1,σ1) and Y=log(Y1)~𝒩(μ2,σ2).From Table4, the GCI for δ indicates that the difference between the two location parameters is statistically significant. Also, for the testing problem H0:δ≥0 versus H1:δ<0, the GPV is 0.007 which indicates that μ1<μ2. Note that this finding corroborates the result given in Krishnamoorthy and Mathew [11], where the authors concluded that μ1 is statistically different from μ2.Table 4
Numerical results for the cloud seeding dataset.
Parameters of interestPoint estimate95% GCIμ13.990406(3.325968, 4.641512)σ11.625515(1.071410, 2.148553)μ25.134187(4.496115, 5.775415)σ21.583602(1.019519, 2.084743)δ−1.143781(−2.073196, −0.2211248)


## 5.2.1. Normal Body Temperature Dataset

This dataset is found in Mackowiak et al. [10]. In this dataset, a total number of 130 patients have been assigned, with 65 males and 65 females. Their body temperatures have been tested and recorded. Furthermore, it is already confirmed that the temperatures in these 2 gender groups are normally distributed. In particular, for the male group, one can consider X~𝒩(μ1,σ1), and for the female group, one can consider Y~𝒩(μ2,σ2). From Table 3, a 95% GCI for δ is (-0.54288915,-0.03725161) and thus, since the interval does not contain 0, there is a significant difference between the two location parameters. By applying (2.5) to the testing problem H0:δ≥0 versus H0:δ<0, the GPV is found to be 0.0133, and this result indicates that the null hypothesis should be rejected at 2% significant level, that is, this confirms that μ1<μ2.Table 3
Numerical results for the normal body temperature dataset.
Parameters of interestPoint estimate95% GCIμ198.1(97.92873, 98.28112)σ1.696(0.5283083, 0.8186700)μ298.39(98.20655, 98.57497)σ20.74(0.5623908, 0.8679169)δ−.2892308(−0.54288915, −0.03725161)


## 5.2.2. Cloud Seeding Dataset

The cloud seeding dataset consists of the amount of rainfall (in acre-feet) which have been recorded. The dataset is given in Krishnamoorthy, and Mathew [11]. For this dataset, 26 clouds were randomly seeded with silver nitrate, and 26 others were unseeded. In the above quoted paper, the authors showed that lognormal model fits the dataset very well. Thus, we assume unseeded cloud group X1~Lognormal(μ1,σ1) and seeded cloud group Y1~Lognormal(μ2,σ2). We set X=log(X1)~𝒩(μ1,σ1) and Y=log(Y1)~𝒩(μ2,σ2).From Table4, the GCI for δ indicates that the difference between the two location parameters is statistically significant. Also, for the testing problem H0:δ≥0 versus H1:δ<0, the GPV is 0.007 which indicates that μ1<μ2. Note that this finding corroborates the result given in Krishnamoorthy and Mathew [11], where the authors concluded that μ1 is statistically different from μ2.Table 4
Numerical results for the cloud seeding dataset.
Parameters of interestPoint estimate95% GCIμ13.990406(3.325968, 4.641512)σ11.625515(1.071410, 2.148553)μ25.134187(4.496115, 5.775415)σ21.583602(1.019519, 2.084743)δ−1.143781(−2.073196, −0.2211248)


## 6. Conclusion

In this paper, we proposed a solution of typical Behrens-Fisher problem in the general setting where two independent samples are from location-scale families. We presented a general statistical method for constructing GPQ and GTV for the difference between two location parameters of location-scale families. The proposed method is based on the minimum risk equivariant estimators which are known to be more general and more efficient than the MLEs. The simulation studies show that the proposed methods provide CIs and tests with high coverage probability and power, and the resulting tests preserve the significance level.The proposed method applies to all members of the location-scale families, as opposed to the methods given in the literature, as Welch's method, which are designed only for the normal case. In addition to this generality, our method is at least as good as Welch's method in the normal Behrens-Fisher problem (see simulation results in AppendixB).



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*Source: 474826-2011-10-18.xml*