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

We are interested in-typical Behrens-Fisher problem in general location-scale families. We present a method of constructing generalized pivotal quantity GPQ and generalized P 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 onmaximum 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.


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 sampleoptimal test and confidence intervals do not exist (see Weerahandi, 1993).To overcome this problem, Tsui and Weerahandi (1989) introduced the concept of generalized P-value (GPV) and generalized test variable (GTV).Further, Weerahandi (1993) developed the concept of generalized pivotal quantity (GPV) and generalized confidence interval (GCI).
These concepts of GPV and GCI can be considered as extensions of the classical P-value and confidence interval.The GCI and GPV have revealed to perform well for some smallsample problems where classical procedures are not optimal.For example, Weerahandi (1993) applied the generalized confidence intervals to the difference in two exponential means and two normal means.In addition, Bebu and Mathew (2007) developed a generalized pivotal quantity for comparing the means and variances of a bivariate log-normal distribution.
The existing literature, however, does not provide a systematic method of constructing GPQ applicable to all families of parametric methods.This paper overcomes these limitations by presenting a method of constructing the GPQ and GTV in two-sample locationscale families.Also, we extend the method in Sprott (2000, Chapter 7) 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 (1979), Gupta and Székely (1994 ) 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 Section 1, we present some background about generalized inference in location and scale family.Section 2 deals with generalized pivotal quantity and generalized test variable.In Section 3, we illustrate the application of the method to some specific location-scale families.Section 4, we present some numerical examples and simulation studies as well as analysis results of two real data sets.Finally, Section 5 gives discussion and concluding remarks.Details and technical results are outlined in the Appendix.

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 generalized pivotal quantity (GPQ), generalized test variable (GTV), and generalized p-value (GPV), the reader is referred to Tsui and Weerahandi (1989), Weerahandi (1993), and Krishnamoorthy, Mathew, and Ramachandran (2007) among others.Let X 1 , ..., X n be iid random variables from the population pdf f x (x|η 1 ).Also, let Y 1 , ...,Y m be iid random variables from the population pdf f y (y|η 2 ).We assume that the two random samples (X 1 , ..., X n ) and (Y 1 , ...,Y m ) are independent.Also, let η = (η 1 , η 2 ) 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 = (θ 1 , θ 2 ) where θ 1 is the parameter of interest and θ 2 is a vector of nuisance parameters.Let X denote the sample space of possible values of (X,Y ) , where X = X 1 , ..., X n , Y 1 , ...,Y m , 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 θ 1 are considered.
First, we are interested in deriving confidence interval estimation of θ 1 .Second, for given θ 0 , we consider the testing problem Then, the function R is said to be a generalized pivotal quantity if 1. given x, y, the distribution of R is free from unknown parameters ; 2. the observed value, defined as R obs = R(x, y, x, y, θ), does not depend on the nuisance parameter θ 2 .
Definition 1.2 The generalized test variable, is defined as a function of (X,Y, x, y, θ), say T (X,Y, x, y, θ), which satisfies the following requirements.
To make the connection between GPQ and GTV, it is noticed that the GTV can be derived from GPQ R(X,Y, x, y, θ).In fact, if R(X,Y, x, y, θ) is a GPQ for θ 1 then, For more details, see Krishnamoorthy, Mathew, and Ramachandran (2006).Further, the generalized p-value for the testing problem (1) is defined as More specifically, for the case where T 1 (X,Y, x, y, θ) = R(X,Y, x, y, θ) − θ 1 , the generalized p-value for the testing problem (1) becomes p = sup Thus, since the distribution of R(X,Y, x, y, θ) is free of any unknown parameters, the generalized p-value at θ 0 can be obtained by using a numerical method or by using Monte Carlo simulation.Below, we consider the case where η = (µ 1 , σ 1 , µ 2 , σ 2 ) , τ (η) = (µ 1 − µ 2 , σ 1 /σ 2 ) .
Thus, we present more explicit form of GPQ and GTV for the difference between two location parameters δ = µ 1 − µ 2 and the ratio between two scale parameters ρ = σ 1 /σ 2 .On one hand, we are interested in deriving generalized confidence interval for δ and ρ.On the other hand, we consider to solve the following testing problems the testing problem and For the testing problems (4) and (5), their corresponding the generalized p-value are and Let δ p and ρ p denote the minimum risk equivariant estimator for δ and ρ respectively.In this notation, the subscript p refers to Pitman estimator.Indeed, in this paper, the loss function under consideration is the quadratic error loss function, and in this case, the minimum risk equivariant estimator is the Pitman estimator.Further, let δobs , ρobs denote the observed values of δ p and ρ p respectively.We close this section by recalling the result which is used in computing μlp , and σlp , l = 1, 2.
Theorem 1.1 Let X 1 , X 2 , ..., X n be iid random sample from location family with pdf f (x|θ ) = σ −1 g ((x − µ)/σ ), where µ and σ are unknown.Also, suppose that there exists an equivariant estimator with finite risk.Then, under quadratic loss function the MRE of µ and σ are respectively For proof, the reader is referred to Schervish (1997, chapter 6) and references therein.
To set up notation, let where Further, let a = (a 1 , . . ., a n ) and let b = (b 1 , . . ., b m ) , where , and for simplify sake, set The following proposition plays a central role in deriving the GPQ and GTV for δ and ρ.
Proposition 2.1 Assume that the two random samples are from the pdfs in (9).Then, conditionally to a, b, the joint pdf of From Proposition 2.1, follows Corollary 2.1 that gives the pdfs of (Z 31 , Z 32 ) and (Z 41 , Z 42 ) conditionally to a, b.
Corollary 2.1 If Proposition 2.1 holds then, conditionally to a, b, where with C given in (12).
Proposition 2.2 If the two samples are from the pdf in (9), then, the GPQ for δ and ρ are Furthermore, the GTV are The proof follows directly from the fact that R 3l and R 4l , l = 1, 2 are GPQ for µ l and σ l , Note that, in general, Monte Carlo simulations are needed in order to generate the distributions of R δ and R ρ .In fact, the distributions of these quantities are derived from Corollary 2.1, that do not lead necessary to the pdf with a closed form.

Illustrative examples
In this section, we provide illustrative examples of the proposed method.More precisely, we apply the method to the two-sample normal families.The goal of this section is to highlight the application of formulas given by Propositions 2.1 and Corollary 2.2.

Two-sample normal case
Let x = (x 1 , . . ., x n ) and let y = (y 1 , . . ., y m ) .From (10), we have Under the model in (19), we apply the method to construct the GPQ for the bivariate normal family.Furthermore, we also illustrate the computation of GCI and GPV, based on the GPQ we constructed.It noticed that, the GTV and GPQ obtained here are similar to that in Bebu and Mathew (2007).
To set up notation, let Further, and Then, using Proposition 2.2 and some computations, the GPQ of ρ is Again, using Proposition 2.2 and some computations, we have and similarly and taking R 31 − R 32 , we get, In this example, similar procedure can be established by replacing MRE by MLE.However, our simulation studies given in Section 4 show that confidence intervals based on MLEs have lower coverage probability as compared to the C.Is from the MRE.Also, the simulation studies show that the tests based on MLEs have lower power as compared to that based on MRE.In addition, below, we give an examples which show that the proposed method is more general and more flexible than that based on MLE.In fact, in the two following examples, the presented method is applied whereas MLEs do not exist.
For the families in (21) and ( 22), the pdf of R 3l and R 4l , l = 1, 2 do not have a closed form and thus, the distribution of R δ and R ρ are obtained numerically throughout Monte-Carlo simulations.
4 Simulation study and data analysis

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 normal distribution, Cauchy distribution, from the distribution in (21) and from the distribution in (22).In order to save space, we report below the empirical coverage probability and the empirical power for the normal family, and for the location-scale family given in (22).Namely, the simulated coverage probabilities of the 95% GCI are presented in Tables 1, 2, and 6 and, the empirical powers of the proposed test are given in Tables 3, 4, 5 and 7, at significance level α = .05.
From Table 1, it is noticed that for small and moderate sample sizes, the coverage probability for ρ is close to 0.95.Also, concerning the GCI of δ , Tables 1, 2 and 6 show that 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 (4), Tables 3, 4, 5 and 7 show that the power function varies with different values of m, n, µ 1 , σ 1 , σ 2 , and It is also noticed that the provided test is consistent.
For comparison purposes, we present in Table 2 the coverage probability obtained by using the Welch's approximation method for the normal case.Also, we present in Tables 3, 4, 5 the empirical powers obtained by using the Welch approximation method for the normal case.We focus on the heteroscedastic case since under homoscedasticity, classical inference provides the uniformly most powerful unbiased test to the testing problem concerning δ .Our numerical findings are summarized as follows.
(i) When the two sample sizes are different.Table 2 shows that the confidence interval from the proposed method is slightly more accurate than that given by Welch method.
(ii) When the two sample sizes are equal.Table 3 shows that the proposed method preserves the nominal level better than the Welch approximation method.Further, for moderate and large sample sizes, Table 3 shows that the power of the Welch method is generally less than or equal to the power of the proposed method.However, for small sample sizes, the opposite conclusion holds.
In conclusion, for the Behrens-Fisher problem with unbalanced sample sizes, the proposed confidence interval is at least as accurate as that given by Welch method.Further, the proposed test is at least as powerful as the Welch approximation test.
In addition, the proposed method has the advantage of being useful for the more general statistical model of two-samples from location-scale family.In fact, from Figure 1, it can be seen that when δ = δ 0 = 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.Power q q q q q q n=5 n=10 n=20 n=50 n=100 Exact Value of delta (sigma1=200, sigma2=200) Power q q q q q q n=5 n=10 n=20 n=50 n=100 Exact Value of delta (sigma1=1, sigma2=2) Power q q q q q q n=5 n=10 n=20 n=50 n=100 Exact Value of delta (sigma1=1, sigma2=200) Power q q q q q q n=5 n=10 n=20 n=50 n=100 Exact Value of delta (sigma1=1, sigma2=2) Power q q q q q q n=5 n=10 n=20 n=50 n=100 Exact Value of delta (sigma1=1, sigma2=200) Power q q q q q q n=5 n=10 n=20 n=50 n=100 (f) Sizes CPR for δ CPR for δ value 1 is included, and thus, the null hypothesis H 0 : ρ = 1 (or H 0 : σ 1 = σ 2 ) is acceptable at 5% significance level.In addition, for testing H 0 : ρ 1, the GPV is 0.6912, which confirms that the null hypothesis H 0 is acceptable.
Further, 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 (6) to the testing problem H 0 : δ 1 versus H 0 : δ < 1, the GPV is found to be 0.0133, and this result indicates that the null hypothesis should be rejected at 2% significant level, i.e. this confirms that µ 1 < µ 2 .

Cloud seeding data set
The cloud seeding data set consists in the amount of rainfall (in acre-feet) which have been recorded.The data set is given in Krishnamoorthy, and Mathew (2003).For this data set, 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 data set very well.
Thus, we assume unseeded cloud group X 1 ∼ Lognormal(µ 1 , σ 1 ) and seeded cloud group From Table9, the point estimate of ρ is 0.9742157, and a 95% GCI for ρ is found to be (0.4540212, 1.3706670), and accordingly, (at 5% significance level) one cannot reject the null hypothesis that the ratio of scale parameters is equal to 1. Further, for the testing H 0 : ρ 1, the corresponding GPV is 0.4548, which indicates that the null hypothesis is acceptable at 5% significance level.Further, the GCI for δ indicates that the difference between the two location parameters is statistically significant.Also, for the testing problem H 0 : δ 0 versus H 1 : δ < 0, the GPV is 0.007 which indicates that µ 1 < µ 2 .Note that this finding corroborates the result given in Krishnamoorthy and Mathew (2003), where the authors concluded that µ 1 is statistically different from µ 2 .

Conclusion
In this paper, we proposed a solution to the 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 C.Is 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 Welch's method which is 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.
where J is the Jacobian matrix.We have Proof From (26), we get directly the conditional joint pdf of ( μ1p , μ2p ) given a, b.By algebraic computations, we verify that the conditional pdf of ( μ1p − µ 1 , μ2p − µ 2 ) corresponds to that stated in the corollary.
(µ 1 , µ 2 , σ 1 , σ 2 ) n m CPR for δ (ρ known) CPR for δ (ρ unknown) CPR for ρ TAB. 3 -Powers for δ (Normal case ; equal samples sizes & unequal scale parameters) TAB. 4 -Powers for δ (Normal case with unequal sample sizes & scale parameters) This data set is found in Mackowiak,Wasserman, and Levine (1992).In this data set, 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 (µ 1 , σ 1 ) and for the female group, one can consider Y ∼ N (µ 2 , σ 2 ).From Table5.20, the 0.95 GCI for ρ is (0.6848705, 1.3106982), in which the TAB.8 -Numerical results for the normal body temperature data set