Generalized inference for the common location. parameter of several location-scale families

Transcription

1 Generalized inference for the common location parameter of several location-scale families Fuqi Chen and Sévérien Nkurunziza Abstract In this paper, we are interested in inference problem concerning the common location parameter of k location-scale families with k 2. More specifically, we study the case where the scale parameters of the families are unknown and possibly heterogeneous. The proposed solution is derived by using generalized inference method. To this end, we present a method of constructing the required generalized pivotal quantity (GPQ) and generalized p-value (GPV) for the common location parameter. The proposed approach is based on the minimum risk equivariant estimators (MRE) which is more general and more efficient than the maximum likelihood estimators (MLEs). Thus, we extend the approaches based on MLEs and conditional inference which have been so far applied to some specific distributions. Also, with intensive simulation studies, we illustrate the performance of the proposed approach in small and moderate sample sizes. Finally, the approach is applied to analyse the normal body temperature. Keywords: GCIs; generalized p-value; location-scale family; MRE; Pitman estimator; University of Windsor, 401 Sunset Avenue, Windsor, Ontario, N9B 3P4. chen111n@uwindsor.ca University of Windsor, 401 Sunset Avenue, Windsor, Ontario, N9B 3P4. severien@uwindsor.ca

2 Introduction Testing the common location parameter of several location-scale families with unknown scale parameters is one of the most interesting statistical inference problems. There are many applications in which this problem is involved. For instance, this situation arises in statistical analysis that combines the information from several independent studies or meta-analysis. Indeed, the meta-analysis is more frequent in clinical trials as well as in social and behavioral sciences. Also, this is commonly seen in many statistical areas or designs such as balanced incomplete block designs, panel models, and some regression models, and in each of these scenarios, practitioners are often interested in inference concerning the common location parameter of several distributions with unknown scale parameters. For other applications and scenarios, we refer to Krishnamoorthy and Lu (2003). In particular, the normal location-scale family is the family which got more attention in statistical literature and for this family many researches have been done since the last century. For instance, one of the methods for constructing approximate confidence interval for the common mean, say µ, is based on the well-known Graybill and Deal (1959) estimator for µ. This approach is extensively discussed in statistical literature. To give some other references, we quote Maric and Graybill (1979), Pagurova and Gurskii (1979), Sinha (1985) among others. Another method used by Fairweather (1972) and Jordan and Krishnamoorthy (1996) consists in constructing the exact confidence interval for µ based on inverting weighted linear combinations of the Student s t statistics and the Fisher-Snedecor s F statistics, respectively. Simulation studies showed the methods provided by Fairweather (1972) and Jordan and Krishnamoorthy (1996) are reliable under the different situations. Further, Yu et al. (1999) verified that only the Fairweather (1972) method always produces nonempty confidence intervals. In this paper, we are interested in generalized inference method concerning the common 2

3 location parameter of several location-scale families. As introduced by Tsui and Weerahandi (1989) and Weerahandi (1993), generalized inference is based on the concepts of generalized test variable (GTV), generalized pivotal quantity (GPQ), generalized P-value (GPV) and generalized confidence interval (GCI). It turns out that the GCI and GPV perform well for some small-sample problems where classical procedures are not optimal (see Weerahandi, 1993, Bebu and Mathew, 2007 among others). In Krishnamoorthy and Lu (2003), the authors propose a procedure based on inverting weighted linear combinations of the generalized pivotal quantities. However, in the quoted paper, the authors considered normal family only. In fact, the existing literature does not provide a systematic method of constructing GPQ applicable to all families of parametric methods. In this paper, we present a method of constructing the GPQ and GTV for the common location parameter of several location-scale families with unknown scale parameters. The suggested method is based on minimum risk equivariant estimator (MRE) that is more efficient and more general than the Maximum likelihood estimator (MLE). Also, the proposed approach is more flexible as compared to that based on MLEs since it is well known that the MLE does not exist in some location and scale families. The rest of this paper is organized as follows. In Section 1, we present some backgrounds about generalized inference. Section 2 deals with generalized pivotal quantity and generalized test variable in location and scale family. Section 3 gives the framework and main result. In Section 4, we illustrate the application of the method to some specific location-scale families. In Section 5, we present some numerical examples and simulation studies as well as analysis results of a real data set. Finally, Section 6 gives discussion and concluding remarks. Details and technical results are outlined in the Appendix. 3

4 1 Background and preliminary results For the convenience of the reader, this section recalls some concepts of generalized inference. For more details about these concepts, we refer to Tsui and Weerahandi (1989), Weerahandi (1993), and Krishnamoorthy et al. (2007) among others. Let g 1,...,g k be pdfs and let (X 11,...,X 1n1 ), (X 21,...,X 2n2 ),..., (X k1,...,x knk ) be k independent samples, and assume that for each i = 1,2,...,k, X i1,...,x ini are iid from the pdfs f i (x µ i,σ i ) = σ 1 i g i ((x µ i )/σ i ) (1.1) where µ 1, µ 2,..., µ k,σ 1,σ 2,...,σ k are unknown parameters, < µ i <, σ i > 0, i = 1,2,...,k. Thereafter, we consider that µ 1 = µ 2 = = µ k = µ. With the above statistical model, we are interested in inference problems concerning the common location parameter µ, with the scale parameters σ i, i = 1,...,k unknown and possibly heterogeneous. Namely, 1. we would like to establish the GCI for µ, based on some GPQ; 2. given a real value µ 0, we would like to derive the GPV for testing H 0 : µ µ 0 versus H 1 : µ < µ 0. (1.2) To simplify the notation, let θ 2 = (σ 1,σ 2,...,σ k ) and let θ = (µ,θ 2 ). Definition 1.1 Let R(X,x,θ) be a function of X,x,θ, with θ = (µ,θ 2 ). The function R(X,x,θ) is said to be a generalized pivotal quantity for µ if 1. given x, the distribution of R(X,x,θ) is free from unknown parameters; 2. the observed value, defined as R(x,x,θ), does not depend on the nuisance parameter θ 2. 4

5 Thereafter, we consider a subclass of GPQ for which R(x,x,θ) is a bijection function in µ. In particular, we consider without loss of generalities that the GPQ satisfies R(x,x,θ) = µ. Definition 1.2 Let R(X,x,θ) be a GPQ for a scaler parameter µ. Then, an equal-tailed (1 α)100% generalized confidence interval (GCI) for µ is [R µ,α/2 (x), R µ,1 α/2 (x)], where the quantity R µ,γ (x) satisfies P[R(X,x,θ) R µ,γ (x)] = γ. (1.3) Also, one-sided generalized confidence bounds are defined in the similar way. Definition 1.3 Let T (X,x,θ) be a function of X,x,θ. The function T (X,x,θ) is said to be a generalized test variable for µ if 1. t = T (x,x,θ) does not depend on θ 2 ; 2. For fixed x,θ, the distribution of T (X,x,θ) is free of the nuisance parameter θ 2 ; 3. For fixed x and θ 2, P[T (X,x,θ) T (x,x,θ)] is stochastically monotone in µ. It is noticed that the GTV can be derived from GPQ R(X,x,θ) by taking T (X,x,θ) = R(X,x,θ) R(x,x,θ). Thus, if R(x,x,θ) = µ, we have T (X,x,θ) = R(X,x,θ) µ. Definition 1.4 Let T (X,x,θ) be GTV and consider the testing problem in (1.2). The generalized p-value (GPV) is defined as p = sup H 0 P[T (X,x,θ) 0]. More specifically, if T (X,x,θ) = R(X,x,θ) µ, the GPV for the testing problem in (1.2) becomes p = sup H 0 P[R(X,x,θ) µ 0]. Also, since the distribution of R(X,x,θ) does not depend on θ, and since P[R(X,x,θ) µ 0] is a decreasing function in µ, we have p = P(R(X,x,θ) µ 0 ). (1.4) 5

6 Note that the relations in (1.3) and (1.4) do not always lead to closed form solutions. Nevertheless, since the distribution of R(X,x,θ) is free of unknown parameters, the GCI and GPV can be obtained using a numerical method or Monte Carlo simulation. Let ˆµ i and ˆσ i denote the MREs for µ i and σ i respectively, for each i = 1,...,k. Further, let ˆµ si, ˆσ si denote the observed values of ˆµ i and ˆσ i respectively. We close this section by recalling a result which is used in computing the MREs ˆµ i and ˆσ i. To this end, let Y 1,...,Y n be iid from the population probability density functions (pdfs) f (x λ,τ) = τ 1 g((x λ)/τ) where τ,λ are unknown parameters, and g is a pdf. Then, under quadratic loss function the MRE of λ and τ are respectively (see Theorem A.3 in the Appendix) ˆλ(y) = ˆτ(y) = 0 0 uv n 3 n g((y i u)/v)dudv v n 2 n g((y i u)/v)dudv / ( 0 / ( 0 ) v n 3 n g((y i u)/v)dudv, ) v n 3 n g((y i u)/v)dudv. (1.5) 2 GCI and PTV for the common µ As mentioned in the introduction, the proposed GPQ and GTV are based on MREs. Let ˆµ i and ˆσ i denote respectively the MREs of µ and σ i based on the i th sample, i = 1,...,k. Further, let ˆµ si and ˆσ si denote respectively the observed values of ˆµ i and ˆσ i, i = 1,...,k. By using the fact that ˆµ i and ˆσ i are equivariant estimator, we conclude that, for each i = 1,2,...,k, ( ˆµ i µ)/ ˆσ i and ˆσ i /σ i are pivotal quantities for µ and σ i respectively. Then, based on the i th sample, i = 1,2,...,k, we consider the following GPQ for µ and σ i respectively, R µi = ˆµ si ˆσ si ( ˆµ i µ)/ ˆσ i, and R σi = ˆσ si ( ˆσ i /σ i ) 1. (2.1) Further, it can be verified that a weighted average of R µi is a GPQ for µ (see also Krishnamoorthy and Lu, 2003). Thus, the proposed GPQ is formally stated in the following proposition. 6

7 Proposition 2.1 If the k samples are from the pdf in (1.1), then, the GPQ for µ is R(X,x,θ) = k W i R µi, with W i = ( n i /R 2 σ i ) ( k i = 1,...,k; where R µi, R σi are given in (2.1). Furthermore, the GTV is T (X,x,θ) = k n j /R 2 σ j ) 1, (2.2) W i R µi µ. (2.3) The proof follows directly from the fact that R µi and R σi are GPQ for µ and σ i respectively. In closing this section, we note that 100γ% GCI is obtained by combining (1.3) and (2.2). Further, in solving the testing problem in (1.2), GPV is obtained by combining (1.4) and (2.2). In general, the equations (1.3) and (1.4) do not have a closed form solution and thus, we use Monte Carlo method, with an algorithm that is given in the following section. 3 Algorithm and main result To set up notation, let b i = ( b i1,...,b i(ni 2)) with bi j = (x i j ˆµ i )/ ˆσ i, i = 1,...,k; j = 1,...,n i. Also, let Z 1i = ( ˆµ i µ)/ ˆσ i, Z 2i = ˆσ i /σ i, and let h i (b) = y n i 1 n i 0 g i ((x + b i j )y)dxdy,i = 1,2,...,k. (3.1) The established algorithm uses extensively the following proposition. Proposition 3.1 If (1.1) holds, then, conditionally to b, the pdf of Z 1i is f 1i (z 1i b) = and the pdf of Z 2i is y n i 1 n i 0 f 2i (z 2i b) = z n i 1 2i n i g i ((z 1i + b i j )y)dy / h i (b), i = 1,2,...,k (3.2) g i ((x + b i j )z 2i )dx / h i (b), i = 1,2,...,k; (3.3) where h i (b) is given in(3.1). 7

8 Proof From Proposition A.3 in the Appendix, it follows directly that conditionally to b, the joint pdf of (Z 1i,Z 2i ) is f ( z 1i,z 2i b ) i = z n i 1 2i n i g i ((z 1i + b i j )z 2i ) / h i (b), (3.4) where h i (b) is given in(3.1). Therefore, conditionally to b, the marginal pdf of Z 1i and Z 2i are given by (3.2) and (3.3) respectively, that completes the proof. Algorithm for the proposed GCI and GPV For given (n 1,...,n k ), and data set x; (i). From (1.5), find ˆµ si (x), ˆσ si (x), the observed values of ˆµ i (X), ˆσ i (X), respectively. (ii). Compute {b i j = ( x i j ˆµ i ) / ˆσi }, i = 1,...,k, j = 1,...,n i. (iii). Generate U 1i U(0,1), i = 1,2,...,k. (iv). For each U 1i, find the quantities Z 1i such that given in (3.2). Z1i f 1i (x b)dx = U 1i, where f 1i (z 1i b) is (v). Generate U 2i U(0,1), i = 1,2,...,k. (vi). For each U 2i, find the quantities Z 2i such that given in (3.3). Z2i f 2i (x b)dx = U 2i, where f 2i (z 2i b) is (vii). By using (2.1), compute R µi and R σi. (viii). By using (2.2), compute W i and R(X,x,θ). (ix). Repeat from the step (iii) to (viii), M times (with M large), and set R l (X,x,θ), the value of R(X,x,θ) obtained at the l th replicate, l = 1,2,...,M. 8

9 (x). Find R θ1,α/2(x) and R θ1,1 α/2(x) as respectively 100α/2 and 100(1 α/2) percentiles of R 1 (X,x,θ),R 2 (X,x,θ),...,R M (X,x,θ). (xi) Using (1.4), estimate the GPV p by ˆp = M 1 M I {Rl (X,x,θ) θ 0 }, where I A denotes the l=1 indicator function of the event A. It is noticed that for the normal sample case, the proposed algorithm corresponds to that in Krishnamoorthy and Lu (2003). Indeed, at normal case, the pdfs (3.2) and (3.3) correspond respectively to the pdfs of Student t and Chi-square distribution with n 1 degrees of freedom. 4 Some illustrative k-sample location-scale families In this section, we highlight the application of Proposition 2.1 along with the formulas (1.3), (1.4) and (3.4). In particular, we apply the proposed method to the logistic scale-location families. Also, we apply the method to the case where the MLEs of the location and scale parameters do not exist. First, we consider the k-sample location-scale normal family in order to highlight the fact that the above algorithm generalizes that in Krishnamoorthy and Lu (2003). 4.1 k-sample from normal distributions Assume that for i = 1,...,k, each X i1,x i2,...,x ini are iid with the pdf given by [ f Xi j (x i j ) = σi 1 (2π) 1/2 exp ( 2σi 2 ) 1 (xi j µ) 2], (4.1) where i = 1,...,k, j = 1,...,n i and µ, σ 1,...,σ k are unknown parameters. Under the model in (4.1), we apply the method to construct the GPQ for the k-sample normal family. We also illustrate the computation of GCI and GPV, based on the constructed GPQ. To this end, let b i j = ( X i j ˆµ i ) / ˆσi, j = 1,2,...,n i 2, i = 1,2,...,k, let b i = 1 n i n i 9 b i j,

10 and let S 2 i = n i (b i j b i ) 2, and s i be the observed value of S i, i = 1,...,k. If X i j N (µ,σ i ), j = 1,...,n i, by using the relation in(1.5) (or see Theorem A.3 in the Appendix), one can verify that based on the i th sample, the MRE for µ is X i, which is the same as MLE. Also, the MRE estimator of σ i is given by ˆσ i (X) = Γ(n i /2)/Γ((n i + 1)/2) 2 1 n i (X i j X i ) 2, and then, b i = 0 and S 2 i = n i b 2 i j = ( 2Γ 2 ((n i + 1)/2) )/ Γ 2 (n i /2). Further, the GPQ R µi as given in (2.1) becomes R µi = ˆµ si ˆσ si (s i / ( ( ) ni ˆµi µ / ) n i (n i 1)) (n i 1) S i ˆσ i = ˆµ si ˆσ si (s i (T ni 1)/ ) n i (n i 1), (4.2) and the GPQ R σi as given in (2.1) can be rewritten as R σi = ˆσ si ( ˆσ i /σ i ) 1 = s i ˆσ si / Xn 2 i 1, i = 1,...,k. (4.3) Therefore, if the data are from normal distributions, the GPQ for µ is given in (2.2), with R µi and R σi given by (4.2) and (4.3), respectively. Thus, from (4.2) and (4.3), we conclude that, for k-sample normal distributions, the proposed method is equivalent to that in Krishnamoorthy and Lu (2003). Below, we consider some other examples which illustrate that the proposed approach is applicable to other members of scale-location families. 4.2 k-sample from Logistic distributions Here we apply the proposed method to k-sample logistic families. In this case, we directly apply the methods described above with the pdf g i, i = 1,...,k set as the standard logistic pdf. That is, the pdfs of Z 1i and Z 2i have the form in (3.2) and (3.3) with g i (x) = exp( x)/[1 + exp( x)] 2, i = 1,...,k. 10

11 4.3 k-sample Location-scale families case where MLE does not exist As mentioned above, the proposed method is also applicable to the case where MLEs do not exist. In order to illustrate this last point, we consider the following example that is based on the result due to Pitman (1979). Let the scale-location families σ 1 i g i ((x µ)/σ i ), i = 1,2,...,k, where g i (x) = ( 2(1 + x )(1 + log(1 + x )) 2) 1, < x <, i = 1,...k. (4.4) As proved in Pitman (1979), MLEs for σ i, µ, i = 1,...,k do not exist. Another illustrative example corresponds to the scale-location family studied in Gupta and Székely (1994). We consider σi 1 g i ((x µ i )/σ i ), where g i (x) = c i (x i log 2 x i ) 1, 0 < x i l i < 1, i = 1,2,...,k with k a constant 0 < l i < 1 and c i = 1/log(l i ) is a constant. Gupta and Székely (1994) proved that for such families, MLEs for the location and scale parameters do not exist. 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 set k = 3 and generate samples from the related distributions of interest. Namely, the simulated coverage probabilities of the 95% GCI are presented in Tables 1-3 and, at significance level α =.05, the simulated powers of the proposed test are given in Tables 4-6. From Tables 1-3, the empirical confidence level of the proposed GCI for µ, is close to the nominal confidence level of 95%. Further, it is noticed that, as the sample size increases, the coverage probability get close to the nominal confidence level (95%). Also, we study the 11

12 Table 1: The coverage probabilities (CPR) of the 95% GCI (Normal samples) (n 1,n 2,n 3 ) (µ, σ 1, σ 2, σ 3 ) CPR (µ,σ 1, σ 2, σ 3 ) CPR (5, 5, 5) (2, 2, 2, 2) (2, 2, 4, 6) (10, 10, 10) (2, 2, 2, 2) (2, 2, 4, 6) (20, 20, 20) (2, 2, 2, 2) (2, 2, 4, 6) (5, 10, 20) (2, 2, 4, 6) (2, 4, 6, 2) (5, 5, 5) (2,0.5,100,500) (2, 2, 100, 200) (20, 20, 20) (2,0.5,100,500) (2, 2, 100, 200) (5,10,20) (2,0.5,100,500) (2,500,100,0.5) Table 2: The coverage probabilities (CPR) of the 95% GCI (Logistic samples) (µ,σ 1,σ 2,σ 3 ) (n 1,n 2,n 3 ) CPR (µ,σ 1,σ 2,σ 3 ) (n 1,n 2,n 3 ) CPR (2, 2, 2, 2) (5, 5, 5) (2, 2, 4, 6) (5, 5, 5) (2, 2, 2, 2) (10, 10, 10) (2, 2, 4, 6) (10, 10, 10) (2, 2, 2, 2) (20, 20, 20) (2, 2, 4, 6) (20, 20, 20) performance of the solution to the testing problem (1.2), for the case where k = 3 and µ 0 = 2. Tables 4-6 show that the power function varies with different values of µ, n i and σ i, i = 1,2,3. More specifically, from the above tables, it can be seen that when µ = µ 0 = 2, the powers are all approximately equal to But as the exact value of µ decreases, the power continually increases to 1, when the distance between µ and µ 0 increases. Also, the power decreases to 0 as the exact value of µ increases. Furthermore, when the exact value of µ is less than µ 0, the power increases as the sample size increases, for each value of µ. Figure 5.1 confirms the monotonicity of the power as well as the consistency of the proposed test. 12

13 Table 3: The coverage probabilities (CPR) of the 95% GCI for µ (family in (4.4)) (µ,σ 1,σ 2,σ 3 ) (n 1,n 2,n 3 ) CPR (µ,σ 1,σ 2,σ 3 ) (n 1,n 2,n 3 ) CPR (2, 2, 2, 2) (5, 5, 5) (2, 2, 4, 6) (5, 5, 5) (2, 2, 2, 2) (10, 10, 10) (2, 2, 4, 6) (10, 10, 10) (2, 2, 2, 2) (20, 20, 20) (2, 2, 4, 6) (20, 20, 20) Table 4: The power function of µ versus sample size (Normal family) (n 1, n 2, n 3 ) (µ, σ 1, σ 2, σ 3 ) Power (µ,σ 1, σ 2, σ 3 ) Power (0, 2, 2, 2) (0, 2, 4, 6) (5, 5, 5) (1, 2, 2, 2) (1, 2, 4, 6) (2, 2, 2, 2) (2, 2, 4, 6) (3, 2, 2, 2) (3, 2, 4, 6) (0, 2, 2, 2) 1 (0, 2, 4, 6) (20, 20, 20) (1, 2, 2, 2) (1, 2, 4, 6) (2, 2, 2, 2) (2, 2, 4, 6) (3, 2, 2, 2) 0 (3, 2, 4, 6) 0 (0, 2, 4, 6) (0, 4, 6, 2) (5, 10, 20) (1, 2, 4, 6) (1, 4, 6, 2) (2, 2, 4, 6) (2, 4, 6, 2) (3, 2, 4, 6) (3, 4, 6, 2) 0 13

14 Table 5: The power function of µ versus sample size (Logistic family) (n 1, n 2, n 3 ) (µ, σ 1, σ 2, σ 3 ) Power (n 1, n 2, n 3 ) (µ,σ 1, σ 2, σ 3 ) Power (0, 2, 4, 6) (0, 2, 4, 6) (5, 5, 5) (1, 2, 4, 6) (20,20,20) (1, 2, 4, 6) (2, 2, 4, 6) (2, 2, 4, 6) (3, 2, 4, 6) (3, 2, 4, 6) 0 Table 6: The simulated powers for µ (the location-scale family in (4.4)) (n 1, n 2, n 3 ) (µ, σ 1, σ 2, σ 3 ) Power (n 1, n 2, n 3 ) (µ,σ 1, σ 2, σ 3 ) Power (0, 2, 4, 6) (0, 2, 4, 6) (5, 5, 5) (1, 2, 4, 6) (20,20,20) (1, 2, 4, 6) (2, 2, 4, 6) (2, 2, 4, 6) (3, 2, 4, 6) (3, 2, 4, 6)

15 Power size: (5,5,5) size: (20,20,20) Power size: (5,5,5) size: (20,20,20) size: (5,10,20) Exact Value Exact Value (a) (b) Power in Normal family (σ 1 = 2, σ 2 = 2, σ 3 = 2) Power in Normal family (σ 1 = 2, σ 2 = 4, σ 3 = 6) Power size: (5,5,5) size: (20,20,20) size: (5,10,20) Power size: (5,5,5) size: (20,20,20) Exact Value Exact Value (c) (d) Logistic family Power in Normal family (σ 1 =.5, σ 2 = 100, σ 3 = 500) Power size: (5,5,5) size: (20,20,20) Exact Value (e) family case in (4.4) 15

16 5.2 Illustrative examples and data analysis In this subsection, we illustrate the application of the proposed method with a data set that correspond to the case where k = 2. In this example, we consider that the samples are taken from normal populations Normal Body Temperature data set This data set is presented in Mackowiak et al. (1992). In this data set, a total number of 130 patients have been assigned, with 65 males and 65 females, and their body temperatures have been 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 1 N (µ 1,σ 1 ) and for the female group, one can consider X 2 N (µ 2,σ 2 ). In addition, it is reasonable to assume that the average body temperature for male and female are equal. This can be confirmed by the fact that 95% GCI for µ 1 µ 2 contains 0. Then, by applying the proposed method, the 95% GCI of common parameter µ is ± for which the value 98.6 is excluded. This contrast with the fact that, for many years, the value of 98.6 has been considered as the normal average body temperature (see Mackowiak et al., 1992 and references therein). However, in the quoted paper, the authors concluded that this value is erroneous. Thus, the obtained GCI corroborates this finding. Also, we consider to test H 0 : µ 98.6 versus H 1 : µ < 98.6 at 5% of significance level. By applying the proposed method, the GPV is 0. Thus, since the GPV is smaller than 0.05, we fail to reject the null hypothesis at 0.05 level of significance. Further, Mackowiak et al. (1992) concluded that the average normal body temperature is 98.2 o F. Accordingly, we consider to test H 0 : µ 98.2 versus H 1 : µ < The proposed GPV is which is greater than 0.05, and hence, we fail to reject H 0 at 0.05 level of 16

17 significance. 6 Conclusion In this paper, we studied an inference problem concerning the common location parameter of several location-scale families where the scale parameters are unknown and possibly unequal. In solving this problem, we presented a general approach for establishing the GPQ and GTV for the common location parameter. In particular, the proposed GPQ and GTV are functions of the MREs which are known to be more general and more efficient than the MLEs. Also, we carried out intensive simulation studies which showed that the proposed approach gives confidence intervals with high coverage probability. Further, the resulting tests have high power, and preserve the significance level. In order to illustrate the application of the proposed method, we analysed the normal body temperature. In particular, our findings corroborate that in Mackowiak et al. (1992) for which the average body temperature of 98.6 o F is erroneous although this value has been used as a standard for many years. In contrast, the value of 98.2 o F, given in Mackowiak et al. (1992), as the average normal body temperature seems to be reliable. Finally, it is noticed that the proposed approach is applicable to all members of the locationscale families, as opposed to the method in Krishnamoorthy and Lu (2003) which is designed only for the normal case. 17

18 A Appendix A.1 Minimum risk equivariant estimator of scale-location parameters In this subsection, we present some results which are useful in deriving MRE. Since we consider the quadratic loss function, the MRE corresponds to Pitman estimator (Pitman, 1939). Theorem A.1 Let X 1,...,X n be the iid random sample from location family with pdf f (x λ) = g(x λ), where λ unknown. Also, consider the loss function L(λ,a) = (λ a) 2 and suppose that there exists equivariant estimator δ 0 with finite risk. Then, the MRE of λ is ( n 1 n ˆλ p (x) = t g(x i t)dt g(x i t)dt). (A.1) The proof follows directly from Theorem 1.20 in Lehmann and Casella (1998, p. 154), and Theorem 6.10 in Schervish (1997, p. 348). Theorem A.2 Let X 1,...,X n be iid random sample from scale family with pdf f (x τ) = τ 1 g(x/τ), where τ is unknown. Also, let the loss function L(τ,a) = (a τ) 2 /τ 2 and suppose that there exists equivariant estimator δ 0 with finite risk. Then, the MRE of τ is ( ˆτ p (x) = t n n 1 g(tx i )dt t n+1 n g(tx i )dt). (A.2) 0 0 For a proof, we refer to Lehmann and Casella (1998, p. 170), and Schervish (1997, p. 352). Theorem A.3 Let X 1,X 2,...,X n be iid random sample from scale-location family with pdf f (x λ,τ) = τ 1 g((x λ)/τ), where λ and τ are unknown. Also, suppose that there exists equivariant estimator δ 0 with finite risk. Then, under quadratic loss function the MRE of λ and τ are respectively ˆλ(x) = ˆτ(x) = 0 uv n 3 n g((x i u)/v)dudv v n 2 n 0 / ( / ( g((x i u)/v)dudv ) v n 3 n g((x i u)/v)dudv, ) v n 3 n g((x i u)/v)dudv.

19 Proof The result is given in Lehmann and Casella (1998, chap. 3). However, for our paper to be self-contained, the main steps of a proof are outlined here. Let δ 0 (X) and δ 1 (X) be equivariant estimators of µ and σ respectively. By Theorem 3.17 in Lehmann and Casella (1998, p. 174), the MRE for µ is ˆµ(x) = δ 0 (x) w(z)δ 1 (x), (A.3) where by relation (3.44) in Lehmann and Casella (1998, p. 175) / w(z) = E[δ 0 (X)δ 1 (X) Z] E [ δ1 2 (X) Z ], (A.4) with Z = (Z 1,Z 2,...,Z n 1 ), where Z i = (X i X n )/(X n 1 X n ), i = 1,2,...,n 2, Z n 1 = (X n 1 X n )/ X n 1 X n. Further, as equivariant estimators of µ and σ, we choose δ 0 (X) = X n and δ 1 (X) = δ 2 with δ 2 = X n 1 X n respectively. One can verify that the transformation from x to (z,δ 0,δ 2 ) has Jacobian δ 2 n 2, and then, for µ = 0, σ = 1, the joint pdf of (Z,δ 0,δ 2 ) is given by f Z,δ0,δ 2 (z,δ 0,δ 2 ) = δ 2 n 2 g( δ 2 + δ 0 ) g(δ 0 ) n 2 g(z i δ 2 + δ 0 ), both for z n 1 = 1 and z n 1 = 1. Then, by some algebraic computations, we get w(z) = 0 x 1 x n v n 1 ug(u) 0 v n g(u) n 1 n 1 g((x i x n )v + u) dudv g((x i x n )v + u) dudv Further, by the change of variables v = 1/s and u = x n v tv, we get w(z) = x n x 1 x n s n 3 t n 0 s n 3 0 g((x i t)/s) dt ds n g((x i t)/s) dt ds Therefore, combining (A.3) and (A.5), we get the first statement of the theorem... (A.5) 19

20 To prove the second statement, note that ˆσ p (X) is the MRE for σ if and only if it is a function of the differences Y i = X i X n, i = 1,2,...,n 1 (see Lehmann and Casella, 1998, p ). Further, the joint pdf of (Y 1,Y 2,...,Y n 1 ) is σ n n 1 g(t/σ) g((y i +t)/σ) dt = σ n+1 n 1 g(u) g((y i /σ) + u) du, and this a joint density of n 1 observations from scale family with the scale parameter σ. Therefore, it suffices to apply Theorem A.2 by replacing t n and n 1 g(u) n g(tx i ) by t n 1 and g(ty i + u) du respectively. Further, we replace y i by x i x n, i = 1,2,...,n 1, and then the desired result follows from the transformation t = 1/s and u = x n s vs, that completes the proof. A.2 Distributions of pivotal quantities Let ˆµ i be the equivariant estimator of µ based on the i th sample, i = 1,...,k. Also, let a i j = X i j ˆµ i, j = 1,...,n i ; i = 1,...,k (A.6) where the k samples (X i1,x 2,...,X ini ), i = 1,...,k are independent from location families with common location parameter µ. Also, let a i = (, a i1,a i2,...,a i(ni 1)) i = 1,2,...,k. Similarly, for the case of scale families, a i j are replaced by c i j = X i j / ˆσ i, j = 1,2,...,n i, i = 1,2,...,k, respectively. Proposition A.1 Assume k random samples are from k independent location families and assume that relation (A.6) holds. Then a = ( a ( 1 k),...,a is ancillary statistic. Furthermore, k the joint pdf of a is f (a n i k ) =... g i (a i j + z i )) dz i. Proof Let e m denote a m-column vector with all entrees equal to 1. We have X i ˆµ i e ni = (X i µ i e ni ) ( ˆµ i µ i )e ni, i = 1,2,...,k. Further, let δ(x i ) = ˆµ i. Since δ(x i ) is equivariant 20

21 estimator, we have ˆµ i µ i = δ(x i ) µ i = δ (X i µ i e ni ). Therefore, since the distribution of X i µ i e ni = ( X i j µ i ), j = 1,2,...,ni does not depend on parameter, we conclude that the distributions of (X i ˆµ i e ni ), i = 1,2,...,k do not depend on parameter, and this proves that a is ancillary statistic. To establish the joint pdf of a, we assume without loss of generality, that σ 1 = σ 2 = = σ k = 1. Also, let us define a ini by X ni = a ini + ˆµ i, i = 1,2,...,k. Then, since ˆµ i, i = 1,2,...,k are equivariant, a ini can be expressed as a function of a i1,...,a i(ni 1), for each i = 1,2,...,k and thus, one can set a ini = T i (a i1,...,a i(ni 1)). Then x i j = a i j + ˆµ i, j = 1,...,n i 1; x ni = a ni + ˆµ i, i = 1,...,k. Let X = ( X 1,...,X k) with Xi = (X i1,x i2,...,x ini ), i = 1,2,...,k, and let x = ( x 1,...,x k) with xi = (x i1,x i2,...,x ini ), i = 1,2,...,k. We have f (x) = k n i g i ( xi j µ i ). Also, let ˆµ = ( ˆµ1, ˆµ 2,..., ˆµ k ). The transformation from x to (a, ˆµ ) has matrix Jacobian J J J = J k x i1 a... i1 with J i =.... x ini a... i1 x i1 a i(ni 1). x ini a i(ni 1) x i1 ˆµ i. x ini ˆµ i = I n i 1 e ni 1 0 1, i = 1,2,...,k; where I m stands for the identity matrix of size m, e m is a m-column vector with all entrees equal to 1. Then, since J = k J i = 1, the joint pdf of (a, ˆµ ) is f ( a, ˆµ ) = k n i g i ( ai j + ˆµ i µ i ). (A.7) Therefore, combining (A.7) and the change of variables z i = ˆµ i µ i, i = 1,2,...,k, we get f ( ( a ) k n i k =... g i (a i j + z i )) dz i, that completes the proof. Proposition A.2 Assume that the k samples are taken from the pdfs in (1.1) with σ i = 1. Then, 21

22 conditionally to a, the joint pdf of ˆµ µ is f ( x a ) = k n i g i ( ai j + x i ) /( ( k... n i k g i (a i j + z i )) dz i ), x R k. Proof From (A.7), the joint pdf of (a, ˆµ µ ) is f (a,x ) = x = (x 1,x 2,...,x k ) R k, that completes the proof. k n i g i ( ai j + x i ), In the similar way, we establish the following proposition that gives the corresponding result for the general case where the k samples are from scale-location family. To this end, let Z 1 = (Z 11,Z 12,...,Z 1k ), Z 2 = (Z 21,Z 22,...,Z 2k ) with Z 1i = ( ˆµ i µ i )/ ˆσ i, Z 2i = ˆσ i /σ i, i = 1,2,...,k. Also, let b = ( b 1,b 2,...,b k) with bi = ( b i1,b i2,...,b i(ni 2)), b i j = ( X i j ˆµ i ) / ˆσi, j = 1,2,...,n i, i = 1,2,...,k. (A.8) Proposition A.3 Assume that k random samples are from k independent scale-location families and assume that relation (A.8) holds. Then b = ( b 1,...,b k) is ancillary statistic. Further, conditionally to b, the joint pdf of (Z 1,Z 2 ) is with k ( b ) = Proof f ( ( x,y b ) k = y n i 1 i n i 0 g i ( (ai j + x i )y i ) ) /k ( b ), x R k, y R +k, ( k w n i 1 i n i k g i ((b i j + z i )w i )) dw i dz i. From the equivariance of ˆµ i, ˆσ i, i = 1,2...,k and by using similar arguments as in the proof of Proposition A.1, we prove that b is ancillary statistic. Further, let ˆµ = ( ˆµ 1, ˆµ 2,..., ˆµ k ) ˆσ = ( ˆσ 1, ˆσ 2,..., ˆσ k ). In the similar ways as in the proof of Proposition A.1, the joint pdf of (b, ˆµ, ˆσ ) is given by f (b, ˆµ, ˆσ k ( ) ) = b ni ˆσ n i 2 i /σ n n i ( ) i i g i ( ˆσbi j + ˆµ i µ i )/σ i. Further, the transformation from (b, ˆµ, ˆσ ) to (b,z 1,z 2 ) has Jacobian k i σ i 2z 2i, and then, the joint pdf of (b,z 1,Z 2 ) is f (b,z 1 k,z 2 ) = b ni z n n i ( i 1 2i g i z2i (b i j + z 1i ) ), that completes the proof. 22

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