Empirical phi-divergence test-statistics for the equality of means of two populations

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1 Epirical phi-divergence test-statistics for the equality of eans of two populations N. Balakrishnan, N. Martín and L. Pardo 3 Departent of Matheatics and Statistics, McMaster University, Hailton, Canada arxiv: v [stat.me] 4 Jan 06 Departent of Statistics and O.R. II Decision Methods), Coplutense University of Madrid, 8003 Madrid, Spain 3 Departent of Statistics and O.R., Coplutense University of Madrid, 8040 Madrid, Spain June 3, 08 Abstract Epirical phi-divergence test-statistics have deostrated to be a useful technique for the siple null hypothesis to iprove the finite saple behaviour of the classical likelihood ratio test-statistic, as well as for odel isspecification probles, in both cases for the one population proble. This paper introduces this ethodology for two saple probles. A siulation study illustrates situations in which the new teststatistics becoe a copetitive tool with respect to the classical z-test and the likelihood ratio test-statistic. AMS 00 Subject Classification: 6F03, 6F5. Keywords and phrases: Epirical likelihood, Epirical phi-divergence test statistics, Phi-divergence easures, Power function. Introduction The ethod of likelihood introduced by Fisher is certainly one of the ost coonly used techniques for paraetric odels. The likelihood has been also shown to be very useful in non-paraetric context. More concretely Owen 988, 990, 99) introduced the epirical likelihood ratio statistics for non-paraetric probles. Two saple probles are frequently encountered in any areas of statistics, generally perfored under the assuption of norality. The ost coonly used test in this connection is the two saple t- test for the equality of eans, perfored under the assuption of equality of variances. If the variances are unknown, we have the so-called Behrens-Fisher proble. It is well-known that the two saple t-test has cone ajor drawback; it is highly sensitive to deviations fro the ideal conditions, and ay perfor iserably under odel isspecification and the presence of outliers. Recently Basu et al. 04) presented a new faily of test statistics to overcoe the proble of non-robustness of the t-statistic.

2 Epirical likelihood ethods for two-saple probles have been studied by different researchers since Owen 988) introduced the epirical likelihood as a non-paraetric likelihood-based alternative approach to inference on the ean of a single population. The onograph of Owen 00) is an excellent overview of developents on epirical likelihood and considers a ulti-saple epirical likelihood theore, which includes the two-saple proble as a special case. Soe iportant contributions for the two-saple proble are given in Owen 99), Adiiri 995), Jin 995), Qin 994, 998), Qin and Zhao 000), Zhang 000), Liu et al. 008), Baklizi and Kibria 009), Wu and Yan 0) and references therein. Consider two independent unidiensional rando variables X with unknown ean µ and variance σ and Y with unknown ean µ and variance σ. Let X,...,X be a rando saple of size fro the population denoted by X, with distribution function F, and Y,...,Y n be a rando saple of size n fro the population denoted by Y, with distribution function G. We shall assue that F and G are unknown, therefore we are interested in a non-paraetric approach, ore concretely we shall use epirical likelihood ethods. If we denote µ = µ and µ = µ+δ, our interest will be in testing H 0 : δ = δ 0 vs. H : δ δ 0, ) being δ 0 a known real nuber. Since δ = µ µ becoes the paraeter of interest, apart fro testing ), we ight also be interested in constructing the confidence interval for δ. In this paper we are going to introduce a new faily of epirical test statistics for the two-saple proble introduced in ): Epirical phi-divergence test statistics. This faily of test statistics is based on phidivergence easures and it contains the epirical log-likelihood ratio test statistic as a particular case. In this sense, we can think that the faily of epirical phi-divergence test statistics presented and studied in this paper is a generalization of the epirical log-likelihood ratio statistic. Let N = +n, assue that and x,...,x, y,...,y n a realization of X,...,X, Y,...,Y n. We denote and Lδ) = n p i q j s.t. Lp,q) = N ν 0,), ),n p i x i µ) = 0, p i =, n p i q j s.t. n q j y j µ δ) = 0, p i = n q j =, p i,q j 0, n q j =, with p i = p i µ) = F x i ) F x ) i, qj = q j µ,δ) = Gy j ) Gyj ) and p = p,...,p ) T, q = q,...,q n ) T. The epirical log-likelihood ratio statistic for testing ) is given by lδ 0 ) = log sup p,qlδ 0 ) sup p,q Lp,q). 3) Using the standard Lagrange ultiplier ethod we ight obtain sup p,q Lδ 0 ), as well as sup p,q Lp,q). For sup p,q Lδ 0 ), taking derivatives on ) ) L = logp i + n logq j +s p i +s n q j λ p i x i µ) λ n n q j y j µ δ 0 ),

3 we obtain and L = 0 p i =, i =,...,, p i +λ x i µ) 4) L = 0 q j =, j =,...,n, q j n+λ y j µ δ 0 ) 5) L µ = 0 λ +nλ = 0. Therefore, the epirical axiu likelihood estiates λ, λ and µ of λ, λ and µ, under H 0, are obtained as the solution of the equations and +λ = x i µ) n n +λ = y j µ δ 0) λ +nλ = 0, 6) log suplδ 0 ) = log log + λ x i µ) ) nlogn n ) log + λ y j µ δ 0 ). 7) p,q In relation sup p,q Lp,q), taking derivatives on we have and ) ) L = logp i + n logq j +s p i +s n q j, 8) L = 0 p i = p i, i =,..., and L = 0 q j =, j =,...,n, 9) q j n log suplp,q) = log nlogn. 0) p,q Therefore, the epirical log-likelihood ratio statistic 3), for testing ), can be written as lδ 0 ) = log ) log + λ X i µ) nlogn n { = ) log + λ Y j µ δ 0 ) Under soe regularity conditions, Jing 995) established that +log+nlogn log + λ X i µ) )+ n log + λ Y j µ δ 0 )) }. ) ) Pr lδ 0 ) > χ,α) = α+on ), where χ,α is the 00 α)-th percentile of the χ distribution. Our interest in this paper is to study the proble of testing given in ) and at the sae tie to construct confidence intervals for δ on the basis of the epirical phi-divergence test statistics. Epirical phi-divergence test statistics in the context of the epirical likelihood have studied by Baggerly 998), Broniatowski and Keizou 3

4 0), Balakhrishnan et al. 03), Felipe et al. 05) and references therein. The faily of epirical phidivergence test statistics, considered in this paper, contains the classical epirical log-likelihood ratio statistic as a particular case. In Section, the epirical phi-divergence test statistics are introduced and the corresponding asyptotic distributions are obtained. A siulation study is carried out in Section 3. Section 4 is devoted to develop a nuerical exaple. In Section 5 the previous results, devoted to univariate populations, are extended to k-diensional populations. Epirical phi-divergence test statistics For the hypothesis testing considered in ), in this section the faily of epirical phi-divergence test statistics are introduced as a natural extension of the epirical log-likelihood ratio statistic given in 3). We consider the N-diensional probability vectors U = N,..., N N )T ) and P = p ν,...,p ν,q ν),...,q n ν)) T 3) where p i, i =,...,, q j, j =,...,n were defined in 4) and 5), respectively, and ν in ). Let P be the N-diensional vector obtained fro P with p i, q j replaced by the corresponding epirical axiu likelihood estiators p i, q j and ν by N. The Kullback-Leibler divergence between the probability vectors U and P is given by D Kullback U, P) = N log N + n p i N N log N ) q j N { } = log p+ n logn q j N { = log + λ x i µ) )+ n log + λ y j µ δ 0 )) }, N where p i =, i =,...,, + λ x i µ) 4) q j =, j =,...,n. n+ λ y j µ δ 0 ) 5) Therefore, the relationship between lδ 0 ) and D Kullback U, P) is lδ 0 ) = ND Kullback U, P). 6) Based on 6), in this paper the epirical phi-divergence test statistics for ) are introduced for the first tie. This faily of epirical phi-divergence test statistics is obtained replacing the Kullback-Leibler divergence by a phi-divergence easure in 6), i.e., T φ δ 0 ) = N φ ) D φu, P), 7) 4

5 where D φ U, P) = ) ) p i N φ N + n ) q j φ N p i N N, ) qj with φ : R + R being anyconvexfunction such that at x =, φ) = 0, φ ) > 0and at x = 0, 0φ0/0) = 0 φu) and 0φp/0) = pli u u. For ore details see Cressie and Pardo 00) and Pardo 006). Therefore, 7) can be rewritten as { T φ δ 0 ) = ) φ p i φ + n ) p i { = φ ) + λ x i µ) φ n q j φ n q j + λ x i µ) )+ n N ) } 8) + λ y j µ δ 0 ) φ + λ y j µ+δ 0 ))) }. If φx) = xlogx x+ is chosen in D φ U, P), we get the Kullback-Leibler divergence and T φ δ 0 ) coincides with the epirical log-likelihood ratio statistic lδ 0 ) given in 6). Let µσ,σ ) be the optial estiator of µ under the assuption of having the known values of σ, σ, i.e. it is given by the shape πx + π)y and has iniu variance. It is well-known that ) X Y δ0 µσ,σ ) = σ +n σ. 9) σ + n σ Siilarly, an asyptotically optial estiator of µ having unknown values of σ, σ, is given by ) X Y δ0 µs,s ) = S +n S, S + n S where S = X i X), S = n n Y j Y) are consistent estiators of σ, σ respectively. In the following lea an iportant relationship is established, useful to get the asyptotic distribution of T φ δ 0 ). Lea Let µ the epirical likelihood estiator of µ. Then, we have µ = µσ,σ )+O p) = µs,s )+O p). Proof. See Appendix 5.3. Theore Suppose that σ <, σ < and ). Then, Proof. See Appendix 5.4. T φ δ 0 ) L n, χ. Reark 3 A α)-level confidence interval on δ can be constructed as CI α δ) = { δ : T φ δ) χ,α}. 5

6 The lower and upper bounds of the interval CI α δ) require a bisection search algorith. This is a coputationally challenging task, because for every selected grid point on δ, one needs to axiize the epirical phi-divergence T φ δ) over the nuisance paraeter, µ, and there is no closed-for solution to the axiu point µ for any given δ. The coputational difficulties under the standard two-saple epirical likelihood forulation are due to the fact that the involved Lagrange ultipliers, which are deterined through the set of equations 6), have to be coputed based on two separate saples with an added nuisance paraeter µ. Such difficulties can be avoided through an alternative forulation of the epirical likelihood function, for which coputation procedures are virtually identical to those for one-saple of size N = + n epirical likelihood probles. Through the transforations v i = w j = ω, x i ω, y j ω δ ω = ω =, 4) and 5) can be alternatively obtained as ) T δ, i =,...,, ω ) T, j =,...,n, p i = + λ T, i =,...,, v i 0) q j = n + λ T, j =,...,n, w j ) where the estiates of the Lagrange ultipliers λ = λ,, λ, ) T are the solution in λ of v i +λ T + n v i w j +λ T w j = 0. Reark 4 In the particular case that = n, the two saples ight be understood as a rando saple of size n fro a unique bidiensional population. In this setting the two saple proble can be considered to be a particular case of Balakrishnan et al. 05). Reark 5 Fu et al. 009), Yan 00) and Wu and Yan 0) pointed out that epirical log-likelihood ratio statistic, lδ 0 ), given in ) for testing ), does not perfor well when the distribution associated to the saples are quite skewed or saples sizes are not large or saple sizes fro each population are quite different. To overcoe this proble Fu et al. 009) considered the weighted epirical log-likelihood function defined by l w p,q) = ω logp i + ω n n logq j, ) with ω = ω =, and obtained the weighted epirical likelihood WEL) estiator as well as the weighted epirical log-likelihood ratio statistic. In order to get the WEL estiator, it is necessary to axiize ) subject to p i = n q j =, 3) p i x i n y j q j = δ 0. 4) 6

7 They obtained that the WEL estiates of p i and q j are given by w p i = + w λ T, i =,...,, v i w q j = n + w λ T, j =,...,n, w j where v i and w j are the sae transforations given in Reark 3 with δ = δ 0 and the estiates of the Lagrange ultipliers w λ = w λ,, w λ, ) T are the solution in w λ of ω v i + w λ T v i + ω n n w j + w λ T w j = 0. Now, if we define the probability vectors w U = ω,..., ),ω n,, n...,, n ) ) T, w P = ω p T,ω q T) T = ω p,...,p ),ω q,...,q n )) T, the weighted epirical log-likelihood ratio test l w δ 0 ) presented in Wu and Yan 0) can be written as l w δ 0 ) = D Kullback w U, w P). 5) The weighted epirical log-likelihood ratio test can be extended by defining the faily of weighted epirical phi-divergence test statistics as S φ δ 0 ) = D φ w U, w P) φ, ) where D φ w U, w P) is the phi-divergence easure between the probability vectors w U and w P, i.e., ) ) n D φ w U, w P) = ω p i φ +ω q j φ p i n q j = ω φ + w λ T ) v i + w λ T + ω n φ + w λ T ) w j n v i + w λ T. w j Taking into account D φ w U, w P) = w λt D w λ +o p N), where D = ω v i v T i + ω n w j w T j n, and based on Theore.. in Wu and Yan 0), we have that S φ δ 0 ) c where c is the second diagonal eleent of the atrix D. L n, χ, 7

8 3 Siulation Study The square of the classical z-test statistic for two saple probles, ) X Y +δ0 tδ 0 ) =, S + n S hasasyptoticallyχ distribution,thesaeastheepiricalphi-divergenceteststatistics,accordingtotheore. In order to copare the finite saple perforance of the confidence interval CI) of δ based on T φ δ) with respect to the ones based on tδ) as well as the epirical log-likelihood ratio test-statistic lδ) given in 3), we count on a subfaily of phi-divergence easures, the so-called power divergence easures φ γ x) = x +γ x γx ) γ+γ), dependent of tuning paraeter γ R, i.e. n p i ) γ + n q j ) γ N γγ +) T γ δ) = log+nlogn+ log p i +n, γ R {0, }, n log q j, γ = 0, n log+nlogn+ p i log p i +n q j log q j, γ =, where p i and q j can be obtained fro 0)-). We analyzed five new test-statistics, the epirical powerdivergence test statistics taking γ {, 0.5, 3,,}. The case of γ = 0 is not new, since the epirical log-likelihood ratio test-statistic lδ) is a eber of the epirical power-divergence test statistics, i.e. lδ) = T γ=0 δ). The CI of δ based on tδ 0 ) with 00 α)% confidence level is essentially the CI of z-test statistic, δ L, δ U ) = x y z α s + n s,x y+z α s + n s ). For T {T γ δ)} γ Λ, Λ = {, 0.5,0, 3,,}, as entioned in Reark 3, since there is no explicit expression for δ L, δ U ) the bisection ethod should be followed. The siulated coverage probabilities of the CI of δ based on T {tδ)} {T γ δ)} γ Λ were obtained with R = 5,000 replications by 00 R R IT r) χ,α), r= with I ) being the indicator function. The siulated expected width of the CI of δ based on T {tδ)} {T γ δ)} γ Λ were obtained with R = 3,000 replications by 00 R R δ r) U r= δ r) L ). The reason why two different values of R were followed is twofold. On one hand calculating δ r) U δ r) L is uch ore tie consuing than IT r) χ,α) and on the other hand for the designed siulation experient the replications needed to obtain a good precision is less for the expected width than for the coverage probability. The siulation experient is designed in a siilar anner as in Wu and Yan 0). The true distributions, unknown in practice, are generated fro: i) X Nµ,σ ), Y Nµ+δ 0,σ ), with µ =, σ = σ =.5, δ 0 = 0; 8

9 ii) X lognoralϑ,θ ), Y lognoralϑ,θ ), with ϑ =., θ = 0.4, ϑ =., θ = 0.. Notice that in case ii) δ 0 = 0 since E[X] = E[Y]. Depending on the saple sizes, six scenarios were considered,,n) {5,30),30,5),30,30),30,60),60,30),60,60)}. Table suarizes the results of the described siulation experient with α = In all the cases and scenarios the narrower width is obtained with T γ= δ), but the coverage probabilities closest to 95% depends on the case or scenario. For the case of the lognoral distribution the CI based on tδ) test-statistic has the closest coverage probability to 95%, but for the case of the noral distribution T γ=/3 δ) and T γ= δ) power divergence based tend to have the closest coverage probability to 95%. In order to copleent this study, the power functions have been drawn through R = 5, 000 replications and taking δ as abscissa. For case i) the power functions exhibit a syetric shape with respect to the center and also a parallel shape, in such a way that the test statistics with better approxiation of the size have worse power. For case ii), fixing the values of the two paraeters of X and changing the two paraeter of Y as ϑ = kϑ, θ = kθ, k = logδ +exp{ϑ + θ }) ϑ + θ, δ is displaced fro δ 0 = 0 to the right when k > and fro δ 0 = 0 to the left when 0 < k < δ > exp{ϑ + θ }). Unlike case i), the power function of case ii) exhibits a different shape on both sides fro the center of abscissa, and the ost proinent differences are on the left hand size. Clearly in case ii), even though the approxiated size for tδ 0 ) is the best one, it has the worst approxiated power function, in particular there is an area of the approxiated power function on the left hand side of δ 0 = 0 with saller value than the approxiated size. Hence, in case ii) the power functions of T {T γ δ)} γ Λ are ore acceptable than the power function of tδ 0 ). Taking into account the strong and weak point of tδ 0 ) in case ii), T γ=/3 δ 0 ) could be a good choice for oderate saple sizes and lδ 0 ) = T γ=0 δ 0 ) for sall saple sizes. 9

10 case i): noral populations n CI coverage width 5 30 T γ= δ) T γ= 0.5 δ) T γ=0 δ) T γ=/3 δ) T γ= δ) T γ= δ) tδ) T γ= δ) T γ= 0.5 δ) T γ=0 δ) T γ=/3 δ) T γ= δ) T γ= δ) tδ) T γ= δ) T γ= 0.5 δ) T γ=0 δ) T γ=/3 δ) T γ= δ) T γ= δ) tδ) T γ= δ) T γ= 0.5 δ) T γ=0 δ) T γ=/3 δ) T γ= δ) T γ= δ) tδ) T γ= δ) T γ= 0.5 δ) T γ=0 δ) T γ=/3 δ) T γ= δ) T γ= δ) tδ) T γ= δ) T γ= 0.5 δ) T γ=0 δ) T γ=/3 δ) T γ= δ) T γ= δ) tδ) case ii): lognoral populations n CI coverage width 5 30 T γ= δ) T γ= 0.5 δ) T γ=0 δ) T γ=/3 δ) T γ= δ) T γ= δ) tδ) T γ= δ) T γ= 0.5 δ) T γ=0 δ) T γ=/3 δ) T γ= δ) T γ= δ) tδ) T γ= δ) T γ= 0.5 δ) T γ=0 δ) T γ=/3 δ) T γ= δ) T γ= δ) tδ) T γ= δ) T γ= 0.5 δ) T γ=0 δ) T γ=/3 δ) T γ= δ) T γ= δ) tδ) T γ= δ) T γ= 0.5 δ) T γ=0 δ) T γ=/3 δ) T γ= δ) T γ= δ) tδ) T γ= δ) T γ= 0.5 δ) T γ=0 δ) T γ=/3 δ) T γ= δ) T γ= δ) tδ) Table : Siulated coverage probability and expected width of 0.95 level CIs of δ for two pupulations. 0

11 0.4 Power function: case noral, =5, n= β d λ=-0.5 λ=0 λ=/3 λ= z-test s.l. 0.4 Power function: case noral, =60, n= β d λ=-0.5 λ=0 λ=/3 λ= z-test s.l. Figure : Power functions for two noral pupulations.

12 0.4 Power function: case log-noral, =5, n= β d λ=-0.5 λ=0 λ=/3 λ= z-test s.l. 0.4 Power function: case log-noral, =60, n= β d λ=-0.5 λ=0 λ=/3 λ= z-test s.l. Figure : Power functions for two log-noral pupulations.

13 4 Nuerical Exaple Yu et al. 00) presented a data set on evaluating gasoline quality based on what is known as Reid vapor pressure, collected by the Environental Protection Agency of the United States. Two types of Reid vapor pressure easureents X and Y are included in the data set. Values of X are obtained by an Agency inspector who visits gas pups in a city, takes saples of gasoline of a particular brand, and easures the Reid vapor pressure right on the spot; values of Y, on the other hand, are produced by shipping gasoline saples to the laboratory for easureents of presuably higher precision at a high cost. The original data set has a double sapling structure, with a subset of the saple units having easureents on both X and Y. Table contains two independent saples of a new reforulated gasoline, one related to X with saple size 30 and the other, to Y with saple size 5. X Field) Y Lab) Table : Field and lab data on Reid vapor pressure for newly reforulated gasoline. One of the assuptions of Yu et al. 00) is that the field easureent X and the lab easureent Y have coon ean µ. The two types of easureents differ, however, in ters of precision. Yu et al. 00) also assued that X, Y) was bivariate noral, which would not be required under our proposed epirical likelihood approach. In Tsao and Wu 006) this exaple was studied on the basis of the epirical log-likelihood ratio test. The 95% CIs of δ based on T {tδ)} {T γ δ)} γ Λ are suarized in Table 3. As in the siulation study, the narrowest CI width is obtained with T γ= δ). In all the test-statistics used to construct the CIs δ 0 = 0 is not contained, so the null hypothesis of equal eans is rejected with 0.05 significance level. CI lower bound upper bound width T γ= δ) T γ= 0.5 δ) T γ=0 δ) T γ=/3 δ) T γ= δ) T γ= δ) tδ) Table 3: Power divergence and z-test based 0.95 level CIs for field and lab data on Reid vapor pressure for newly reforulated gasoline. 3

14 5 Further extensions 5. Extension of the diension for the rando variable Let X,...,X and Y,...,Y n be two utually independent rando saples with coon distribution function F and G respectively. Assuing that X i and Y j take values in R k and E[X i ] = µ, Cov[X i ] = Σ, i =,...,, E[Y j ] = µ, Cov[Y j ] = Σ, j =,...,n, with µ = µ and µ = µ+δ, our interest is in testing H 0 : δ = δ 0 vs. H : δ δ 0, 6) where δ 0 R k and known. The epirical likelihood under H 0 is Lδ) = n p i q j s.t. p i x i µ) = 0 k, p i =, n q j y j µ δ) = 0 k, n q j =, and in the whole paraeter space, Lp,q) = n p i q j : p i = n q j =, p i,q j 0, with p i = F x i ) F x ) i, qj = Gy j ) Gy j ) and p = p,...,p ) T, q = q,...,q n ) T. The epirical log-likelihood ratio statistic for testing 6) is given by lδ 0 ) = log sup p,qlδ 0 ) sup p,q Lp,q). Based on Lagrange ultiplier ethods, sup p,q Lδ 0 ) is obtained for p i = +λ T x, i =,...,, 7) i µ) q j = n+λ T, j =,...,n, 8) y j µ δ 0 ) where λ T +nλt = 0. The epirical axiu likelihood estiates λ, λ and µ of λ, λ and µ, under H 0, can be obtained as the solution of x i µ) +λ T xi µ) = 0 k n y j µ δ) +λ T y j µ δ) = 0 k n On the other hand sup p,q Lp,q) is obtained for λ T +nλ T = 0 k. p i =, i =,..., and q j =, j =,...,n. 9) n 4

15 After soe algebra, we obtain lδ 0 ) = log suplδ 0) suplp,q) = n log + λ T ) Y j µ δ 0 ) = Under soe regularity conditions, it follows that log +log+nlogn log + λ T ) X i µ) nlogn log + λ T X i µ) )+ n log + λ T Y j µ δ)) ). 30) Pr lδ 0 ) < χ k,α) = α+on ), where χ k,α is the α-th order quantile of the χ k distribution. Let P = be the estiate the probability vector p N,..., p ν, q N ),..., q n N ) ) T P = p ν,...,p ν,q ν),...,q n ν)) T, where p i and q j are obtained fro 7) and 8) replacing λ, λ and µ by λ, λ and µ, respectively. In this k-diensional case, the Kullback-Leibler divergence between the probability vectors U and P is given by D Kullback U, P) = N log N p i N { = log N + n N log N n q j N + λ T X i µ) )+ n Therefore, the relationship between lδ 0 ) and the Kullback-Leibler divergence is ) log + λ T Y j µ δ)) }. lδ 0 ) = ND Kullback U, P). 3) Based on 3) the faily of epirical phi-divergence test statistics are defined as with Therefore the expression of T φ δ 0 ) is T φ δ 0 ) = N φ ) D φu, P), ) D φ U, P) = p i +n φ N + n p +n q n jφ i +n T φ δ 0 ) = φ ) = φ ) { { p i φ p i ) + n + λ T X i µ) n q j φ n q j N n +n q j ) } φ + λ T ) X i µ) + n + λ T φ + λ T Y j µ δ 0 )) }. 3) Y j µ δ) ). 5

16 A result siilar to the one given in Lea for the k-diensional case is µ = Σ +nσ ) Σ X +nσ Y δ )) +Op k ), where X = X i and Y = n n Y i. Finally, based in this result it is possible to establish T φ δ 0 ) L n, χ k. 5. Extension of the test-statistic using the Rényi s divergence Rényi 96) introduced the Rényi s divergence easure as an extension of the Kullback-Leibler divergence. Unfortunately this divergence easure is not a eber of the faily of phi-divergence easures considered in this paper. Menéndez et al. 995, 997) introduced and studied theh,phi)-divergence easures in order to have a faily of divergence easures in which the phi-divergence easures as well as the Rényi divergence easure are included. But not only the Rényi divergence easure is included in this new faily but another iportant divergence easures not include in the faily of phi-divergence easures are included. For ore details about the different divergence easures included in the h,phi)-divergence see for instance, Pardo 006). Based on the h,phi)-divergence easures between the probability vectors U and P, defined in ) and 3) respectively, we can consider the following faily of epirical h,phi)-divergence test statistics for the two-saple proble considered in ) T h φ δ 0 ) = ht φ δ 0 )), 33) where h is a differentiable increasing function fro [0, ) onto [0, ) with h0) = 0 and h 0) > 0. If we consider in 33), and we get hx) = logaa )x+), a 0, aa ) φx) = xa ax ), a 0,, aa ) T Rényi δ 0 ) = Tφ h δ 0) = aa ) logaa )T φδ 0 )+) { = aa ) log ) a n + λ x i µ) + } ) a + λ y j µ+δ 0 )), i.e., the epirical Rényi s divergence test statistics for testing ). For a = 0 and a =, we get li T Rényiδ 0 ) = ND Kullback U,P) a 0 and It is clear that li T Rényiδ 0 ) = ND Kullback P,U). a ht φ δ 0 )) = h0)+h 0)T φ δ 0 )+ot φ δ 0 )). 6

17 Therefore T h φ δ 0) h 0) L n, χ. In the sae way can be established for the proble considered in 6) that where with T φ δ 0 ) defined in 3). T h φ δ 0) h 0) L n, χ k, T h φ δ 0) = ht φ δ 0 )) Acknowledgeent. This research is partially supported by Grants MTM fro Ministerio de Econoia y Copetitividad Spain). References [] Adiari, G. 995). Epirical likelihood confidence intervals for the difference between eans Statistica, 55, [] Baggerly, K. A. 998). Epirical likelihood as a goodness-of-fit easure. Bioetrika, 85, [3] Baklizi, A. and Kibria, B.M. G. 009). One and two saple confidence intervals for estiating the ean of skewed populations: an epirical coparative study. Journal of Applied Statistics, 36, 6, [4] Balakrishnan, N, Martin, N. and Pardo, L. 05). Epirical phi-divergence test statistics for testing siple and coposite null hypotheses. Statistics, 49, [5] Basu, A., Mandal, A. Martin, N. and Pardo, L. 05). Robust tests for the equality of two noral eans based on the density power divergence. Metrika, 78, [6] Bhattacharyya, A. 943). On a easure of divergence between two statistical populations defined by their probability distributions. Bulletin of the Calcutta Matheatical Society, 35, [7] Broniatowski, M. and Keziou, A. 0). Divergences and duality for estiating and test under oent condition odels. Journal of Statistical Planning and Inference, 4, [8] Cressie, N. and Pardo, L. 00). Phi-divergence statisitcs. In: Elshaarawi, A.H., Plegorich, W.W. editors. Encyclopedia of environetrics, vol. 3. pp: , John Wiley and sons, New York. [9] Cressie, N. and Read, T. R. C. 984). Multinoial goodness-of-fit tests. Journal of the Royal Statistical Society, Series B, 46, [0] Felipe, A. Martín, N., Miranda, P. and Pardo, L. 05). Epirical phi-divergence test statistics for testing siple null hypotheses based on exponentially tilted epirical likelihood estiators. arxiv preprint arxiv:

18 [] Fu, Y., Wang, X. and Wu, C. 008). Weighted epirical likelihood inference for ultiple saples..journal of Statistical Planning and Inference, 39, [] Jing, B. Y. 995). Two-saple epirical likelihood ethod. Statistics and Probability Letters, 4, [3] Liu, Y., C. Zou, and Zhang, R. 008). Statistics and Probability Letters, 78, [4] Menéndez, M. L., Morales, D., Pardo, L. and Salicrú, M. 995). Asyptotic behavior and statistical applications of divergence easures in ultinoial populations: A unified study. Statistical Papers, 36, 9. [5] Menéndez, M. L., Pardo, J. A., Pardo, L. and Pardo, M. C. 997). Asyptotic approxiations for the distributions of theh, φ)-divergence goodness-of-fit statistics: Applications to Rényi s statistic. Kybernetes, 6, [6] Owen, A. B. 988). Epirical likelihood ratio confidence interval for a single functional. Bioetrika, 75, [7] Owen, A. B. 990). Epirical likelihood confidence regions. The Annals of Statistics, 8, [8] Owen, A. B. 99). Epirical Likelihood for linear odels. The Annals of Statistics, 9, [9] Owen, A. B. 00). Epirical Likelihood, Chapan and Hall/CRC. [0] Pardo, L. 006). Statistical Inference Based on Divergence Measures. Chapan & Hall/ CRC Press, Boca Raton, Florida. [] Qin, J. 994). Sei-paraetric likelihood ratio confidence intervals for the difference of two saple eans. Annals of the Institute of Statitical Matheatics, 46, 7 6. [] Qin, J.998). Inferences for case-control and seiparaetric two-saple densisty ratio odels. Bioetrika, 85, [3] Qin, J. and Lawless, J. 994). Epirical likelihood and general estiating equations. The Annals of Statistics,, [4] Qin, Y. and Zhao, L. 000). Epirical likelihood ratio intervals for various differences of two populations. Chinese Systes Sci. Math. Sci., 3, [5] Rényi, A. 96). On easures of entropy and inforation. Proceedings of the Fourth Berkeley Syposiu on Matheatical Statistics and Probability,, [6] Shara, B. D. and Mittal, D. P. 997). New non-additive easures of relative inforation. Journal of Cobinatorics, Inforation & Systes Science,, 33. [7] Tsao, M. and Wu, C. 006). Epirical likelihood inference for a coon ean in the presence of heteroscedasticity. The Canadian Journal of Statistics, 34,,

19 [8] Yan, Y. 00). Epirical likelihood inference for two-saple probles. Unpublished aster s thesis, Departent of Statisticsd and Actuarial Science, University of Waterloo, Canada. [9] Yu, P.L.H., Su, Y. and Sinha, B. 00). Estiation of the coon ean of a bivariate noral population. Annals of the Institute of Statistical Matheatics, 54, [30] Wu, C. and Yan, Y. 0) Epirical likelihood inference for two-saple problles. Statistics and Its Inference, 5, [3] Zhang, B. 000). Estiating the treatent effect in the two-saple proble with auxiliary inforation. Nonparaetric Statistics,, Appendix 5.3 Proof of Lea In a siilar way as in Hall and Scala 990), we can establish λ = λ µ) = σ ) X µ +Op ) and λ = λ µ) = σ ) Y µ δ0 +Op n ). Now applying that we have σ λ µ)+nλ µ) = 0, ) X µ +Op )+nσ ) Y µ δ0 +Op ) = 0. 34) Solving the equation for µ we have the enunciated result. 5.4 Proof of Theore First we are going to establish ) ) X µ φ p i φ = +o p ) 35) ) p i σ ) ) n Y µ δ0 φ nq j φ = n +o p ). 36) ) nq j σ ) If we denote W i = λ µ)x i µ) we have φ p i = φ+w i ). A Taylor expansion gives φ+w i ) = φ)+φ )W i + φ )Wi +o Wi ). On the other hand p i = +λ µ)x i µ) = = W i +Wi +W +o W ) i. i 9

20 Then But φ ) ) φ p i φ = Wi ) p i φ +Wi ) +o W )) i φ )Wi +o Wi ) ) = φ ) φ ) Wi + o W ) i φ ) Wi 3 o W ) i Wi + φ ) Wi 4 + o W ) i W i + φ ) o Wi ) W i + o Wi ) ) ) o W i o ) W = i φ ) o see page 0 in Owen 00)), and = Wi + φ o W ) i W 3 i o W ) i Wi + Wi 4 ) + φ o W ) i + o W ) i W ) i + φ o W ) ) i o W ) i. λ µ)x i µ) ) = φ ) = o p )o O p )) = o p ), because λ µ) = O p / ) X i µ) = o p ) X i µ) o λ µ) ) applying the strong law of large nubers. Wi 3 λ µ) 3 X i µ 3 = O p 3/ )o /) = o p ), because X i µ 3 = o /), by Lea.3 in page 8 in Owen 00). o ) Wi Wi o λ µ)) λ µ) X i µ 3 = o O p ) O p / )o 3/ ) = o p ). Wi 4 λ µ) 4 X i µ 4 O ) p Z X i µ 3 = O ) p o / )O / ) = o p ), because Z = ax i X i µ = O / ) applying Lea. in page 8 in Owen 00). φ ) o ) ) Wi o W i φ ) λ µ) 4 X i µ 4 = o p ). Therefore ) φ p i φ = Wi ) p +o p) i = λ µ)x i µ) +o p ) = σ 4 ) X µ X i µ) +o p ) ) X µ = +o p ). σ 0

21 In a siilar way we can get Therefore, n φ nq j φ ) nq j ) ) Y µ δ0 = n +o p ). { T φ δ 0 ) = ) φ p i φ + n n q j φ ) p i n q j σ ) } Applying 34), and nσ ) ) X µ Y µ δ0 = +n +o p ). σ σ ) ) Y µ δ0 = σ X µ +Op ) ) X µ T φ δ 0 ) = σ ) ) X µ Y µ δ0 +op ) σ ) X µ X ) = µ Y + µ+δ0 +op ) σ ) X µ X ) = Y +δ0 +op ) = nσ σ Y µ δ0 ) X Y +δ0 ) +op ). Fro 9) we have Y µ δ 0 = Y X σ +n Y δ0 ) σ σ + n σ δ 0 Therefore, Now we have, = σ ) T φ δ 0 ) = nσ Y X δ0 σ + σ n X +Y δ0 ) σ + σ n. = n )) σ X Y δ0 +n σ = +n σ + n +n σ n +n ) ) X Y +δ0 σ σ +nσ +n X Y δ0 )) +op ). ) +o p ) X µ ) L N0,σ ), n Y µ δ0 ) ) L n N0,σ ). and n +n n +n X µ ) L n, N0, ν)σ ). Y µ+δ0 ) ) L n, N0,νσ ),

22 where is such that ). Hence n ) L X Y +δ0 +n n, N0, ν)σ +νσ), fro which is obtained +n σ + n +n σ n ) L X Y +δ0 +n N0,) n, and now the result follows.