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Generalized con&dence intervals for the ratio of
means of two normal populations
Jack C. Lee
a;∗, Shu-Hui Lin
baInstitute of Statistics and Graduate Institute of Finance, National Chiao Tung University, Hsinchu, Taiwan
bNational Taichung Institute of Technology, Taichung 404, Taiwan Received 16 September 2001; accepted 16 January 2003
Abstract
Based on the generalized p-values and generalized con&dence interval developed by Tsui and Weerahandi (J. Amer. Statist. Assoc. 84 (1989) 602), Weerahandi (J. Amer. Statist. Assoc. 88 (1993) 899), respectively, hypothesis testing and con&dence intervals for the ratio of means of two normal populations are developed to solve Fieller’s problems. We use two di9erent proce-dures to &nd two potential generalized pivotal quantities. One procedure is to &nd the generalized pivotal quantity based directly on the ratio of means. The other is to treat the problem as a pseudo Behrens–Fisher problemthrough testing the two-sided hypothesis on , and then to construct the 1 − generalized con&dence interval as a counterpart of generalized p-values. Illustrative ex-amples show that the two proposed methods are numerically equivalent for large sample sizes. Furthermore, our simulation study shows that con&dence intervals based on generalized p-values without the assumption of identical variance are more e=cient than two other methods, especially in the situation in which the heteroscedasticity of the two populations is serious.
c
2003 Elsevier B.V. All rights reserved.
Keywords: Fieller’s theorem; Generalized con&dence interval; Generalized p-values; Generalized pivotal quantity; Heteroscedasticity; Pseudo Behrens–Fisher problem; Ratio estimation
1. Introduction
Much attention has been paid to Fieller’s problems, because they occurred frequently in many important research areas such as bioassay and bioequivalence. In bioassay problem, the relative potency of a test preparation as compared with a standard is esti-mated by (i) the ratio of two means for direct assays, (ii) the ratio of two independent
∗Corresponding author. Tel.: +886-3-5728746; fax: +886-3-5728745.
E-mail address:jclee@stat.nctu.edu.tw(J.C. Lee).
0378-3758/$ - see front matter c 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0378-3758(03)00141-1
normal random variables for parallel-line assays and (iii) the ratio of two slopes for slope-ratio assays. In biological assay problems (Fieller, 1954; Finney, 1978) and bioequivalence problems (Chow and Liu, 1992; Berger and Hsu, 1996), one is in-terested in the relative potency of two drugs or treatments. Traditionally,Fieller (1944, 1954)provides a widely used general procedure for the construction of con&dence in-tervals (often called Fieller’s theorem) for the ratio of means (also discussed by Rao, 1973; Finney, 1978; Koschat, 1987 and Hwang, 1995). Under homoscedasticity case,
Koschat (1987) has also shown that within a large class of sensible procedures the Fieller solution is the only one that gives exact coverage probability for all parame-ters. However, the conventional procedures are often restricted to the assumption of a common variance or pairwise observations for mathematical tractability. Thus, the ex-act approaches to Fieller’s problems under the unequal variances assumption have also been intensively investigated. Consider the following problem: Let X =(X1; X2; : : : ; Xn1)
and Y = (Y1; Y2; : : : ; Yn2) be two independent sets of observations for the potency of
a standard drug and a new drug, respectively. Assume that Xi are independently and
identically distributed as N( 1; 12); Yi are independently and identically distributed as
N( 2; 22), where 1 and 2 are the true potencies. The problemis to determine, with
any desired probability, the range of values for the ratio of means = 2= 1, which is
the relative potency of the new drug to the standard.
Under the assumption of identical variance, Fieller (1954)constructed a con&dence interval based on the statistic
T =( JY − JX) (1 n2+ 2 n1)S2 ; (1.1)
where JX = (1=n1)ni=11 Xi; JY = (1=n2)nj=12 Yj and S2= (ni=11 (Xi− JX)2+nj=12 (Yj−
JY)2)=(n
1+ n2). It is obvious that (n1+ n2− 2)=(n1+ n2)T has the Student’s t
dis-tribution with (n1+ n2− 2) degrees of freedom. Solving the inequality
: (n1+ n2− 2) n1+ n2 |T| ¡ t1− 2 = : | Jy − Jx| 6 t1− 2 (n1+ n2)(1=n2+ 2=n1)s2 n1+ n2− 2 ; (1.2)
where t1−=2 is the (1 − =2)th quantile of the t distribution, the exact 1 − con&dence
interval for will be obtained.
On the other hand, if variances are related to the means, such as 2
i = (c + i)k 2
with c+ i¿ 0; i=1; 2 and k is known, Cox (1985)provided a interval estimate based
on the statistic T∗=(N − 2)l2 aS∗ ; (1.3) with l = (c + 2)(c + Jx) − (c + 1)(c + Jy); a = (c + 2)2((c + 1)k=n1) + (c + 1)2((c + 2)k=n2) and S∗= 2=ni=11 (Xi− JX)2= 12+ n2
the Fisher-Snedecor’s F distribution with 1 and n1+ n2− 2 degrees of freedom. De&ne
= (c + 2)=(c + 1), the 100(1 − )% con&dence interval for is obtained by solving
the inequality
: (N − 2)laS∗ 26 F1−(1; n1+ n2− 2)
: (1.4)
For c = 0 and k = 2, the 100(1 − )% con&dence interval for = 2= 1 is based on
solving the quadratic inequality : ( Jy − Jx)26F1−(1; n1+ n2− 2) n1+ n2− 2 s2 2 n1=(n1+ n2) + 2 s21 n2=(n1+ n2) ; (1.5) with s2
1= (1=n1)ni=11 (xi− Jx)2 and s22= (1=n2)ni=12 (yi− Jy)2, respectively.
In this article, we propose two di9erent exact approaches based on generalized p-values and generalized con&dence intervals, as de&ned by Tsui and Weerahandi (1989),Weerahandi (1993), respectively, to construct con&dence intervals for the ratio of means of two normal populations under heteroscedasticity. The lack of exact con&-dence intervals in many applications can be attributed to the statistical problems involv-ing nuisance parameters. The possibility of exact con&dence interval can be achieved by extending the de&nition of con&dence interval. To generalize the de&nition of con-&dence intervals, &rst examine the properties of interval estimates obtained by the conventional de&nition. To &x ideas, consider a randomsample X = (X1; X2; : : : ; Xn)
froma distribution with an unknown parameter . Let A(X) and B(X) be two statistics satisfying the equation
P[A(X) 6 6 B(X)] = ; (1.6)
where is a prespeci&ed constant between 0 and 1. Let a = A(x) and b = B(x) be the observed values of the two statistics, then, in the commonly used terminology, [a; b] is a con&dence interval for with the con&dence coe=cient . For example, if = 0:95, then the interval [a; b] obtained in this manner is a 95% con&dence interval. This approach to constructing interval estimates is conceptually simple and easy to implement, but in most applications involving nuisance parameters it is not easy or impossible to &nd A(x) and B(x) so as to satisfy (1.6) for all possible values of the nuisance parameters. The idea in generalized con&dence intervals is to make this possible by making probability statements relative to the observed sample, as done in Bayesian and nonparametric methods. In other words, we allow the functions A(·) and B(·) to depend not only on the observable randomvector X but also on the observed data xobs.
We will brieOy introduce the general theory and provide our &rst procedure for &nding the generalized pivotal quantity based directly on the ratio in Section 2. The equal-tail con&dence intervals are included as well. In Section 3, we tackle the prob-lemas a pseudo Behrens–Fisher problemthrough the testing of two-sided hypothesis on , with the interval construction treated as a counterpart of generalized p-values.
It is interesting to note that these two procedures are numerically equivalent for large sample sizes. Through the procedure presented in Section 3, the interval proposed by
Cox (1985)with c = 0; k = 2 can be viewed as an approximation of our method when n1=n2. Both procedures developed in this article get more precise interval than those of
Fieller’s and Cox’s when serious heteroscedasticity is present. Two numerical examples are illustrated in Section4 comparing the proposed methods with other methods in the presence of heteroscedasticity. A simulation study is conducted to calculate the coverage probabilities in di9erent scenarios in Section 5, and &nally some concluding remarks are given in Section6.
2. Generalized pivotal quantity based directly on the ratio of means 2.1. Notations and theory
Let X be an observable randomvector having a density function f(X|), where = (; ) is a vector of unknown parameters, is a parameter of interest, and is a vector of nuisance parameters. Let " be the sample space of possible values of X and # be the parameter space of . A possible observation from X is denoted by x, where x ∈ ", and the value of X actually observed is denoted by xobs.
We are interested in &nding interval estimates of based on observed values of X. The problemis to construct generalized con&dence intervals of the form[A(xobs);
B(xobs)], where A(xobs) and B(xobs) are functions of xobs.
The conventional approach to constructing con&dence intervals is based on the notion of a pivotal quantity R=r(X; ) with the property that for given we can &nd a region C such that
P(r(X; ) ∈ C) = (2.1)
for all . We then de&ne #(X) = { | r(X; ) ∈ C}. Since P{ ∈ #(X)} = P{r(X; )
∈ C} = ; #(X) is a -level con&dence region and we are “100 % con&dent” that is
in the observed region #(xobs).
Weerahandi (1993) extended the de&nition of a pivotal quantity as follows. Let R = r(X; x; ) be a function of X and possibly x and as well. Then R is said to be a generalized pivotal quantity if it has the following two properties:
(i) For any &xed x ∈ "; R has a probability distribution Px free of unknown
param-eters.
(ii) If X = x, then r(x; x; ') does not depend on , the vector of nuisance parameters. Using (i), for given we can &nd, for any &xed x, a computable region C(x) such that
Px{r(X; x; ) ∈ C(x)} = : (2.2)
By (ii), for any x whether r(x; x; ) ∈ C(x) holds depends only on , and not on the nuisance parameters , so we may de&ne the 100 % generalized
con&dence set
#(xobs) = { | r(xobs; xobs; ) ∈ C(xobs)}: (2.3)
This region is a realization of a randomsubset
#(X) = { | r(X; X; ) ∈ C(X)} (2.4)
of values. From(2.2) we obviously do not have the analogue of (2.1), and hence can-not claim P{ ∈ #(X)} = . However, from(2.2) it does follow that P{r(X1; X2; ) ∈
C(X2)} = if X1 and X2 are independent of one another, each distributed as X. Thus
the generalized con&dence set (2.4) intuitively corresponds to using the same X twice: viewing it as a future, unobserved variable, and also conditioning on its observed value in the data.
It is noted that a generalized pivotal quantity in interval estimation is the counterpart of generalized test variables in signi&cance testing of hypotheses proposed byTsui and Weerahandi (1989). If the formof a p-value for a one-sided test is readily available, a generalized con&dence interval for can be deduced directly fromits power function. 2.2. The procedure
Let X1; X2; : : : ; Xn1 and Y1; Y2; : : : ; Yn2 be randomsamples fromN( 1; 21) and
N( 2; 22), respectively, where i and 2i are unknown with i = 1; 2. For the univariate
Fieller–Creasy problem, we want to construct intervals for the parameter = 2= 1.
First, we will &nd a generalized pivotal quantity, R, based on the following su=cient statistics JX = 1n 1 n1 i=1 Xi; JY = 1n 2 n2 j=1 Yj; S2 1 = n1 i=1(Xi− JX)2 n1 ; and S 2 2= n2 i=1(Yi− JY)2 n2 :
Since a generalized pivotal can be a function of all unknown parameters, we can construct R based on the randomquantities Z1 =√n1( JX − 1)= 1 ∼ N(0; 1); Z2=
√n
2( JY − 2)= 2 ∼ N(0; 1); U1= n1S12= 21 ∼ "2n1−1 and U2= n2S22= 22 ∼ "2n2−1, whose
distributions are free of unknown parameters. Using ≡ JY − Z2 2= √n 2 JX − Z1 1=√n1 = JY − Z2S2=√U2 JX − Z1S1=√U1;
we can de&ne the following potential generalized pivotal R(X; Y ; x; y; 1; 2; 21; 22) = Jy − Z2s2= √ U2 Jx − Z1s1=√U1 = Jy − T2s2=√n2− 1 Jx − T1s1=√n1− 1;
where s1; s2; Jx; Jy are the observed values of S1; S2; JX; JY, respectively. Note that T1 ∼
Now consider the problemof constructing lower con&dence bounds for . Since the observed value of R is , the following probability statement will lead to a right-sided 100(1 − )% con&dence interval, 1 − = P{R ¿ c} = P T2¿ √ n2− 1 s2 Jy + c T1√ns1 1− 1 − Jx T1¿ √ n1− 1 s1 Jx ×P T1¿ √ n1− 1 s1 Jx + P T26 √ n2− 1 s2 Jy + c T1√ns1 1− 1 − Jx T1¡ √ n1− 1 s1 Jx ×P T1¡ √ n1− 1 s1 Jx (2.5) where T1 ∼ t(n1− 1) and T2 ∼ t(n2− 1). It is evident that [c1−; ∞) is the
de-sired 100(1 − )% generalized con&dence interval for , where c1− is the value of c
satisfying (2.5) for a speci&ed value of 1 − .
Similarly, the 100(1 − )% upper con&dence bound c
1− for can be obtained
through 1 − = P{R 6 c 1−} = P T26 √ n2− 1 s2 Jy + c 1− T1√ns1 1− 1 − Jx T1¿ √ n1− 1 s1 Jx ×P T1¿ √ n1− 1 s1 Jx + P T2¿ √ n2− 1 s2 Jy + c 1− T1√ns1 1− 1 − Jx T1¡ √ n1− 1 s1 Jx ×P T1¡ √ n1− 1 s1 Jx : (2.6)
Likewise, the 100(1 − )% equal tail con&dence interval for can also be derived through &nding c1−=2 and c1−=2 in (2.5) and (2.6), respectively.
The underlying family of distributions is invariant under the common scale transfor-mations
( JX; JY; S1; S2) → (k JX; k JY; kS1; kS2) and ( 1; 2) → (k 1; k 2);
where k is a positive constant. Obviously, the parameter of interest is una9ected by any change of scale, and therefore, the statistical problemis invariant. Furthermore, the observed value of the statistic R does not depend on the data, any scale invariant
generalized con&dence region of can be constructed from R alone (see Weerahandi, 1993, Theorem3.1).
3. Generalized pivotal quantity de ned through the testing procedure
In this section, we solve the Fieller–Creasy problemthrough the testing procedure with the interval estimation obtained as a counterpart of generalized p-values. Suppose X1; X2; : : : ; Xn1 and Y1; Y2; : : : ; Yn2 are randomsamples fromN( 1; 21) and N( 2; 22),
respectively, where i and 2i are unknown with i = 1; 2. Consider the problemof
signi&cance testing of hypotheses concerning the parameter = 2= 1. Since H0: =0
versus H1: = 0 can be transformed to H0∗: + = 0 versus H1∗: + = 0 where + = 2−
10, that is, we can treat this testing problemas a pseudo Behrens–Fisher problem. In
view of the fact that JY −0JX is distributed as N(+; ( 22=n2)+20( 21=n1)) which depends
on the parameter of interest + and the nuisance parameter ( 2
2=n2) + 20( 21=n1), the
following potential pivotal quantity can be de&ned as in Tsui and Weerahandi (1989), R∗(X; Y ; x; y; 1; 2; 12; 22) =( JY − 0JX) − + 2 2=n2+ 20 21=n1 s2 2 22 n2S22 + 2 0 s 2 1 21 n1S12 + +; (3.1) where JX = 1 n1 n1 i=1 Xi; JY = 1n 2 n2 j=1 Yj; S2 1 = n1 i=1(Xi− JX)2 n1 and S 2 2 = n2 i=1(Yi− JY)2 n2
are summary statistics, and s1and s2are the observed values of S1 and S2, respectively.
It is noted that the observed value of R∗is robs= Jy−0Jx, where Jx and Jy are the observed
values of JX and JY, respectively. Furthermore, the distribution of R∗ is the same as
Z(s2
2=U2) + 20(s21=U1)++, where Z ∼ N(0; 1); U1=n1S12= 12∼ "2n1−1; U2=n2S22= 22∼
"2
n2−1, and the randomvariables Z; U1; U2 are independent. Since R∗ is stochastically
increasing in +, the geneneralized p-values appropriate for testing the left-sided null hypothesis of the form H0: + 6 0 versus H1: + ¿ 0 is
p = P{R∗¿ r obs|+ = 0} = EB Gn1+n2−2 −√n1+ n2− 2( Jy − 0Jx) s2 2=(1 − B) + 20s21=B ;
where Gn1+n2−2 is the cdf of the Student’s t distribution with n1 + n2 − 2
variable, B =U U1 1+ U2 ∼ beta n1− 1 2 ; n2− 1 2 :
In particular, the p-value for testing point null hypotheses of the form H0: + = 0 is
p = 2EB Gn1+n2−2 −√n1+ n2− 2| Jy − 0Jx| s2 2=(1 − B) + 20s21=B : (3.2)
Two sided con&dence intervals can be deduced from(3.2). A generalized 100(1 − )% con&dence interval for can be derived by solving
1 − = EB P |Tn1+n2−2| 6 √ n1+ n2− 2| Jy − Jx| s2 2=(1 − B) + 2s21=B ; (3.3)
where T has a Student’s t distribution with n1+ n2− 2 degrees of freedomand EB
denotes the expectation with respect to beta((n1− 1)=2; (n2− 1)=2). Eq. (3.3) can also
be expressed as 1 − = EB HF1; n1+n2−2 (n1+ n2− 2)( Jy − Jx)2 s2 2=(1 − B) + 2s21=B ; (3.4)
where HF1; n1+n2−2 is the cdf of the F distribution with 1; n1+n2−2 degrees of freedom.
It is interesting to note that if 2
i = i2 ; i = 1; 2, Cox’s con&dence interval in (1.5)
can be obtained from(3.4) by replacing B with its expected value 1
2 when n1= n2.
Thus, in a way, Cox’s result can be treated as an approximation of our method when n1= n2. According to Cox’s procedure with c = 0; k = 2, the 1 − con&dence interval
is based on solving the quadratic inequality : ( Jy − Jx)26F1−(1; n1+ n2− 2) n1+ n2− 2 s2 2 1 − ˆB+ 2 s2 1 ˆB ;
with ˆB = n2=(n1+ n2), which is equal to E(B) when n1= n2. Comparing (3.4) with
Fieller’s con&dence interval in (1.2), we see that separate estimates s2
1 and s22 are used
for 2
1 and 22, respectively, rather than a pooled estimate s2 for the common variance
2. Also, the con&dence interval is obtained via evaluating the expectation with respect
to a beta distribution. Consequently, the proposed procedure will be more general and useful in getting a decent interval when serious heteroscedasticity is present.
It is noted that for the Behrens–Fisher problem,Tsui and Weerahandi (1989)derived the generalized test variable, which is similar to R∗, by the methods of invariance and
similarity. Consider the equivalent problem of constructing interval estimates based on the three randomquantities, R∗; U1, and U2. Recall that the distributions of each
of these randomvariables is free of unknown parameters. Moreover, the observed value of U1 and U2 depend on the nuisance parameter 1 and 2, respectively, but
Tsui and Weerahandi (1989), all con&dence intervals similar in both 1 and 2 can be
generated using R∗ alone.
4. Illustrative examples
Two examples are given to illustrate the advantages of our proposed methods for setting limits on the ratio of means of two normal populations. The objective of these examples is to show how one will fail to get a decent con&dence interval, such as Fieller’s method, when the variances are unequal.
4.1. Example 1
The data in Table 1 is taken from Jarvis et al. (1987) and Pagano and Gauvreau (1993, p. 254) to measure the relative level of carboxyhemoglobin for a group of nonsmokers and a group of cigarette smokers. The purpose of this example is to analyze the relative carboxyhemoglobin level for two large groups of nonsmokers and cigarette smokers, where 1 and 2 are the true means of carboxyhemoglobin levels
for nonsmokers and cigarette smokers, respectively. The summary data is provided in Table 1 and the interval limits as well as interval widths for four procedures are demonstrated in Table 2.
It is found that the procedures without the assumption of equal variance is better than Fiellers. Thus, the procedures based on a common variance will be given at the cost of wider interval estimates when the population variances are di9erent. Moreover, our procedures developed in Sections 2 and 3 are numerically equivalent and both procedures yield shorter intervals than the other methods.
4.2. Example 2
The second example is to construct con&dence intervals for the ratio of two propor-tions. Let x = (x1; : : : ; xn1) and y = (y1; : : : ; yn2) be two independent sets of
observa-tions for the potency of a standard drug and a new drug, respectively. Suppose X is
are independently and identically distributed as B(1; .) and Y
is are independently and
identically distributed as B(1; /). We are interested in the interval estimation for the ratio of proportions =/=.. In this case, we assume the sample sizes are large enough, so the binomial distribution can be adequately approximated by the normal distribution. Suppose Jx = 0:38; Jy = 0:52, then the alternates for s2
1 and s22 are ((n1− 1)=n1) Jx(1 − Jx)
and ((n2− 1)=n2) Jy(1 − Jy), respectively. It is noted that the MLE for the ratio of two
proportions is ˆ = Jy= Jx = 1:37. We will compare four di9erent procedures with di9erent pairs of sample sizes (n1; n2) = (50; 50); (80; 50); (100; 30). The 95% con&dence
inter-vals and the corresponding con&dence widths for are shown in Table 3. It is noted that in this mild heteroscedasticity example, Fieller’s and Cox’s methods perform well in the case of equal sample size, but their performances deteriorate as the di9erence of sample sizes increases. Again, for large sample sizes, our proposed methods are
Table 1
Carboxyhemoglobin for nonsmokers and smokers groups, percent
Group ni Jxi s2
i
Nonsmokers 121 1.3 1.704
Smokers 75 4.1 4.054
Table 2
95% con&dence interval for 2= 1
Procedure Interval limits Interval width
Section2Eqs. (2.5) and (2.6) (2:57; 3:95) 1.38
Section3Eq. (3.4) (2:57; 3:95) 1.38
Fieller (1954) (2:44; 4:40) 1.97
Cox (1985) (2:52; 4:14) 1.62
Table 3
95% con&dence interval for /=.
Procedure n1= 50; n2= 50 n1= 80; n2= 50 n1= 100; n2= 30
Limits Width Limits Width Limits Width
Eqs. (2.5) and (2.6) (0:88; 2:27) 1.39 (0:91; 2:06) 1.15 (0:83; 2:08) 1.25 Eq. (3.4) (0:88; 2:27) 1.39 (0:91; 2:06) 1.15 (0:83; 2:08) 1.25
Fieller (1954) (0:88; 2:27) 1.39 (0:93; 2:22) 1.29 (0:83; 2:61) 1.73
Cox (1985) (0:88; 2:26) 1.38 (0:93; 2:21) 1.28 (0:88; 2:60) 1.72
numerically equivalent and they perform reasonably well comparing with the other methods in all cases.
5. A simulation study
A simulation study is conducted for calculating the coverage probabilities in di9erent combinations of sample sizes and population variances. Two sets of normal data are generated with 1= 2= 2 and 95% coverage probabilities are calculated based on
1000 replicates. The results are demonstrated in Table 4. We &nd that Fieller’s pro-cedure has good coverage probabilities when the data are generated fromtwo normal populations with identical variance, but its performance deteriorates as the degree of heteroscedasticity increases. Cox’s method perform poorly in the situations in which the smaller sample sizes are associated with larger variances. On the other hand, our procedures performquite well even when the population variances are di9erent.
Table 4
95% Comparison of coverage probabilities for 2= 1 (1 − = 0:95)a
n1 : n2 1: 2 Fieller Cox Eq. (3.4) Eqs. (2.5) and (2.6)
10:10 1:1 0.947 0.951 0.954 0.936 10:10 1:2 0.944 0.947 0.956 0.954 10:10 1:3 0.938 0.946 0.950 0.956 10:10 1:4 0.836 0.940 0.948 0.954 10:10 1:5 0.696 0.939 0.950 0.946 10:5 1:1 0.956 0.953 0.956 0.956 10:5 1:2 0.874 0.871 0.960 0.962 10:5 1:3 0.764 0.849 0.958 0.958 10:5 1:4 0.672 0.836 0.953 0.962 10:5 1:5 0.580 0.799 0.954 0.952 5:10 1:1 0.951 0.945 0.958 0.938 5:10 1:2 0.971 0.976 0.959 0.960 5:10 1:3 0.888 0.982 0.948 0.966 5:10 1:4 0.636 0.979 0.933 0.958 5:10 1:5 0.404 0.981 0.928 0.956
aBased on 1000 replicates in each combination.
6. Concluding remarks
In this article, we propose two di9erent exact generalized approaches based on generalized p-values and generalized con&dence intervals to solve the well-known Fieller-Creasy problem, which is widely used in many important research areas such as bioassay and bioequivalence. Under homogeneous case, Fieller’s solution gives ex-act coverage probability for all parameters. Unfortunately, in the presence of serious heteroscedasticity, the methods under the restriction of identical variance cannot yield decent con&dence intervals. Through the proposed methods in this article, an exact 1 − generalized con&dence intervals for the ratio of two means can be obtained un-der unequal variances and unequal sample sizes. According to our &ndings, the existing procedures ignoring the mild heteroscedasticity will perform well. However, they will performvery poorly in the situation in which serious heteroscedasticity is present. Thus our proposed methods are very valuable in practice, especially when the two variances are quite di9erent.
Acknowledgements
We would like to thank the Editorial Board Member (EOBM) and two referees for their kindly help and constructive comments which led to a substantial improvement of the paper.
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