TEST OF HOMOGENEITY OF PARALLEL SAMPLES FROM LOGNORMAL POPULATIONS WITH UNEQUAL VARIANCES S. E. Ahed, R. J. Tokins and A. I. Volodin Departent of Matheatics and Statistics University of Regina Regina, Saskatchewan, CANADA S4S 0A October 8, 00 Abstract Consider ( ) independent rando saples fro lognoral populations with ean paraeter θ 1, θ respectively. A large saple test for the hoogeneity of the ean paraeters is developed. An estiator and a confidence interval are proposed for the coon ean paraeter. The asyptotic distribution of the proposed test-statistic under the null hypothesis as well as under local alternative is derived. Key Words and Phrases: Coon ean, Cobination of lognoral odels, Asyptotic tests and confidence interval, Asyptotic power. 1. INTRODUCTION Let the rando variable Y be distributed norally with ean µ and variance σ ; then X = e Y of X is given by will have a lognoral distribution. The probability distribution function f(x) = { } 1 xσ π exp (ln x µ), 0 < x < σ 1
Evidently, the lognoral odel is related to the noral distribution in the sae way that the Weibull is related to the extree value distribution. Both noral and lognoral odels have received considerable use in lifetie and reliability probles. In edical and engineering applications, one often encounters rando variables X such that logarith of X has a noral distribution. This lognoral odel is coonly used in edicine and econoics, the basic process under consideration leads to phenoena which are often a ultiplication of factors. Also, as stated in Cheng (1977), reliability studies indicate that any sei-conductor devices follow lifetie distributions, which are well represented by the lognoral. We refer to an edited volue by Crow and Shiizu (1988) for a coprehensive review of the subject s theory and applications. Suppose X j1, X j,, X jnj is a rando saple fro a two-paraeter lognoral distribution with ean θ j and variance τj, denoted by Λ(θ j, τj ), corresponding to the noral distribution with ean µ j and variance σj, j = 1,,,. Thus, θ j = e µ j+ σ j, τ j = e µ j+σj e σj 1. Define Y ji = ln(x ji ), 1 i. If µ j and σ j are unknown, then the axiu likelihood estiator (MLE) of (µ j, σ j ) is (ˆµ j, ˆσ j ) with ˆµ j = 1 Y ji, ˆσ j = 1 i=1 (Y ji ˆµ j ). (1.1) i=1 Hence, the axiu likelihood estiators (MLE) of (θ j, τ j ) are ˆθ j = eˆµ j+ 1 ˆσ j, ˆτ j = e ˆµ j+ˆσ j eˆσ j 1. (1.) The sapling distribution of ˆθ j is given in the following lea. Lea AT (Ahed and Tokins, 1995) For each j = 1,,,, nj (ˆθj θ j ) D N (0, ν j ) as tends to, D eans convergence in distribution and ν j = σj (1 + σj /) exp{µ j + σj }. We consider the estiation and testing of the lognoral eans when ultiple-saples are available. For exaple, the data ay have been acquired at a different tie or space. Also, in any experiental situations it is a coon practice to replicate an experient. In these situations, we often encounter the proble of pooling independent estiates of a paraeter obtained fro different sources. Each such estiate is reported
as a nuber with an estiated standard error. In this case two probles are to be considered. First, can all these estiates be considered to be hoogeneous, i.e., are they estiating the sae paraeter? Second, if the estiates are hoogeneous, what is the best way of cobining the to obtain a single estiate? We will address these questions in an orderly anner. The focal point of this investigation is to develop an inference procedure when several parallel saples are available. In a large-saple classical setup, we propose a test of the hoogeneity of the lognoral eans. Further, a point estiator and interval estiator of the coon paraeter is given.. MULTIPLE SAMPLE PROBLEMS In this section we discuss estiation and testing procedures to provide inforation about properties of the ean paraeters based on several rando saples taken fro lognoral populations..1. Pooling of Estiates In any real occasions it is desirable to cobine the individual saple estiates to obtain a cobined or pooled estiate of coon paraeter θ. It is known that the variance of a linear cobination l j ξ j of the rando variables ξ 1,..., ξ, subject to l j = 1 is iniized by choosing l j = σj σk σ j is the variance of the ξ j, j = 1,...,. Hence, the following theore is proposed. Theore.1 Let X j Λ(θ, τ j ), j = 1,,, and suppose that observations fro a saple of size are available for each population. Then a cobined saple estiate of θ which has iniu variance aong the class of the unbiased estiators of θ which are linear functions of ˆθ 1,... ˆθ is given by, θ = ˆθj ˆν j, ˆν j ˆν j = ˆσ j (1 + ˆσ j /) exp{ˆµ j + ˆσ j }. Then, θ n is approxiately norally distributed with ean θ and asyptotic variance [ 1 /ν j ] 1. 3
The proof of the above theore follows fro Lea AT (stated in the previous section). It is concluded fro Theore.1 that θ provides a good estiate for a coon θ based on several saples. More iportantly, it is not necessary for the ν j to be equal in this case. We refer to Rao (1981, pp. 389-391) for further discussions on the topic... Test of Hypothesis There are any situations when one ust ake a decision that is based on an unknown paraeter s value(s). On such occasions a test of the hypothesis about the paraeters ay be ore appropriate. We consider two classes of testing probles for lognoral data: (i) siple null versus global alternative, and (ii) test for hoogeneity...1. Test of siple null versus global alternative Using failiar atrix notation, let θ = (θ 1,, θ ) be a 1 be vector of paraeters and ˆθ = (ˆθ 1,, ˆθ ) be the axiu likelihood estiator vector of θ. Suppose it is desired to test the siple hypothesis H o : θ = θ o, θ o = (θ1, o θ, o, θ) o (.1) against the global alternative H a : θ θ o. Define n = n 1 + n +... + n. It is natural to construct a test statistic for the null hypothesis, which is defined by the noralized distance of ˆθ fro θ o. Hence, define T 1 = n(ˆθ θ o ) S 1 1 (ˆθ θ o ), n n S 1 = Diag ˆν 1,, ˆν. n1 n Note that the test statistics T 1 can be rewritten as By using Lea AT we arrive at the following theore. ) (ˆθj θj T 1 = (.) ˆν j Theore.: Let n and also /n approches to a constant for any j = 1,...,. Then under the null hypothesis in (.1), the test statistic T 1 follows a chisquare distribution with degrees of freedo. 4
Proof. Follows fro the Lea AT and the fact that T 1 is the su of squares of independent asyptotically noral rando variables. Thus, when the null hypothesis is true, the upper α-level critical value of T 1, by C n,α, ay be approxiated by the central χ distribution with degrees of freedo. Note that σ j is a function of θ j ; i.e., σ j = (ln θ j µ j ). Hence, under the null hypothesis σ j = σ o j = (ln θ o j ˆµ j ) and in this case we will have ˆν j = ˆν o j = σ o j (1 + σ o j /) exp{ˆµ j + σ o j }. Thus, we can define another test statistic for the proble at hand as follows: T 1 = n(ˆθ θ o ) S 1 1 (ˆθ θ o ), n S 1 = Diag ˆν 1, o n, ˆν o. n1 n Note that the test statistics T can be rewritten as ) (ˆθj θ o T1 j = n ˆν j o j Under H o, for large n, T 1 will have a χ distribution with degrees of freedo..3. Test of Hoogeneity In this section we focus on developing a testing ethodology for the hoogeneity of lognoral eans when saples are pooled. Let us suppose that two or ore saples are available for which a coon value of θ is assued. The statistical proble is to test the following null hypothesis of hoogeneity of the ean paraeters: the coon value of θ is unknown. Define θ = ( θ, θ) H o : θ 1 = θ = θ = θ, (.3) = θ1, 1 = (1,, 1). We propose the following test statistic to test the null hypothesis in relation (.): T = n(ˆθ θ1 ) S 1 (ˆθ θ1 ), n n S = Diag ν 1,, ν, n1 n 5
and ν j = σ j (1 + σ j /) exp{ˆµ j + σ j }, σ j = ln( θ j ˆµ j ). And again, we can rewrite test statistics T without of atrix notations as (ˆθj θ ) T =, (.4) ν j In an effort to derive the null distribution of T we consider the asyptotic distribution of soe rando variables related to the proposed test statistic. Define U n = n(ˆθ θ1 ). It can be seen that U n = n Cˆθ C = I 1ˆω J D, Theore.3: ( n1 D = Diag,, n ), J = I 1 1 ν 1 ν D, and ˆω = nj For large n, under the null hypothesis in (.), the test statistic T is distributed as a chi-square distribution with ( 1) degrees of freedo. Proof. Follows fro the Lea AT and the fact that T is the su of squares of asyptotically independent asyptotically noral rando variables. As the consequence of the above theore, under the null hypothesis and for large n, for given α, the critical value of T ay be approxiated by χ 1,α, the upper 100α% point of the chi-square distribution with ( 1) degrees of freedo..4. Power of the Tests It is iportant to note that, for a fixed alternative that is different fro the null hypothesis, the power of both test statistics proposed earlier will converge to one as n. This follows fro the fact that test statistics tends to infinity if θ θ 0 (cf. the siilar aguent given in Sen and Singer (1993, pages 37-38)). Thus, to study the asyptotic power properties of T 1 and T, we ust confine ourselves to a sequence of local alternatives {K n }. When θ is the paraeter of interest, such a sequence ay be specified by K n : θ = θ o + ξ. n 1 6 ν j
ξ is a vector of fixed real nubers. Evidently, θ approaches θ o at a rate to n 1/. Stochastic convergence of ˆθ to θ ensures that ˆθ p θ under local alternatives as well. Hence, nonnull distributions and the power of the proposed test statistic can be deterined under the local alternatives. Theore.4: Under the local alternatives and as n we have the following distributional result: n 1/ {ˆθ θ o } D N (ξ, Γ 1 ), ( ν1 Γ 1 = Diag,, ν ). ω 1 ω Theore.5: Under the local alternatives and as n we have the following distributional result: n 1/ {ˆθ θ1 } D N (Jξ, Γ ), and ( ν1 Γ = Diag,, ν ) C, ω 1 ω C = I 1 ω JD, ( ω1 D = Diag,, ω ), J = I 1 1 ν 1 ν D, and ω = ωj. ν j Proof of the both theores can be obtained using the general contiguity theory (cf. Roussas (197)). By thores.4 and.5, a su of squares of asyptotically independent and asyptotically noral rando variables with the unit variance and nonzero eans has the asyptotic distribution noncentral χ -square with the corresponding paraeter of noncentrality. Thus, under local alternatives, test statistics T 1 and T will have asyptotically a noncentral chi-square distribution with and 1 degree of freedo respectively, and noncentrality paraeters Θ 1 = ξγ 1 1 ξ, Θ = (Jξ) Γ 1 (Jξ), respectively. Hence, using a noncentral chi-square distribution, one can do the power calculations of the proposed test statistics..4. Interval Estiation 7
Let z α be the usual percentile point such that 1 Φ z α = α, Φ( ) is the cuulative distribution function for a standard noral rando variable. Noting that, P r ˆθ j z α ( ˆνj ) 1/ θ j ˆθ j + z α 1/ ˆνj converges to 1 α as. Hence intervals having 1 α coverage probability for θ j can be expressed as ˆθ j ± z α 1/ ˆνj. (.5) If the null hypothesis H o : θ 1 = θ = = θ is not rejected, it ay be of interest to obtain a 100(1 α)% confidence interval about the coon value of θ. A 100(1 α)% confidence interval about θ ay be obtained by using the cobined data. Note that, ( 1/ ( ) 1/ 1 P r z α θ θ ( / ν j )) θ 1 + z α ( / ν j ) converges to 1 α as n. Thus, a 100(1 α)% confidence interval about coon paraeter θ ay be obtained as follows: ( 1/ 1 θ ± z α. (.6) ( / ν j )) Clearly this interval will provide shorter confidence interval than that based on individual estiates, for any given α. 3. Concluding Rearks A large saple analysis is presented when lognoral eans are cobined. A test of the hoogeneity of the eans is presented, and a point and interval estiator of the coon ean paraeter is also provided. As a word of caution, the statistical procedures based on cobined estiates are sensitive to departure fro the null hypothesis. Therefore, soe other alternatives to pooled estiator should be considered. Furtherore, the proposed procedures involve nonlinear functions of asyptotic noral estiators that ay not be well approxiated by a noral law unless the saple sizes are large. Acknowledgeents The work was supported by grants fro the Natural Sciences and Engineering Research Council of Canada. 8
REFERENCES Ahed, S. E. and R. J. Tokins (1995). Estiating lognoral eans under uncertain prior inforation. Pakistan Journal of Statistics, 11, 67-9. Cheng, S. S. (1977). Optial replaceent rate of devices with lognoral failure distribution. IEEE Trans. Reliability R-6, 174-178. Crow, E. L. and Shiizu, K. (1988). Lognoral Distributions. Marcel Dekker: New York. Rao, C. R. (1981). Linear Statistical Inference and its Applications (second edition). Wiley Eastern Liited: New Delhi. Roussas, G. G. (197) Contiguity of Probability Measures, Cabridge University Press. Sen, P.K. and Singer, J.M. (1993) Large Saple Methods in Statistics, Chapan and Hall: New York - London. 9