Variance-Stabilizing and Confidence-Stabilizing. Transformations for the Normal Correlation Coefficient. with Known Variances

Transcription

1 Variace-Stabilizig ad Cofidece-Stabilizig Trasformatios for the Normal Correlatio Coefficiet with Kow Variaces Bailey K. Fosdick ad Michael D. Perlma Techical Report o. 610 Departmet of Statistics Uiversity of Washigto Seattle, WA March 6, 013

2 Abstract Fosdick ad Raftery 01 revisited the classical problem of iferece for a bivariate ormal correlatio coefficiet ρ whe the variaces are kow. They cosidered several frequetist ad Bayesia estimators, the former icludig the maximum likelihood estimator MLE, but did ot obtai the stadard errors of these estimators or cofidece itervals for ρ. Here we preset a ew variace-stabilizig trasformatio y for the MLE i the kow-variace case. Adjustig y appropriately accordig to the sample size produces a cofidece-stabilizig trasformatio y that provides more accurate iterval estimates for ρ tha the MLE, as does Fisher s classical z trasformatio for the MLE i the ukow-variace case. Iterestigly, the z trasform applied to the MLE for the ukow-but-equal-variace case performs well i the kow-variace case for smaller values of ρ. Both these methods are also useful for comparig two or more correlatio coefficiets i the kow-variace case; hypothesis testig i this case is also discussed.

3 1. Itroductio Let x i, y i, i = 1,...,, be i.i.d. observatios from a bivariate ormal distributio with correlatio coefficiet ρ ad variaces σ x ad σ y. We cosider the followig three icreasigly restrictive covariace models: Model 1: σx ad σy both ukow; Model : σx/σ y aσ, a kow, σ ukow; Model 3: σx ad σy both kow. Without essetial loss of geerality we assume throughout that the meas are both 0, that a = 1 so σx = σy = σ i Model, ad that σx = σy = 1 i Model 3. These well-kow models form the basis for may textbook examples ad exercises. Models 1 ad are full expoetial families with complete sufficiet statistics W 1 s x, s y, y ad W s x + s y, y respectively, where s x = 1 x i, s y = 1 y i, y = 1 xi y i. 1 Both Models 1 ad admit explicit maximum likelihood estimators MLE for their ukow parameters: Model 1: ˆρ = r 1, ˆσ x = s x, ˆσ y = s y; Model : ˆρ = r, ˆσ = 1 + s y; where r 1 = y, r = y. s y s x + s y The geometric mea of s x ad s y i r 1 is replaced by the arithmetic mea i r, so r < r 1. Model 3 is a curved expoetial family with log likelihood fuctio l ρ = c log1 ρ + s y 1 ρ + ρy 1 ρ, 3 cf. Stuart ad Ord 1991, Ch. 18. Here W is miimal sufficiet but ot complete; ote that 1

4 W is two-dimesioal while the ukow parameter ρ is oe-dimesioal. The MLE ˆρ is r 3, the maximizig root of the cubic equatio ρ1 ρ ρs x + s y ρ y = 0. 4 I this paper we focus o Model 3, which is less ameable to exact aalysis but was recetly ecoutered by Fosdick ad Raftery 01 whe comparig fertility rate forecast errors betwee developed ad udeveloped coutries. They cosidered several variats of three frequetist estimators for ρ, as well as several Bayesia estimators. The frequetist estimators were r 1, r 3, ad the empirical estimator r 4 = y 5 s trucated to lie i [ 1, 1], where the estimated stadard deviatio estimates ad s y i r 1 ad r are replaced by the kow values 1. However, Fosdick ad Raftery did ot discuss the stadard errors of their estimators. It is well kow 1 that as, r1 ρ N 0, 1 ρ uder Model 1, 6 r ρ N 0, 1 ρ uder Model, 7 r3 ρ N 0, 1 ρ uder Model 3, ρ r4 ρ N 0, 1 + ρ uder Model 3. 9 Because Model 3 is regular sufficietly smooth, the MLE r 3 is amptotically optimal for this model, that is, has smallest amptotic variace amog all amptotically ormal estimators. Whereas the MLE r 3 is optimal for large samples sizes uder Model 3, Fosdick ad Raftery s iterest was i the case of small ad moderate sample sizes. I Table 1 the mea-squared errors 1 E.g. Lehma 1983, 19 p.441, Problems 6..0 ad 6.5.9, Stuart ad Ord 1991, Ch. 18. The result 7 follows from 6 because r r 1 = O p 1 uder Model by a stadard Taylor expasio argumet.

5 of r 1, r, r 3, ad r 4 have bee obtaied,3 by simulatio. Here r 3 performs best uless ρ is small, where r may be somewhat better, while r 4 performs poorly i most cases. These effects become more marked as the sample size icreases, which is i agreemet with the orderig of the amptotic variaces i 6 9. Table 1: Root MSE 1000; r 4 trucated to [-1,1]. largest stadard error of estimates is 0.4. Estimator ρ r r r r r r r r r r r r I additio to its large MSE the estimator r 4 sometimes falls outside the admissible rage [ 1, 1], while the Bayesia estimators are ot ameable to frequetist iterval estimatio; these will ot be discussed further here. As is the case for r 1, covergece to ormality is also slow for r ad r 3, so for small or moderate sample sizes the amptotic variaces i 6 8 do ot provide good approximatios for their stadard errors Table. Fisher s celebrated z trasformatio greatly improves the ormal approximatio to the distributio of r 1 uder Model 1 so is ivaluable for iferece about ρ i this case see.1. Iterestigly, the exact distributio of zr is ea to specify uder Model see. ad therefore uder Model 3, where it provides estimates ad tests that perform well for smaller values of ρ see.4 ad Sectio 3. The MSE is show for a uiform rage of ρ values rather tha ρ values, because it is ρ rather tha ρ that idicates the stregth of the relatioship betwee x ad y. 3 Usig the formulas for r 3 give i F&R, % of estimates did ot satisfy the cubic equatio i 4 withi ±1e 8. For these simulatios, the default optimize procedure i R was used to obtai the estimate r 3 that maximizes the likelihood. 3

6 Table : Ratio of empirical variace to amptotic variace. Estimator ρ r r r r r r r r r For Model 3, a ew variace-stabilizig trasformatio y for the MLE r 3 is preseted i.3. Ulike the z-trasform, the y-trasform must be adjusted for the sample size to stabilize the cofidece coverage of r 3. Whe this is doe the resultig cofidece-stabilized trasformatio y r 3 provides more precise cofidece itervals uder Model 3 tha itervals based o r 1 ad r for moderate ad large values of ρ. For small values, itervals based o zr are preferable. This is demostrated via simulatio i.4. The use of zr, yr 3, ad y r 3 for comparig two or more correlatio coefficiets uder Model 3 is outlied i Sectio 3. F&R also cosidered the problem of testig H 0 : ρ = 0 idepedece vs. the oe-sided alterative H 1 : ρ > 0 uder Model 3. They cosidered the tests that reject H 0 for large values of r 1, r 3, ad r 4 ad their variats, as well as several Bayes tests, ad approximated the sigificace levels of these tests by Mote Carlo simulatio. It is well kow that the r 1 test is exact for this problem. I Sectio 4 we ote that the r test is also exact ad has a iterestig although limited optimality property. Also we show that the r 4 test is locally most powerful for alteratives ρ 0 ad derive the amptotically most powerful test also exact for alteratives ρ 1. O the basis of umerical power comparisos, the r resp., r 3 test is recommeded if small resp., large alterative values of ρ are expected.. Cofidece itervals for ρ uder Model 3. 4

7 .1. Cofidece itervals based o the Model 1 MLE r 1. Let g γ deote the upper γ- quatile of the stadard ormal Gaussia distributio. From 6, for sufficietly large r 1 ± 1 r 1 g α/ 10 is a approximate 1 α cofidece iterval 4 for ρ uder the urestricted Model 1, hece for Models ad 3. Ufortuately the sample size required for the accuracy of the ormal approximatio 6 depeds o the ukow ρ, but this is remedied by Fisher s celebrated z-trasformatio for r 1 cf. Aderso 1984, 4..3, give by the idefiite itegral zρ = dρ 1 ρ = 1 log 1 + ρ ρ This is a variace-stabilizig trasformatio that satisfies [zr1 zρ] N0, 1 1 with faster covergece to ormality tha 6 see Table 3. This provides the approximate 1 α cofidece iterval for ρ give by z 1 zr 1 ± g α/, 13 valid for Model 1 hece for Models ad 3. Note that z 1 r = tahr. A exact cofidece iterval for ρ uder Model ad therefore Model 3 is readily obtaied from the classical Studet t-distributio of the sample regressio coefficiet of y i give x i e.g. Stuart ad Ord 1987, eq. 16.9: 1 r1 sx s y ρ 1 r 1 t As is well kow for 13, usig rather tha results i a coverage probability somewhat closer to the omial value 1 α. 5

8 This provides the exact 1 α cofidece iterval 5 1 r1 r 1 ± 1 t 1;α/ 15 for ρ uder Models ad 3, where t 1;γ deotes the upper γ-quatile of the t 1 distributio. Although the iterval 15 is exact, it is ot a fuctio of the miimal sufficiet statistic s x + s y, y for Model 3 so may be iefficiet; this is cofirmed i.4... Cofidece itervals based o the Model MLE r. Because r has the same amptotic ormal distributio 6 as r 1, r ± 1 r 1 g α/ 16 is a approximate 1 α cofidece iterval for ρ uder Model, hece uder Model 3. Furthermore, the z-trasformatio also applies to r i this case, yieldig the same ormal approximatio: 1 [zr zρ] N0, 1; 17 this gives aother approximate 1 α cofidece iterval for ρ uder Model, hece Model 3: z 1 zr ± g α/ It is perhaps less well kow that Fisher s z-trasformatio i fact applies exactly to r uder Model. The orthogoally-trasformed radom vectors u i 1 = x i + y i, i = 1,...,, 19 1 x i y i v i 5 By regressig x i o y i, a secod exact 1 α cofidece iterval is obtaied by iterchagig ad s y i See Footote 1. Also, the etries for zr i Table 3 show that its variace is approximated much better by 1/ 1 tha by 1/, hece its use i 17. 6

9 have the zero-mea bivariate ormal distributio with covariace matrix 1 + ρσ ρσ σ u 0 0 σv. 0 Thus u i v i, i = 1,...,, ad u u i σuχ, v vi σvχ, 1 u v = 1 + r σ u 1 + ρ F, F,. 1 r 1 ρ σ v Take logarithms to obtai the exact relatio zr zρ + 1 log F, zρ + Z, 3 where Z deotes Fisher s Z distributio with ad degrees of freedom cf. Stuart ad Ord 1987, This yields the followig exact 1 α cofidece iterval for ρ: z 1 zr ± Z ;α/, 4 valid uder Model hece Model 3. Here Z ;γ deotes the upper γ-quatile of Z, which ca be expressed i terms of the γ-quatile of F,. Ulike r 1, r is a fuctio of the miimal sufficiet statistic W s x + s y, y for Model 3 so it may be expected to produce more efficiet estimates tha r 1 i this case. This should be most oticeable whe ρ is small, sice r < r 1 ; see Tables 1 ad..3. Cofidece itervals based o the Model 3 MLE r 3. From 8, for sufficietly large 1 r3 r 3 ± g α/ r 3 is a approximate 1 α cofidece iterval for ρ uder Model 3. As for r 1, however, the ormal approximatio 8 for r 3 is iaccurate uless the sample size is large see Table. This suggests 7

10 seekig a variace-stabilizig traformatio y for r 3 uder Model 3. Startig from 8 ad applyig Mathematica we obtai yρ = 1 + ρ = 1 log dρ 6 1 ρ [ 1 ] + ρ log 1 + ρ + ρ, 1 ρ 1 + ρ ρ 1 + ρ + 1 ρ so that [yr3 yρ] N0, 1 as. 7 Thus a approximate 1 α cofidece iterval for ρ valid uder Model 3 is give by 7 y 1 yr 3 ± g α/. 8 It follows from 6 that y0 = 0, yρ = y ρ atimmetry, ad y ρ 1 so y is strictly icreasig o 1, 1. I fact y ρ z ρ by comparig 6 ad 11, so y icreases faster tha z. This is most marked i the tails, as see from Figure 1: yρ zρ for 0 ρ.5, while 1 < yρ/zρ i ρ for ρ >.5. This form of y is eeded to stabilize the variace of r 3 because its amptotic variace is smaller tha that of r 1 for larger values of ρ, cf. 6 ad 8. Table 3 shows, however, that whe the sample size is small or moderate, y is ot etirely successful at stabilizig the variace of r 3 for ρ ear 0. For example, whe = 10 ad ρ = 0, the actual variace of yr 3 is 33% greater tha the amptotic approximatio 1/ give by 7. Furthermore, i Table 5 it is see that the coverage probability of the cofidece iterval 8 based o yr 3 may deviate oticeably from the omial value 1 α for α =.05 whe ρ is small. This suggests makig a multiplicative adjustmet y ρ m ρ yρ 9 of the y-trasformatio such that m ρ < 1 for ρ ear 0 so that y ρ will icrease slower tha yρ for ρ i that regio, while m ρ 1 for larger values of ρ. 7 Like Varzr 1, Varyr 3 is better approximated by 1 tha by 1 ; see Table 3. 8

11 Table 3: Empirical variace. Estimator ρ zr zr yr y r zr zr yr y r zr zr yr y r This ca be accomplished by a ad hoc choice for m ρ of the form m ρ = 1 a /10 e b yρ c, 30 where a, b, ad c are positive costats chose as described below. Like y, y 0 = 0 ad y ρ is atimmetric ad strictly icreasig o 1, 1. It is see i Figure 1 that as desired, y ρ icreases slower tha yρ for ρ ear 0 ad y ρ yρ outside that regio. Trasformatios Trasformatios ear ρ=0 Trasformatios for ρ z y y ρ ρ ρ Figure 1: Trasformatios z, y, ad y. 9

12 Because y r 3 yr 3 = O p 1, y r 3 has the same amptotic distributio as yr 3, amely [y r 3 y ρ] N0, 1 as, 31 so y 1 y r 3 ± g α/ 3 is also a approximate 1 α cofidece iterval for ρ uder Model 3. We choose a, b, ad c to miimize the maximum differece betwee the empirical coverage probabilities of 3 ad the omial values 1 α across the rages of, α, ad ρ cosidered i Table 4. Specifically, we search for the triple a, b, c that satisfies a, b, c = argmi a,b,c max N, α L, ρ R [ 1 α Pr y ρ y r 3 ± g ] α/, where N = {10, 0, 40}, L = {.01,.05,.1}, ad R = {0,.1,.3,.5,.7,.9}. Due to the o-covexity of the optimizatio problem, first we chose the best three triples a, b, c o the grid A B C, where A = B = C = {.1,.,...,.9, 3.0}. Next, three improved triples were obtaied usig the Nelder-Mead 1965 optimizatio procedure iitialized at each of the first three chose triples. The best of the three improved triples was selected for y : a = 0.403, b = 1.091, ad c = See the Appedix for further details. With this choice of a, b, c for y, Table 4 shows that the coverage probabilities for the 1 α cofidece iterval 3 usig y r 3 are acceptably close to the omial value 1 α for α A, eve though its variace remais somewhat below the omial value 1/ whe ρ = 0 Table 3. For this reaso y might better be called a cofidece-stabilizig trasformatio, rather tha variace-stabilizig..4. Compariso of the iterval estimators for ρ. Table 5 shows the coverage probabilities for the ie 1 α cofidece itervals for ρ preseted i.1,., ad.3 whe α = 0.05, i.e., 95% cofidece. Oly five of these, marked by, attai or adequately approximate the omial 95% level: the exact itervals 15 based o r 1 ad 4 based o zr, ad the approximate 10

13 Table 4: Coverage probabilities for the 1 α-cofidece itervals based o y r 3. α ρ itervals 13 based o zr 1, 18 based o zr, ad 31 based o the modified y-trasform y r 3 but ot the approximate iterval 8 based o yr 3 itself. The average half-widths of these five iterval estimators are show i Table 6 for α =.10,.05,.01. Of these five, 18 ad 4, both based o zr, are most precise for small values of ρ while 31 based o y r 3 is most precise for itermediate ad large values of ρ. The rage of ρ values for which 31 is preferable to 18 ad 4 expads as the sample size icreases, i accordace with the smaller amptotic variace of r 3 except at ρ = 0, as see i 7 ad Comparig two or more correlatios whe the variaces are kow. Suppose that samples of sizes k, k = 1,..., q, are draw from q bivariate ormal distributios with ukow populatio correlatios ρ k ad kow variaces. Just as the z-trasformatio zr 1 is useful for combiig or comparig two or more sample correlatio coefficiets uder Models 1 ad cf. Sedecor ad Cochra 1967, 7.7, the y-trasform yr 3 or its modificatio y r 3 ca be used for these purposes uder Model 3, the kow-variace case. The z-trasform zr ca also be used, especially if small values of the correlatios are expected. i Suppose that ρ 1 = = ρ q ρ ad that it is desired to estimate this commo ρ. If the sample sizes k are large, we ca weight the y-trasforms of the Model 3 MLEs r 1 3,..., r q 3 accordig to their amptotic iverse variaces 1,..., q to obtai the followig weighted 11

14 Table 5: Coverage probabilities of 95% cofidece itervals for ρ largest stadard error The itervals marked have exact or approximate 95% coverage for all three samples sizes. Cofidece Iterval ρ r 10 r 1 ± 1 g α/ z 1 zr 1 ± g α/ r r 1 ± 1 1 t 1;α/ r r ± g α/ z 1 zr ± g α/ z 1 zr ± Z ;α/ r3 r 3 ± g 1+r α/ y 1 yr 3 ± g α/ y r 3 ± g α/ y 1 1 r 0 r 1 ± 1 g α/ z 1 zr 1 ± g α/ r 1 ± y 1 1 r 1 1 t 1;α/ r r ± g α/ z 1 zr ± g α/ z 1 zr ± Z ;α/ r3 r 3 ± g 1+r α/ y 1 yr 3 ± g α/ y r 3 ± g α/ r 40 r 1 ± 1 g α/ z 1 zr 1 ± g α/ r 1 ± y 1 1 r 1 1 t 1;α/ r r ± g α/ z 1 zr ± g α/ z 1 zr ± Z ;α/ r3 r 3 ± g 1+r α/ y 1 yr 3 ± g α/ y r 3 ± g α/

15 Table 6: Average half-width of the five 1 α-cofidece itervals marked i Table 5 largest stadard error The smallest half-width for each combiatio of, ρ ad α is i bold. α Cofidece Iterval z 1 zr 1 ± g α/ 1 r r 1 ± 1 1 t 1;α/ z 1 zr ± g α/ 1 y 1 z 1 zr ± Z ;α/ y r 3 ± g α/ 0 z 1 zr 1 ± g α/ 1 r r 1 ± 1 1 t 1;α/ z 1 zr ± g α/ 1 z 1 zr ± Z ;α/ y r 3 ± g α/ y 1 40 z 1 zr 1 ± g α/ 1 r r 1 ± 1 1 t 1;α/ z 1 zr ± g α/ 1 z 1 zr ± Z ;α/ y r 3 ± g α/ y z 1 zr 1 ± g α/ 1 r r 1 ± 1 1 t 1;α/ z 1 zr ± g α/ 1 z 1 zr ± Z ;α/ y r 3 ± g α/ y 1 0 z 1 zr 1 ± g α/ 1 r r 1 ± 1 1 t 1;α/ z 1 zr ± g α/ 1 z 1 zr ± Z ;α/ ρ

16 y 1 y r 3 ± g α/ 40 z 1 zr 1 ± g α/ r 1 ± z 1 zr ± g α/ 1 y 1 1 r 1 1 t 1;α/ z 1 zr ± Z ;α/ y r 3 ± g α/ z 1 zr 1 ± g α/ 1 r r 1 ± 1 1 t 1;α/ z 1 zr ± g α/ 1 z 1 zr ± Z ;α/ y r 3 ± g α/ y 1 0 z 1 zr 1 ± g α/ 1 r r 1 ± 1 1 t 1;α/ z 1 zr ± g α/ 1 z 1 zr ± Z ;α/ y r 3 ± g α/ y 1 40 z 1 zr 1 ± g α/ 1 r r 1 ± 1 1 t 1;α/ z 1 zr ± g α/ 1 z 1 zr ± Z ;α/ y r 3 ± g α/ y estimator for yρ: ȳ w r 3 = q k=1 k yr k 3 q k=1 k N yρ, 1 q. 33 k=1 k This provides a approximate 1 α cofidece iterval for yρ, which is the iverted to obtai 14

17 a approximate 1 α cofidece iterval for ρ: y 1 ȳ w r 3 ± g α/ q k=1 k. 34 If the sample sizes are small or moderate, however, the ȳ w r 3 should be replaced by ȳ,w r 3 = q k=1 k y kr k 3 q k=1 k N ȳ,w ρ, 1 q, 35 k=1 k where ȳ,w ρ = m,w ρ yρ [ q ] k=1 k m kρ q yρ. 36 k=1 k The 35 provides a approximate 1 α cofidece iterval for ȳ,w ρ, which is strictly icreasig i ρ hece ca be iverted to provide a approximate 1 α cofidece iterval for ρ: ȳ 1,w ȳ,w r 3 ± g α/ q k=1 k. 37 If the sample sizes are equal, i.e., 1 = = q, the 35 ad 37 simplify to ȳ r 3 = 1 q q k=1 y r k 3 N y ρ, 1, 38 q ȳ 1 ȳ r 3 ± g α/. 39 q Lastly, if it is expected that the commo ρ is small, the the fidigs i.4 suggest that yr k 3 be replaced by the z-trasforms zr k of the Model MLEs to obtai a more precise estimate of ρ. Thus 33 would be replaced by recall 17 z w r = q k=1 k 1zr k q k=1 k 1 N zρ, 1 q, 40 k=1 k 1 which provides a approximate 1 α cofidece iterval for zρ that is the iverted to obtai 15

18 a approximate 1 α cofidece iterval for ρ: z 1 z w r ± g α/ q k=1 k Of course, if the uderlyig bivariate ormal data is available from the q populatios, the these data should be combied ito a sigle sample of size q from which a more efficiet estimate of the commo ρ ca be obtaied by the methods of Sectio. However, the weighted estimates obtaied here would still be useful for testig homogeeity of the ρ k as ow described. ii The homogeeity hypothesis H 0 : ρ 1 = = ρ q is equivalet to homogeeity of the y- trasforms: yρ 1 = = yρ q. To test this agaist the geeral alterative, if the sample sizes are large we may reject H 0 for large values of the weighted chi-square statistic T y,w = q k=1 k yr k 3 ȳ w r 3, 4 distributed approximately as χ q 1 uder H 0. If the samples sizes are ot large but equal, i.e., 1 = = q, the H 0 is equivalet to homogeeity of the modified y trasforms, i.e., y ρ 1 = = y ρ q, so T y,w ca be replaced by the statistic T y = q k=1 y r k 3 ȳ r 3, 43 distributed approximately as χ q 1 uder H 0. However, if the sample sizes are ot equal the H 0 is ot equivalet to homogeeity of y 1ρ 1,..., y qρ q, so the weighted test statistic T y,w = q k=1 k y kr k 3 ȳ,w r 3 44 is ot ecessarily appropriate for testig H 0. Fially, if it is expected that ρ 1,..., ρ q are small, the we would reject H 0 for large values 16

19 of the weighted chi-square statistic T z,w = q k=1 k 1 zr k z w r, 45 also distributed approximately as χ q 1 uder H Testig ρ = 0 bivariate idepedece uder Model 3. Fosdick ad Raftery 01 also discussed the problem of testig ρ = 0 vs. ρ > 0 i a sigle bivariate ormal populatio with kow variaces. They cosidered tests that reject H 0 for large values of r 1, r 3, ad r 4 ad their variats, as well as several Bayes tests, determiig the critical of these tests by Mote Carlo simulatio. Here we add a few observatios about these tests ad some others for this testig problem. Exact tests for idepedece are well kow for Models 1 ad. Uder Model 1 the test that rejects ρ = 0 for large values of r 1 is the uiformly most powerful ubiased UMPU test for ρ = 0 vs. ρ > 0 Lehma 1986, Whe ρ = 0 it follows from 14 that 1 r 1 1 r 1 t 1, 46 from which the exact ull distributio of this test is readily obtaied. Uder Model the problem of testig ρ = 0 vs. ρ > 0 is equivalet to the problem of comparig two ormal variaces, i.e., testig σ u = σ v vs. σ u > σ v recall. The F -test that rejects ρ = 0 if 1+r 1 r u v > F,;α is UMPU level α uder Model Lehma 1986, 5.3. Therefore this test is exact for testig ρ = 0 vs. ρ > 0 uder Model 3 as well, ad should perform reasoably well there. Note that σ u + σ v = uder Model 3. Uder Model 3, it follows from 3 that the pdf of x i, y i does ot have mootoe likelihood ratio MLR ad that o uiformly most powerful UMP test exists for ρ = 0 vs. ρ > 0. I fact, for a fixed alterative ρ 1 > 0 the most powerful level α test rejects ρ = 0 iff y > ρ 1s x + s y + c α 47 17

20 where c α is chose to attai size α. Thus the locally most powerful LMP level α test for alteratives ρ 1 0 rejects H 0 iff r 4 = y > c α, whereas the amptotically most powerful AMP level α test for alteratives ρ 1 1 rejects H 0 iff y s x s y > c α, equivaletly, iff v < χ ;1 α recall 19 ad 1. Sice these two tests are differet, o UMP test exists uder Model 3. The exact test based o r has a iterestig albeit limited optimality property uder Model 3. If we set c α = 0 i 47, it follows that the test that rejects ρ = 0 if r > ρ 1 is the MP test of its size for the fixed alterative ρ 1 > 0. By 3 this size is give by αρ 1 Pr[r > ρ 1 ρ = 0] = 1 F, e zρ 1, 48 where F, deotes the cdf of the F, distributio. Values of αρ 1 are show i Table 7. Table 7: Size αρ 1 of the MP test of ρ = 0 vs ρ = ρ 1 that rejects whe r > ρ 1. ρ e-4 1.3e e-7 <1e e-5 8.3e-8 <1e-10 <1e-10 Uder Model 3 the powers of the size.05 tests for ρ = 0 vs. ρ > 0 based o r 1, r, r 3, r 4 the LMP test ad v the AMP test are compared via simulatio 8 i Table 8. The LMP AMP test is domiated i power by the other three tests for all except very small very large values of the alterative ρ 1, so is ot recommeded. The r 1 test is domiated by the r test but oly slightly, which suggests that for testig purposes ot much power is gaied from the kowledge that the variaces are equal. For sample size = 10 the r test domiates the r 3 test for ρ 1.3 while the reverse is true whe ρ 1.5. For sample sizes 0 ad 40 this crossover occurs for ρ 1.1,.3. Thus we recommed either the r or r 3 test uder Model 3, depedig o whether small values or large 8 The size.05 critical values for the tests based o r 3 ad r 4 also were obtaied by simulatio, whereas the exact critical values were used for the other three tests. 18

21 Table 8: Power whe testig ρ = 0 vs ρ > 0 for various alteratives ρ 1 at the α = 0.05 sigificace level ρ Test Rejectio Criterio r 1 > t 1;α r 1 1+r 1 r > F,;α r 3 > c α, r 4 > c α, v < χ ;1 α r 1 > t 1;α r 1 1+r 1 r > F,;α r 3 > c α, r 4 > c α, v < χ ;1 α r 1 > t 1;α r 1 1+r 1 r > F,;α r 3 > c α, r 4 > c α, v < χ ;1 α values of the alterative ρ 1 are of most iterest. Because the r test is exact, it is easier to apply tha the r 3 test, so the r test might be recommeded for Model 3 o this basis. Remark. Uder Model 3, E ρ s x + s y = for all ρ, but s x + s y is ot a acillary statistic. Noetheless it might be of iterest to cosider the coditioal test based o the coditioal distributio of y give s x + s y. Refereces Aderso, T. W A Itroductio to Multivariate Statistical Aalysis, d editio. Wiley, New York. Fosdick, B. K. ad Raftery, A. E. 01. Estimatig the correlatio i bivariate ormal data with kow variaces ad small sample sizes. The America Statisticia, 661: Lehma, E. L Theory of Poit Estimatio. Wiley, New York. 19

22 Lehma, E. L Testig Statistical Hypotheses, d editio. Wiley, New York. Nelder, J. A. ad Mead, R A simplex method for fuctio miimizatio. Computer Joural, 7: Sedecor, G. W. ad Cochra, W. G Statistical Methods. Iowa State Uiversity Press, Ames, Iowa. Stuart, A. ad Ord, J. K. 1987, Kedall s Advaced Theory of Statistics, Volumes 1 ad. Oxford Uiversity Press, New York. Appedix: Choosig a, b, ad c for y The y ρ trasformatio has the form y ρ = 1 a /10 e b yρ c yρ. The costats a, b, ad c are defied as a, b, c = argmi fa, b, c, where a,b,c [ fa, b, c = max 1 α Pr y ρ y r 3 ± g ] α/, N, α L, ρ R N = {10, 0, 40}, L = {0.01, 0.05, 0.1}, ad R = {0, 0.1, 0.3, 0.5, 0.7, 0.9}. The procedure used to search for a, b, c is outlied below; the results associated with each step are also icluded. 1. Fid the three triples ã, b, c o the grid A B C that result i the smallest values of fã, b, c, where A = B = C = {0.1, 0.,...,.9, 3.0}. ã b c fã, b, c

23 . Iitialize the Nelder-Mead optimizatio algorithm i R at each of the ã, b, c triples foud i Step 1 ad obtai three improved triples a, b, c. Iitial Values Improved Values ã b c a b c fa, b, c From the three improved triples obtaied i Step, select that which results i the smallest value of fa, b, c. a, b, c = 0.403, 1.091,