Joural of Statistical Research 26, Vol. 37, No. 2, pp. 43-55 Bagladesh ISSN 256-422 X EDGEORTH SIZE CORRECTED, AND TESTS IN THE FORMATION OF THE PRELIMINARY TEST ESTIMATOR Zahirul Hoque Departmet of Statistics Faculty of Busiess ad Ecoomics Uited Arab Emirates Uiversity, P.O. Box 7555, Al Ai, UAE E-mail: zahirul-hoque@yahoo.com Shahjaha Kha Departmet of Maths ad Computig Uiversity of Souther Queeslad, Toowoomba, Australia Email: khas@usq.edu.au & Baki Billah Departmet of Epidemiology ad Prevetive Medicie Moash Uiversity, Clayto, Victoria, Australia Email: Baki.Billah@med.moash.edu.au Full Referece: Hoque, Z., Kha, S. ad Billah, B. 23). Prelimiary test estimators for the multivariate ormal mea based o the modified, ad tests, Joural of Statistical Research, 37, 43-55. summary This paper defies the prelimiary test estimator PTE) of the uivariate ormal mea uder the origial as well as the Edgeworth size corrected ald ), likelihood ratio ) ad Lagrage multiplier ) tests. The bias ad mea squared error ) fuctios of the estimators are derived. The coflicts amog the biases ad the s of the PTEs uder the three origial ad the size corrected tests have bee obtaied. It is foud that istead of the origial, ad tests, the use of the Edgeworth size corrected, ad tests i the formatio of the PTEs reduces the coflict amog the biases ad s of the estimators remarkably. Keywords ad phrases: Prelimiary test estimator; ald, likelihood ratio ad Lagrage multiplier tests; bias; mea squared error; ad coflict. Correspodig author
AMS Classificatio: Primary 62H2, Secodary 62G5. Itroductio The ormal distributio is the most widely used statistical model for may real life pheomea, ad hece the estimatio of its ukow mea is very importat. The most commoly used estimator of the mea of a ormal distributio is the maximum likelihood estimator. This estimator is exclusively based o the sample iformatio ad posses some good statistical properties. Ofte credible o-sample prior iformatio about the value of the mea is available from previous experiece or expert kowledge. Iclusio of such iformatio i the formatio of the estimator is likely to improve some of the statistical properties of the estimator. The o-sample prior iformatio ca be expressed i the form of a ull hypothesis. The value of the parameter specified by such a ull hypothesis is kow as the restricted estimator RE). Ulike the MLE. the RE is biased. But uder the ull hypothesis it performs better tha the MLE. As we are ot sure about the accuracy of the prior iformatio, the performace of RE is i doubt. Uder the alterative hypothesis, the MLE performs better whe we use the sample iformatio oly, ad the RE performs better whe the ull hypothesis is true. Therefore, a estimator that combies the sample ad o-sample prior iformatio is likely to perform better tha both MLE ad RE. Prelimiary test estimator, origially proposed by Bacroft [], combies both sample ad o-sample prior iformatio i its defiitio. By defiitio the PTE ivolves a appropriate test statistic to remove the ucertaity i the ull hypothesis. Traditioally, the likelihood ratio ) test statistics is used i the defiitio of the PTE. But there are alterative tests available i the literature to test the same hypothesis. Tests such as ald ) ald [4]), ad Lagrage multiplier ) Rao []) tests ca be used to defie PTE. Billah [3], ad i a series of papers Billah ad Saleh [4], [5] ad [6]) itroduce the, ad tests i the formatio of the SPTE ad PTE of the regressio vector of liear multiple regressio model. Berdt ad Savi [2] show that the systematic iequality relatio exists amog the three test statistics. The exact samplig distributios of these test statistics are complicated. I practice, the critical regios are determied based o their commo asymptotic distributio. Uder the ull hypothesis, the three test statistics are asymptotically distributed as chi-square. Eva ad Savi [8] show that i small sample the three tests lead to coflictig coclusios. The previous studies show that the performace of the SPTEs uder these coflictig tests are also coflictig. As a result, the practitioers of the prelimiary test estimator might be i a dilemma as to which of the three tests should be used to achieve the optimal statistical properties. 2
To reduce the coflict amog the tests Eva ad Savi [8] proposes some modificatios for the three test statistics. They use the degrees of freedom correctio Gallat [9]) for the ad tests, ad Edgeworth correctio see Rotheberg []) for the test. The study of Evas ad Savi [8] shows that the amout of coflict amog the modified tests is smaller tha that amog the origial tests. But the amout of coflict amog the modified tests is still substatial. The coflict amog the three tests with or without modificatio is due to the fact that the tests do ot have the correct sigificace level. Based o the Edgeworth expasios of the exact distributios of the test statistics, ad uder the ull hypothesis Evas ad Savi [8] cosider some correctio factors for the chi-square critical values which make the sigificace levels of each test correct to certai order. For details of the correctios readers may see Rotheberg []. The tests with the adjusted critical values are kow as the size corrected tests. The amout of coflict amog the size corrected tests is egligible cf. Evas ad Savi [8]. I this study we itroduce the size corrected, ad tests i the formatio of the PTEs of the uivariate ormal mea. Our study reveals that the amout of coflict amog the properties of the PTEs uder the size corrected tests is very small, practically egligible. Therefore, the practitioers of the PTE ca use ay of the three size corrected tests, rather tha the origial ad modified tests, i the formatio of the PTE. This will guard the users agaist the risk of coflictig decisios that may arise while usig the origial or modified tests i the defiitio of the PTE. The orgaizatio of the paper is as follows. The tests ad the PTEs are preseted i sectio 2. The bias fuctios of the PTEs are studied i sectio 3. Sectio 4 deals with the of the PTEs. Fially, some cocludig remarks are provided i sectio 5. A brief derivatio of the bias ad fuctios is give i the appedix. 2 The Tests ad the Estimators Let x, x 2,, x be a radom sample of size from a uivariate ormal populatio with ukow mea µ ad variace σ 2. The above sample observatios ca be expressed i the followig form X = µ + e 2.) where X is a vector of sample observatios, =,..., ) is a vector of - tuple of s, µ is the ukow mea, ad e = e,..., e ) is a vector of idepedet error compoets distributed as N, σ 2 I ). Here I is a idetity matrix of order. Based o the sample iformatio the maximum likelihood estimator of µ is defied as µ = ) X = X 2.2) where X is the sample mea. The samplig distributio of µ is ormal with mea µ ad variace σ 2 /. Hece, the MLE of µ is ubiased, ad it s mea squared error 3
is the same as it s variace. The MLE of σ 2 is give by σ 2 = X µ ) X µ ) 2.3) which is biased, ad the ubiased estimator of σ 2 ca be obtaied by replacig the deomiator of 2.3) with ). The distributio of the ubiased estimator of σ 2 is scaled chi-square with ) d.f. Followig Fishers recipe, the o-sample prior iformatio is expressed i the form of the ull hypothesis as follows: H : µ = µ 2.4) where µ is the suspected value of the ukow mea µ. To remove the ucertaity i the prior iformatio a appropriate statistical test is performed to test the ull hypothesis i 2.4) agaist the alterative H : µ µ. 2.5) The above hypotheses i 2.4) - 2.5) ca be tested by usig the three origial, ad test statistics = F 2.6) = l + F ) 2.7) F = 2.8) + F respectively. Uder the alterative hypothesis the distributio of F is o-cetral F with ad ) d.f., ad o-cetrality parameter 2 = µ µ ) 2 /σ 2. The asymptotic distributio of the above three test statistics are the same, ad is chisquare with d.f. cf. Evas ad Savi [8]). I this study, we defie the PTE of µ based o the asymptotic distributio of the three test statistics. The PTE of µ uder the test statistics i 2.6) - 2.8) are defied as ˆµ PT = µ µ ˆµ) I < χ 2 α) 2.9) ˆµ PT = µ µ ˆµ) I < χ 2 α) 2.) ˆµ PT = µ µ ˆµ) I < χ 2 α) 2.) where χ 2 α is the critical value of the test at sigificace level α, I ) is the idicator fuctio which assumes value uity whe the iequality i the argumet holds, ad otherwise. 4
The modified form of the, ad test statistics are = F 2.2) = 3/2) l + F ) 2.3) F = 2.4) + F respectively. The asymptotic distributio of the modified test statistics are also chisquare with d.f. Usig Edgeworth expasios of the exact distributios of the test statistics uder the ull hypothesis the adjusted critical values of the test statistics i 2.2) - 2.4) are obtaied as { } C = χ 2 α + χ2 α + 2.5) 2 ) C = χ 2 α 2.6) { } C = χ 2 α + χ2 α 3. 2.7) 2 ) The tests with the modified test statistics ad the adjusted critical values are kow as the size corrected tests. Evas ad Savi [8] show that the amout of coflict amog the size corrected tests is egligible. Based o the size corrected tests the PTEs of µ are defied as ˆµ PT * = µ µ ˆµ) I < C ) 2.8) ˆµ PT * = µ µ ˆµ) I < C ) 2.9) ˆµ PT * = µ µ ˆµ) I < C ) 2.2) where C ) is the critical value of the respective size corrected test, ad I ) is the idicator fuctio as defied earlier. 3 The Bias Fuctio I this sectio we state the bias fuctios of the PTEs of µ uder the origial as well as the Edgeworth size corrected, ad tests. The coflict amog the biases of the PTEs are obtaied ad preseted graphically. Some aalytical commets o the biases are provided at the ed of this sectio. The bias fuctios of the estimators based o the origial tests are give by Bˆµ PT ; µ) = δ G 3, C ; ) 3.) Bˆµ PT ; µ) = δ G 3, C ; ) 3.2) Bˆµ PT ; µ) = δ G 3, C ; ) 3.3) 5
respectively where C i = )χ2 α 2i+), C i = )eχ 2 α / ) 2i+) ad C i = )χ2 α 2i+) χ 2 ), α i =, 2; G p,q a; b) is the distributio fuctio of the o-cetral F variable with p, q) d.f., o-cetrality parameter b, ad evaluated at a. See appedix for the derivatio of the above bias fuctios. The bias fuctios of the estimators of µ based o the three size corrected tests i 2.8) - 2.2) are give by Bˆµ PT *; µ) = δ G 3, C ; ) 3.4) Bˆµ PT *; µ) = δ G 3, C ; ) 3.5) Bˆµ PT *; µ) = δ G 3, C ; ) 3.6) respectively where C i = χ2 αχ 2 α+2 ) 22i+) ), C i = )eχ2 α / 3/2) ) 2i+) ad C i = χ 2 α χ2 α +2 5) ) 2i+){2 ) χ 2 αχ 2 α+2 5)}, i =, 2; G p,qa; b) is the distributio fuctio of the o-cetral F variable with p, q) d.f., o-cetrality parameter b ad evaluated at a. From Figure it is observed that for = the biases of the PTEs uder both origial ad size corrected tests are zero regardless of the sample size. As deviates from zero, the biases grow larger up to some moderate value of, ad the the biases decrease ad merge to zero at some large values of. Iterestigly, i both origial ad size corrected cases, the same iequality relatio exists amog the biases of the PTEs as their tests couterparts. Uder both origial ad size corrected tests, as the sample size icreases, the coflict amog the biases of the PTEs decreases. This is due to the fact that the coflict amog the tests is more ievitable for small sample, ad hece amog the PTEs. The origial, ad tests do ot have the correct sigificace level. The Edgeworth size correctio esures that the tests have sigificace level correct to order /. As a result, the use of the size corrected tests i the formatio of the PTEs cosiderably reduces the coflict amog the biases of the three PTEs as compare to the use of the three origial tests. 4 The Mea Squared Error Fuctio This sectio states the fuctios of the PTEs of µ uder both origial ad Edgeworth size corrected, ad tests. The amout of coflict amog the s of the three PTEs are obtaied ad preseted i both graphical ad umerical forms. Some aalytical commets o the s are also provided i this sectio. 6
2.5 = ad origial tests 2.5 2 = ad size corrected tests Bias.5 Bias.5.5 2 3 2 3.4.2 = 3 ad origial tests.4.2 = 3 ad size corrected tests Bias.8.6 Bias.8.6.4.4 2 3 2 3 Figure : The bias of the PTE of µ for α =.5 ad varyig. The fuctios of the PTEs of µ uder the origial, ad tests are obtaied as ˆµ PT ; µ) = σ2 [ G3, C ; ) + 2 {2G 3, C ; ) G 5, C 2 ; )}] 4.) ˆµ PT ; µ) = σ2 [ G3, C ; ) + 2 {2G 3, C ; ) G 5, C 2 ; )}] 4.2) ˆµ PT ; µ) = σ2 [ G3, C ; ) + 2 {2G 3, C ; ) G 5, C 2 ; )}] 4.3) 7
where C i = )χ2 α 2i+), C i = )eχ 2 α / ) 2i+) ad C i = )χ2 α 2i+) χ 2 α ), i =, 2; G p,q a; b) is the distributio fuctio of the o-cetral F variable with p, q) d.f., o-cetrality parameter b ad evaluated at a. See appedix for the derivatio..7.6.5 α =.5 ad origial tests.8.6 α =.5 ad size corrected tests.4.3.4. 2 3 4 2 3 4.3 5.5 α =. ad origial tests.3 5.5 α =. ad size corrected tests...5 2 3 4.5 2 3 4 Figure 2: The of the PTE of µ for =, σ = ad varyig α. The fuctios of the PTEs of µ uder the size corrected, ad tests 8
are obtaied as ˆµ PT *; µ) = σ2 [ G3, C ; ) + 2 { 2G 3, C ; ) G 5, C 2 ; )}] 4.4) ˆµ PT *; µ) = σ2 [ G3, C ; ) + 2 { 2G 3, C ; ) G 5, C 2 ; )}] 4.5) ˆµ PT *; µ) = σ2 [ G3, C ; ) + 2 { 2G 3, C ; ) G 5, C 2 ; )}] 4.6) respectively where C i = χ2 α χ2 α +2 ) 22i+) ), C i = )eχ2 α / 3/2) ) 2i+) ad C i = χ 2 αχ 2 α+2 5) ) 2i+){2 ) χ 2 αχ 2 α+2 5)}, i =, 2; G p,qa; b) is the distributio fuctio of the o-cetral F variable with p, q) d.f., o-cetrality parameter b ad evaluated at a. From Tables ad 2, ad Figures 2 ad 3 it is observed that for ay fixed, α, ad σ, from zero to some small value of say ) the of the PTE uder the Edgeworth size corrected tests is less tha that of the PTE uder the origial tests. For >, the of the PTE uder the origial tests is less tha that of the PTE uder the size corrected tests. However, the coflict amog the s of the PTEs uder the size corrected tests is less tha that uder the origial tests. Moreover, the amout of coflict amog the PTEs uder the size corrected tests is egligible. For example, whe =, α =., σ = ad = 3, the amout of coflict amog the s uder the origial tests is 2. O the other had, for same, α, σ ad, the amout of coflict amog the PTEs uder the size corrected tests is oly.. The performace of the PTE depeds o the level of sigificace of the test o the ull hypothesis. For a optimum choice of the level of sigificace readers may see Chiou ad Saleh [7]. As the sample size icreases, the of the PTEs decreases regardless of the test used i the formatio of the estimator. Similarly, the amout of coflict amog the PTEs also decreases with the icrease i sample size. For example, whe α =.5, σ =, =, = 2, the origial tests used i the formatio of the PTEs, the amout of coflict amog their s is.2. O the other had, for the same values of α, σ, ad tests used, the amout of coflict amog the s of the PTEs is decreased by.86 for doublig the sample size. 5 Coclusio e have defied the PTE of µ uder the three origial as well as the size corrected tests. The coflictig bias ad mea squared error fuctios of the estimators are 9
.7.6.5 = ad origial tests.8.6 = ad size corrected tests.4.3.4. 2 3 4.35.3 5.5..5 = 2 ad origial tests 2 3 4.35.3 5.5..5 = 2 ad size corrected tests 2 3 4 2 3 4 Figure 3: The of the PTE of µ for α =.5 σ = ad varyig. derived. The fidigs of this study show that the coflict amog the PTE uder the, ad tests with or without modificatio) is because of the fact that the tests do ot have the correct sigificace level. For the Edgeworth size corrected tests the coflict is reduced sigificatly. The bias ad based coflicts reflect the same behaviours. Our fidigs also illustrate the dager of sizeable coflict i usig the origial, ad tests i the formatio of the PTE. e observe that whe Edgeworth correctio is applied o the three tests it ot oly reduces the coflict of test decisios for testig hypotheses o µ, but also reduces the coflict amog the PTEs, based o such correctios, with respect to both bias ad criteria. Although use of such size corrected tests does ot remove the coflict of the PTEs altogether, it reduces the coflict sigificatly. The size of the coflict is so egligible that it ca be igored for almost all practical problems. Therefore, use of ay of the size corrected, ad tests i the formatio of the PTE would lead to almost idetical bias ad for the PTEs. Thus the users may use
the liberty to use ay of the three size corrected tests without riskig of icreased coflict i the performace of the PTEs. Refereces [] Bacroft, T.A. 944). O biases i estimatio due to the use of the prelimiary test of sigificace. A. Math. Statist., 5, 9-24. [2] Berdt, E. ad Savi, N.E. 977). Coflict Amog Criteria for Testig Hypotheses i the Multivariate Liear Regressio Model, Ecoometrica, 45, 263-278. [3] Billah, M.B. 997). Small Sample Studies of the,, ad Based SPTEs of Regressio Parameters, Joural of Statistical Research, 32), 93-8. [4] Billah, M.B. ad Saleh, A.K.Md.E. 998). Coflict Betwee Pretest Estimators Iduced by Three Large Sample Tests Uder a Regressio Model ith Studets t-error, The Statisticia, 474), 593-66. [5] Billah, M.B. ad Saleh, A.K.Md.E. 2a). Performace of Three Large Sample Tests i the Actual Formatio of the Pretest Estimators for Regressio Model ith t-error, Joural of Applied Statistical Sciece, 93), 237-25. [6] Billah, M.B. ad Saleh, A.K.Md.E. 2b). Performace of the PTE Based o the Coflictig, ad Tests i Regressio Model, Advaces o Methodological ad Applied Aspects of Probability ad Statistics, N. Balakrisha ed.), Gordo ad Breach Publishig Compay, 263-276. [7] Chiou, P. ad Saleh, A.K.Md.E. 22). Prelimiary test cofidece sets for the mea of a multivariate ormal distributio, Joural of Propagatio i Probability ad Statistics, 2, 77-89. [8] Evas, G.B.A. ad Savi, N.E. 982). Coflict Amog the Criteria Revisited: the, ad tests, Ecoometrica, 53), 737-748. [9] Gallat, A.R. 975). Seemigly Urelated Noliear Regressio, Joural of Ecoometrics, 3, 35-5. [] Rao, C.R. 947). Large Sample Tests of Statistical Hypotheses Cocerig Several Parameters with Applicatios to Problems of Estimatio, Proceedigs of the Cambridge Philosophical Society, 44, 5-57. [] Rotheberg, T.J. 977). Edgeworth Expasios for Multivariate Test Statistics, orkig Paper i Ecoomic Theory ad Ecoometrics IP-255, Ceter for
Research i Maagemet Sciece, Istitute of Busiess ad Ecoomic Research, Uiversity of Califoria, Berkeley. [2] Saleh, A.K.Md.E. 26). Theory of Prelimiary Test ad Stei-Type Estimatio with Applicatios, iley, New York. [3] Savi, N.E. 976). Coflict Amog Testig Procedures i a Liear Regressio Model ith Autoregressive Disturbaces, Ecoometrica, 44, 33-35. [4] ald, A. 943). Tests of Hypotheses Cocerig Several Parameters he the Number of Observatios is Large, Trasactio of the America Math Society, 54, 426-482. 2