# EDGEWORTH SIZE CORRECTED W, LR AND LM TESTS IN THE FORMATION OF THE PRELIMINARY TEST ESTIMATOR

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4 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

5 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 ) 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

7 2.5 = ad origial tests = ad size corrected tests Bias.5 Bias = 3 ad origial tests.4.2 = 3 ad size corrected tests Bias.8.6 Bias 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

8 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 α =.5 ad origial tests.8.6 α =.5 ad size corrected tests α =. ad origial tests α =. ad size corrected tests 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

12 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, [4] ald, A. 943). Tests of Hypotheses Cocerig Several Parameters he the Number of Observatios is Large, Trasactio of the America Math Society, 54,

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