POWER AND SIZE OF NORMAL DISTRIBUTION AND ITS APPLICATIONS

Transcription

1 Jurnal Ilmah Matematka dan Penddkan Matematka (JMP) Vol. 9 No., Desember 07, hal. -6 ISSN (Cetak) : ; ISSN (Onlne) : ; POWER AND SIZE OF NORMAL DISTRIBION AND ITS APPLICATIONS Bud Pratkno Department of Mathematcs, Faculty Mathematcs and Natural Scences Jenderal Soedrman Unversty bpratkto@gmal.com Jajang Department of Mathematcs, Faculty Mathematcs and Natural Scences Jenderal Soedrman Unversty Setanngsh Department of Mathematcs, Faculty Mathematcs and Natural Scences Jenderal Soedrman Unversty R. Sudarwo Open Unversty, UPBJJ-, Purwokerto, Indonesa ABSTRACT. The research studed power and sze of normal dstrbuton and ts applcatons on lnear regresson model. The power and sze formulas are derved, and the unrestrcted test (), restrcted test () and pre-test test (PTT) are used. The recommendaton of the test s gven by choosng mamum power and mnmum sze, and also graphcal analyss. The result showed that the power and sze for large standard devaton ( ) tend to be dentcal and flat. In smulaton study, the graphs of the,, and PTT are stll smlar to the prevous research (Pratkno, 0), where the PTT tend to le between and. Keyword: power, lnear regresson model, and sze. ABSTRAK. Rset n mengkaj power dan sze dstrbus normal dan aplkasnya pada model regres lner. Formula power dan sze dturunkan, dan unrestrcted test (), restrcted test () dan pre-test test (PTT) dgunakan. Rekomendas uj tersebut dberkan dengan memlh power maksmum dan sze mnmum, dan juga analss grafk. Haslnya menunjukkan bahwa power and sze untuk smpangan baku besar ( ) cenderung dentk dan mendatar. Dalam kajan smulas, grafk,, dan PTT mash mrp dengan rset sebelumnya (Pratkno, 0), d mana PTT cenderung berada d antara dan PTT. Kata kunc: power, model regres lner, dan sze

2 Bud Pratkno d.k.k.. Introducton Normal dstrbuton s often called as Gaussan dstrbuton. The probablty densty functon (pdf) of varabel random X wth parameter mean ( ) and varance ( ) s defned as f ( ) e,, and 0. () To compute the pdf and cumulatve dstrbuton functon (cdf) n the equaton (), R-code s used. Due to ths dffcultes, here, the values of the power for rejectng null hypothess (H 0 ) under alternatf hypothess (H ) s also computed usng R- code. The power and sze n term of unvarate normal dstrbuton havel already studed by many authors, such as Pratkno (0), Khan (005) and Yunus (00). Moreover, Pratkno (0) used the power to compute the values of the power of the tests: unrestrcted test (), restrcted test () and pre-test test (PTT) n testng ntercept usng non-sample pror nformaton (NSPI) on model regresson lnear model. Many authors have already contrbuted n developng ths research area especally n estmaton area, such as Khan (005, 008), Khan and Saleh (997, 005, 008), Khan and Hoque (003), and Saleh (006). The values of the power and sze are very sgnfcance crtera n testng, and pre-test test (PTT) on regresson model (RM). Therefore, we used power and sze to test, dan PTT by choosng mamum power and mnmum sze of them. Furthermore, ths research s focused to derve fromula of the power and sze of the normal dstrbuton and compute them usng R-code. To more clear, we also dd graphcal analyss of the power and sze the, and PTT. The research presented the ntroducton n Secton. The reconstructon of the power and sze formula of the normal dstrbuton are gven n Secton. A smulaton study and ther (power and sze) applcatons on the multvarate smple regresson model (MSRM) model are presented n Secton 3, and Secton 4 descrbed concluson

3 Power and Sze of Normal Dstrbuton 3. Power and Sze Normal Dstrbuton Cummulatve dstrbuton functon (cdf) of the normal dstrbuton wth parameter and for and random vaable X s gven as P( X ) f ( ) d e d () To compute the equaton (), R-code s used. Ths s due to the probablty ntegral of the cdf of the equaton () s dffcult. Here, we need numercal analyss and theory of the Maclourn seres. To compute the values of the power and sze for two-sde hypothess versus, the cdf s used. The power (π(µ)) and sze (α ) formulas are then gven as, respectvely, 0! 0! ( u) P(reject H0 under H : ) e d = e d e d ! 0! P(reject H0 under H 0 : 0) e d = e d e d For hypothess versus, 4, range 0 to 4, and, we then got the power s 0,8370, and for the power s 0, 384. Smlarly, the value of the sze under H 0 for we presented graphs of the power and sze for several (3) (4) s 0, Furthermore, as follows

4 4 Bud Pratkno d.k.k. (a) = (b) = (c) =4 (d) = Fgure. The graphs of power and sze of the normal dstrbuton for some selected Fgure. showed that the power tend to be close to sze for to 6, on and when the values of the ncreases. The graphs are smlar to the curve of the normal dstrbuton,.e. the curve tend to be leptocurtc for large the, and they wll be flat for small the. 3. A Smulaton Study of the Power dan Sze on Generate Data Followng Pratkno (0), then the power and sze of the, and PTT on multvarate smple regresson model (MSRM) for 4 dependent varables, namely,, and, wth level of sgnfcance 0,05 n two-sde hypothess H0 : 0 0 versus H0 : 0 0, the graphs of the, and PTT s then obtaned as follows.

5 kuasa kuasa kuasa kuasa Power and Sze of Normal Dstrbuton 5 (a) =0.5 (b) =0.5 PTT, =0. PTT, = (c) =0.5 (d) =0.5 PTT, =0.5 PTT, = Fgure. Power of the,, and PTT for 0,, 0,3, 0,5, 0,7, = 0,5 From Fgure., we see that PTT les between and, t means the PTT can be an alternatve choce of the test. In the contect of maksmum power, the PTT stll follows the prevous research, Pratkno (0). Smlarly, we also got that the alternatve choce of the mmnmu sze of under H Concluson The research studed power and sze of normal dstrbuton and ts applcatons on regresson lnear model. The power and sze formulas of the normal dstrbuton are then reconstructed, and they are used to compute some tests, namely test (), restrcted test () and pre-test test (PTT). The result showed that the power and sze for large standard devaton ( ) tend to be dentcal and flat. In smulaton study, the graphs of the,, and PTT are stll smlar to the prevous research,.e. the PTT tend to le between and. References Khan, S., Estmaton of Parameters of The Multvarate Regresson Model wth Uncertan Pror Informaton and Student-t Errors, Journal of Statstcal Research, 39() (005), 79-94

6 6 Bud Pratkno d.k.k. Khan, S., Shrnkage Estmators of Intercept Parameters of Two Smple Regresson Models wth Suspected Equal Slopes, Communcatons n Statstcs-Theory and Methods, 37 (008), Khan, S. and Saleh, A. K. Md. E., Shrnkage Pre-test Estmator of The Intercept Parameter for A Regresson Model wth Multvarate Student-t Errors, Bometrcal Journal, 39 (997), -7. Khan, S. and Hoque, Z., Prelmnary Test Estmators for The Multvarate Normal Mean Based on The Modfed W, LR and LM Tests, Journal of Statstcal Research, 37 (003), Khan, S. and Saleh, A. K. Md. E., Estmaton of Intercept Parameter for Lnear Regresson wth Uncertan Non-Sample Pror Informaton, Statstcal Papers, 46 (005), Khan, S. and Saleh, A. K. Md. E., Estmaton of Slope for Lnear Regresson Model wth Uncertan Pror Informaton and Student-t Error, Communcatons n Statstcs-Theory and Methods, 37(6) (008), Pratkno, B., Tests of Hypothess for Lnear Regresson Models wth Non Sample Pror Informaton, Dssertaton, Unversty of Southern Queensland, Australa, 0. Saleh, A. K. Md. E., Theory of Prelmnary Test and Sten-Type Estmaton wth Applcatons, John Wley and Sons, Inc., New Jersey, 006. Yunus., R. M., Increasng Power of M-test through Pre-testng, Unpublshed Ph.D. Thess, Unversty of Southern Queensland, Australa, 00.