A Comparison of Two Approximate One-sided Confidence Intervals of a Mean

Size: px

Start display at page:

Download "A Comparison of Two Approximate One-sided Confidence Intervals of a Mean"

Virginia Manning
5 years ago
Views:

1 Official Statistics i Hoour of Daiel Thorbur, A Comariso of Two Aroximate Oe-sided Cofidece Itervals of a Mea Per Gösta Adersso 1 Oe-sided Wald itervals, costructed from asymtotic argumets, usually exhibit coverage robabilites which are far from the omial cofidece levels if the uderlyig distributio is ot ormal. Two recetly roosed rocedures for adjustig the Wald iterval leadig to differet, but related cofidece itervals will here be reseted ad comared. These itervals take ito accout ad adjust for skewess, leadig to coverage robabilites closer to the omial levels. Key words: Correlatio; ivotal; skewess. 1. Itroductio I Jue 2000 I atteded the secod Iteratioal Coferece o Establishmet Surveys (ICES II) i Buffalo, New York. The mai urose of my articiatio was to reset the aer A balaced adjusted cofidece iterval rocedure alied to fiite oulatio samlig, which was a joit work with Olle Nerma. The discussat Phil Kott soo became ethusiastic about the adjustmet idea ad immediately after the coferece we iitiated a based collaboratio. The first result was Two-sided coverage itervals for small roortios based o survey data, which was reseted by Phil at the Federal Committee o Statistical Methodology Research Coferece (FCSM), Arligto, Virgiia i We the cotiued with a modificatio of the adjusted iterval ivolvig calculatio of degrees of freedom for the uderlyig ivotal, thus makig the iterval t-based. This resulted i the aer Two-sided coverage itervals for a estimator of a small roortio based o a simle radom cluster samle (ot ublished). Our cooeratio the ufortuately faded, mostly due to lack of time. Phil moved o (together with Ya Liu) ad roduced Evaluatig alterative oe-sided coverage itervals for a biomial roortio i I, o the other had, ublished i 2004 Alterative cofidece itervals for the total of a skewed biological oulatio, which is based o the same idea as the Buffalo aer, with a real-data alicatio which I got from Timothy Gregoire, who also articiated i that coferece. The geeral adjustmet aroach the got full attetio i A simle correlatio adjustmet rocedure alied to cofidece iterval costructio (2009). Now our aths cross agai, it seems, sice Phil s achievemets (with Ya Liu) Oesided coverage itervals for a roortio estimated from a stratified radom samle ad 1 Deartmet of Statistics, Swedish Busiess School, Örebro Uiversity, Örebro, Swede. ergosta@gmail.com

2 62 Official Statistics i Hoour of Daiel Thorbur Seedig u the asymtotics whe costructig oe-sided coverage itervals with survey data cotai, besides a mixture of Edgeworth ad Taylor exasios, the use of the same alterative variace estimator as for the adjustmet i e.g., Adersso (2009). This leads to cofidece iterval exressios closely related to what is obtaied i Adersso (2009). The itetio of this aer is to exlicitly show how the resultig itervals relate both i theory ad by simulatios for the case where we cosider itervals for a mea based o i.i.d. cotiuous radom variables. 2. The Two Alterative Procedures Cosider a i.i.d. radom samle X 1 ;:::;X from some cotiuous distributio with mea m ad fiite variace s 2. (I fact, to be o the safe side for the rocedures to be reseted, we assume a fiite sixth momet for the distributio of X). The stadard Wald-iterval is based o the ivotal X 2 m S= ffiffiffi ð1þ where S is the samle stadard deviatio {1=ð 2 1Þ} P i21 ðx i 2 XÞ 2 1=2, as a estimator of s. O the assumtio of aroximate ormality of (1), we get from observed values x 1 ;:::;x the 100(1 2 a) % uer ad lower bouded itervals x þ zs= ffiffi ffiffiffi ad x 2 zs= ð2þ where z is the 1 2 a -quatile of the stadard ormal distributio, i.e., z ¼ F 21 ð1 2 aþ. (Alteratively, the corresodig quatile from the t-distributio with 2 1 degrees of freedom is used). This aroximate ormality assumtio is ofte dubious though, eve if it would hold reasoably well for X. For istace, uderlyig skewess of X causes X ad S to be correlated. Refiemet of the Wald iterval may be obtaied by usig Edgeworth exasios ad/or adjustig for the iduced correlatio. Followig Adersso (2009), we first ote that X is ucorrelated with S 2 = 2 Kð X 2 mþ, where K ¼ Covð X; S 2 =Þ=Varð XÞ. We also have that VarðS 2 = 2 Kð X 2 mþþ ¼ VarðS 2 =Þ 2 K 2 Varð XÞ. Now, K ¼ðm 3 = 2 Þ=ðs 2 =Þ ¼gs=, where m 3 is the third cetral momet of X ad g ¼ m 3 =s 3=2 is the skewess coefficiet of X. K eeds to be estimated ad ^K ¼ ^gs=; where ^g ¼ ^m 3 =s 3=2 P ad ^m 3 ¼ 1=ð 2 3 þ 2=Þ i¼1 ðx i 2 XÞ 3 (ubiased estimator of m 3 ). The ANNE cofidece iterval 2 is the derived by ivertig the ivotal (assumed to be aroximately ormally distributed) 2 Actually Phil later amed the iterval, as reseted by me at the Buffalo coferece, the AN (Adersso- Nerma) iterval. A associate editor thought, however, that this should stad for Asymtotically Normal, so ANNE (ANdersso-NErma) is ow the official ame. (As it turs out, Ae is also the ame of Olle Nerma s wife!).

3 Adersso: Oe-sided Cofidece Itervals of a Mea 63 X 2 m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S 2 = 2 ^Kð X 2 mþ leadig to the uer ad lower bouded itervals 21; x þ z 2 2 ^gs þ z s r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ffiffi 1 þ z 2 4 ^g 2 ð3þ ð4þ ad x þ z 2 2 ^gs 2 z s rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ffiffi 1 þ z 2 4 ^g 2 ; 1 ð5þ As oe would exect, emirical ivestigatios i e.g., Adersso (2009) have show that the adjusted ivotal (3) has less skewess tha (1), ad this meas cofidece itervals which have coverage robabilities closer to the chose cofidece levels. Kott ad Liu cosider a fiite oulatio samlig situatio, but their aroach also works for the reset situatio. It starts with a combiatio of a oe-term Edgeworth exasio for X ad a first order Taylor aroximatio of F(z), leadig to X P 2 m s= ffiffi # z z 2 t < FðzÞ 6 where t ¼ E X 2 mþ 3 =Varð XÞ 3=2 ¼ K=ðs= ffiffiffi Þ, the skewess coefficiet of X. Takig squares ad exadig the yields P ðm 2 XÞ z 2 3 Kðm 2 XÞ # z 2 s 2 Þ < FðzÞ ad if we estimate s 2 = by S 2 = 2 ^Kð X 2 mþ,the followig uer/lower bouded itervals are obtaied: 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 21; x þ 1 6 þ z 2 ^gs 3 þ z s ffiffi 1 þ 1 6 þ z 2 ^g 2 A ð6þ 3 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x þ 1 6 þ z 2 ^gs 3 þ z s ffiffi 1 þ 1 6 þ z 2 ^g 2 3 ; 1 A The ANNE ad Kott-Liu iterval exressios thus are closely related; ad if we look at other reviously suggested itervals, we see that the Kott-Liu iterval is i its first art exactly the same as the iterval due to Abramovitch ad Sigh (1985), which is a refiemet of a iterval by Johso (1978). The uer boud of the Abramovitch ad Sigh iterval is x þ 1 6 þ z 2 ^gs 3 þ z s ffiffi ð7þ

4 64 Official Statistics i Hoour of Daiel Thorbur Now, the ANNE ad Kott-Liu iterval-bouds share the same structure: x þ A ^gs ^ z s rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi 1 þ A ^g 2 2 Sice z 2 =ð2þ. ð1þ=ð6þþðz 2 Þ=ð3Þ; for z. 1 (which holds for ay reasoable cofidece level), we ca immediately coclude that the coverage robability, for the uer bouded ANNE Iterval is higher. The coverse is true for a lower bouded iterval, sice it ca be show that, for a fixed samle, the lower boud (like the uer boud) is a icreasig fuctio of A. 3. A Simulatio Study I order to emirically illustrate roerties of cofidece itervals for the mea m a modest simulatio study was coducted. Samles were geerated from three ositively skewed distributios: chi-square ðx 2 ð f ÞÞ with f ¼ 4; f ¼ 1 ad f ¼ 0:5 degrees of freedom. The skewess coefficiet g for these distributios were: 1.4, 2.8 ad 4.0. The chose samle sizes were 10, 50, ad 100. For each combiatio of distributio ad samle size, 10,000 samles were geerated usig MATLAB, versio 6.0. The cofidece itervals studied are the Wald itervals (2), the Kott-Liu itervals (6) ad (7) ad the ANNE itervals (4) ad (5). The omial level for lower/uer edoit itervals was 95% throughout. As dislayed i Table 1, the Wald itervals have for most cases highly iaccurate coverage robabilities, which to a great deal is a effect of skewess of the corresodig ivotal (1), caused by the correlatio betewee X ad S. As exected, the Kott-Liu ad ANNE itervals both have substatially higher/lower coverage for uer/lower bouded itervals tha the Wald itervals. While ot beig as coservative as the Wald itervals for Table 1. Coverage rates give as ercetages of the Wald, Kott-Liu ad ANNE lower/uer edoit cofidece itervals with omial level 95%, for samles geerated with sizes ¼ 10; 50 ad 100 from x 2 ð4þ; x 2 ð1þ ad x 2 ð0:5þ distributios. Give is also the correlatio coefficiet r for X ad S 2. For each combiatio of distributios ad samle sizes, 10,000 samles were geerated ¼ 10 ¼ 50 ¼ 100 Lower Uer Lower Uer Lower Uer x 2 ð4þ, g ¼ 14 rð X; S 2 Þ¼0:63 Wald Kott-Liu ANNE x 2 ð1þ, g ¼ 2:8 rð X; S 2 Þ¼0:76 Wald Kott-Liu ANNE x 2 ð0:5þ, g ¼ 4:0 rð X; S 2 Þ¼0:78 Wald Kott-Liu ANNE

5 Adersso: Oe-sided Cofidece Itervals of a Mea 65 the lower bouded case, the Kott-Liu itervals have (with oe excetio) a coverage above the omial level 95%. The lower bouded ANNE itervals, o the other had always have coverage less tha 95%, ad eve though the ANNE itervals rovide the best uer limit coverage, we rarely reach the omial level. 4. Cocludig Remarks Comared to Wald s iterval, the Kott-Liu ad ANNE itervals have, for the cases studied here, coverage robabilites closer to the omial levels. I order to imrove uo these results, we may furthermore estimate effective degrees of freedom ad make the alterative itervals t-based istead of z-based, as doe i Kott ad Liu (2009a). (I Adersso (2009) the ANNE iterval is actually t-based with 2 1 degrees of freedom). Prelimiary results i this directio are romisig. 5. Refereces Abramovitch, L. ad Sigh, K. (1985). Edgeworth Corrected Pivotal Statistics ad the Bootstra. The Aals of Statistics, 13, Adersso, P.G. (2004). Alterative Cofidece Itervals for the Total of a Skewed Biological Poulatio. Ecology, 85, Adersso, P.G. (2009). A Simle Correlatio Adjustmet Procedure Alied to Cofidece Iterval Costructio. The America Statisticia, 63, Adersso, P.G. ad Nerma, O. (2001). A Balaced Adjusted Cofidece Iterval Procedure Alied to-fiite Poulatio Samlig. Proceedigs from ICES II Iteratioal Coferece o Establishmet Surveys, Buffalo, N.Y., U.S.A. Johso, N.J. (1978). Modified t Tests ad Cofidece Itervals for Asymmetrical Poulatios. Joural of the America Statistical Associatio, 73, Kott, P.S. ad Liu, Y.K. (2009a). Oe-sided Coverage Itervals for a Proortio Estimated from a Stratified Radom Samle. Iteratioal Statistical Review, 77, Kott, P.S. ad Liu, Y.K. (2009b). Seedig u the Asymtotics whe Costructig Oesided Coverage Itervals with Survey Data. Techical Reort. Kott, P.S., Adersso, P.G., ad Nerma, O. (2001). Two-sided Coverage Itervals for Small Proortios Based o Survey Data. Proceedigs from FCSM (Federal Committee o Statistical Methodology Research Coferece), Arligto, VA, U.S.A. Liu, Y.K. ad Kott, P.S. (2007). Evaluatig Alterative Oe-sided Coverage Itervals for a Biomial Proortio. Submitted to the Joural of Official Statistics.

Confidence Intervals

Confidence Intervals Cofidece Itervals Berli Che Deartmet of Comuter Sciece & Iformatio Egieerig Natioal Taiwa Normal Uiversity Referece: 1. W. Navidi. Statistics for Egieerig ad Scietists. Chater 5 & Teachig Material Itroductio