Interval Estimation for a Linear Function of. Variances of Nonnormal Distributions. that Utilize the Kurtosis

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1 Appled Mathematcal Scences, Vol. 7, 013, no. 99, HIKARI Ltd, Interval Estmaton for a Lnear Functon of Varances of Nonnormal Dstrbutons that Utlze the Kurtoss Srma Suwan Department of Appled Statstcs, Faculty of Appled Scence Kng Mongkut s Unversty of Technology North Bangkok, Thaland, 1050 srmasuwan@hotmal.com Sa-aat Nwtpong Department of Appled Statstcs, Faculty of Appled Scence Kng Mongkut s Unversty of Technology North Bangkok, Thaland, 1050 snw@kmutnb.ac.th Copyrght 013 Srma Suwan and Sa-aat Nwtpong. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Abstract Confdence nterval for a lnear combnaton of varances from k ndependent non-normal populatons that utlze kurtoss va the method of varance estmates recovery (the MOVER) and ts applcaton to a general lnear functon of parameters presented by Zou et al.[6] s proposed. Our method wll compare to the exstng confdence nterval va Monte Carlo smulaton. The coverage probablty and the average nterval wdth are used to assess the confdence ntervals. Mathematcs Subject Classfcaton: 6F5 Keywords: Mnmum mean squared error, The MOVER, Kurtoss, Interval estmaton, Lnear combnaton

2 4910 Srma Suwan and Sa-aat Nwtpong 1 Introducton Let X 11,...,X 1n1,X 1,...,X n,x k1...x kn k,be m contnuous ndependent samples, each sample beng dentcal ndependent wth any dstrbuton functon G(x), mean μ, varance σ and fnte fourth moments γ 4 for 1,,,k. The n sample means and varances are X Xj n, 1,,...,k and n j 1 j S (X X ) (n 1), 1,,...,k, respectvely. In the sectons that follow, we present a new hybrd method for makng nference about a confdence nterval k for lnear functon of populaton varances c 1 σ, where k and the c are coeffcent n the lnear functon. j1 The proposed nterval estmaton procedures.1 Asymptotc varance estmators (General approach).1.1 An unbased populaton varance estmator It s well known that the usual unbased estmate of varance ss, 1,,,k and ts varance, avalable n statstcal lterature, are gven by 4 4 Var(S ) [ γ4 (n 3) / (n 1)] σ / n, where γ 4 μ / σ and μ s the populaton fourth central moment. For samples suffcently large provded the populaton fourth moment s fnte, the sample varance s asymptotcally normally dstrbuted wth mean E( S )and varance V( S ). A smple large-sample procedure for constructng a 100 (1-α) % confdence nterval for varance can be obtaned as S S σ, (1) 1+ z α/ [ γ4 (n 3) / (n 1)]/ n 1 z α/ [ γ4 (n 3) / (n 1)]/ n n where γ 4 (X j1 j X ) / ns and z α / be a crtcal z-value. Another approach s to make use of the MBBE of varance n the smlar pattern of (1)..1. The MBBE of varance An mproved estmator of the varance that utlzes the kurtoss was ntally derved by Seals and Intarapanch [] and later generalzed by Wencheko and Chpoyera [5]. The estmator has the form S W w(n-1) S where the weght, w [(n+1)+( γ 4 3)n -1 (n-1)] -1 s an optmal value that mnmzes the MSE ( S W ) and γ 4 s the kurtoss. Wencheko et al., [5] defned ths estmator of varance as

3 Interval estmaton for a lnear functon of varances 4911 the mnmum mean-squared error best based estmator (MBBE). Snce the relatve effcency (RE) of the MBBE s larger than 1, thus, mplyng that the MBBE s always more effcent than the usual unbased estmator S of varance. Ths statstc s of nterested, n the present paper, we ntended to deal wth the MBBE of varance by adjustng a kurtoss estmaton procedure usng trmmed mean (and later let s call the adjusted MBBE of varance), then makng use of the two based estmators, the adjusted MBBE and the MBBE and the usual unbased estmator S to establsh the asymptotc confdence ntervals for a lnear functon of ndependent varances of nonnormal dstrbutons that utlzes the kurtoss va the method of varance estmates recovery, the MOVER and ts smple applcaton to a general lnear functon of parameters(zou et al.,[6]). The basc dea s constructon of confdence ntervals for a lnear functon of varances that nvolves usng the readly avalable method of Donne and Zou [3] and Zou et al.[6] to combne confdence ntervals based on separate samples. The MBBE of varance s of the form Sw w (n 1)S S [{(n + 1)/(n 1)} + ( γ4 3)/n ] and E(S w) w(n 1) σ, 1,,,k, w 1 [(n + 1) + ( γ 3)(n 1) / n ], 0 < w < 1 where 4 [ ] 4 MSE(S w) E(S w σ ) w (n 1) Var(S ) + (n 1)w 1 σ, where γ 4 s the kurtoss. For large n, when randomly samplng from any dstrbuton wth a fnte fourth moment, and By the central lmt theorem, The MBBE of varance s approxmately standard normal wth E(S ) and MSE(S ).Consequently, an approxmate two sded 100 (1-α)% confdence nterval for the varance may be gven as α/ 4 α/ 4 n γ 4 (X j1 j X ) / ns, z α / be a crtcal z-value and L S {1+ z [{ γ (n 3)/(n 1)}/n ] + [1 1/w (n 1)] }, U S {1 z [{ γ (n 3)/(n 1)}/n ] + [1 1/w (n 1)] }, () where ŵ [(n 1) ( 3)(n 1)/n ] γ The adjusted MBBE of varance Snce an estmate of MSE(S w ) wll requre an estmate of kurtoss, and t s well n known that a usual kurtoss estmate γ 4 (X j1 j X ) / ns, 1,, k, was badly based n samplng from nonnormal populatons, an alternatve adjusted kurtoss estmate then has been used, and s of the form: ' n γ 4 (X j 1 j m ) / ns,where m s a trmmed mean wth trm proporton equal to 1 n 4. Note that we used the trmmed mean n place of mean as suggested by Bonett [1] because the trmmed mean not only tends to provde a w w

4 491 Srma Suwan and Sa-aat Nwtpong better kurtoss estmate but also tends to mprove the accuracy of the nterval estmaton for leptokurtc (heavy-taled) or skewed dstrbutons. Ths adjusted MBBE estmator of varance (adjusted MBBE) yelds the two sded 100(1-α) % confdence nterval for varance: ' ' α/ 4 L S {1+ z [{ γ (n 3)/(n 1)}/n ] + [1 1/w (n 1)] } ' ' α/ 4 U S {1 z [{ γ (n 3)/(n 1)}/n ] + [1 1/w (n 1)] }, (3) ' n where γ 4 (X j m ) / ns, z α / be a crtcal z-value, and j 1 ŵ ' ' 1 [(n + 1) + ( γ 3)(n 1) / n ]. 4. Approxmate Intervals for a lnear functon of varances..1 The MOVER and ts applcatons [6] Suppose we would lke to construct two sded 100 (1-α) % confdence nterval, denoted by ( L, U ) for θ 1 +θwhere θ1, θdenote any two nterested parameters and θ, 1, are estmators of θ. Zou et al.,[6] have extended the argument of Donner and Zou [3] to a lnear functon of parameters by regardng θ 1 +θand θ1 θ as c1θ 1 + cθ, where c1andc are constants, hence, the nterval can be wrtten as L c1θ 1+ cθ c1θ 1 mn(c1l 1,c1u 1) + cθ mn(cl,cu ) and U c1θ 1+ cθ + c1θ 1 max(c1l 1,c1u 1) + cθ max(cl,cu ) (4) Ther further extenson s to use a mathematcal nducton applcaton n order to derved a generally 100(1-α) % confdence nterval for lnear functons of k parameters c 1 θ, 1,,,k,where k, θ denote any nterested parameters and c are coeffcent n the lnear functon, as defned k k L cθ cθ mn(cl 1 1,cu ) k k U cθ + cθ max(cl 1 1,cu ). (5).. The ntervals estmaton for a lnear functon of varances k Let s defned a lnear functon of varances as c,k 1 σ where c are known constants. Snce there are at least three ntervals for a sngle varance as n

5 Interval estmaton for a lnear functon of varances 4913 equaton (1), () and (3), and to obtan a confdence nterval for lnear functons of varances va equaton (5) we should have k separate confdence lmts for the asymptotc varance estmates σ, 1,,..., k (.e.,(l 1,u 1 ),,(l k, u k )), thus the three dstnct hybrd confdence ntervals for a lnear functons of varances can then be easly computed as they all have closed form solutons. Hence, the 100(1-α)% tradtonal MOVER and ts applcaton confdence nterval for a lnear functon k of varances c 1 σ that arse from each equaton (1), () and (3) are respectvely, as follows,. Namely U1: k k k k L c S c S mn(c l,c u ), U c S + c S max(c l,c u ), where (l, u ), 1,,..., k denote an avalable (1-α)100% confdence ntervals for σ, 1,, k gven by equaton (1).. Namely M1: k k k k L c σ c σ mn(c l,c u ), U c σ+ c σ max(c l,c u ), where (l, u ), 1,,..., k denote an avalable (1-α)100% confdence ntervals for σ, 1,, k gven by equaton (), where w σ MBBE S w (n 1)S, 1 ŵ [(n + 1) + ( γ 4 3)(n 1) / n ] n and γ 4 (X j X ) / ns.. Namely M: j1 k k k k L c σ c σ mn(cl,c u ),U c σ + c σ max(cl,cu ), where (l, u ), 1,,..., k denote an avalable (1-α)100% confdence ntervals for σ, 1,, k gven by equaton (3), where ' ' ' 1 w [(n + 1) + ( γ4 3)(n 1)/n ], σ adjustedmbbe w (n 1)S, ' n γ 4 (X j 1 j m ) / ns and m s a trmmed mean wth trm proporton equal to 1 n 4.

6 4914 Srma Suwan and Sa-aat Nwtpong 3 Smulaton Results 3.1 Method A smulaton study was carred out to nvestgate the performance of the 95% confdence lmts for a lnear functon of varances. The 10,000 sets of varance values were randomly sampled from a varety of dstrbutons. The U1, M1 and M were used to compute the coverage probabltes () and the average nterval wdths () for each of 10,000 sets of varance values and for varous balanced and unbalanced sample szes. The smulaton programs were wrtten n R and executed on an Intel computer. 3. Results k In constructng confdence ntervals for lnear functons of varances c 1 σ, k, the performance of those wth no dfference among varances for a varety of non-normal dstrbutons are frst nvestgated n terms of coverage probabltes and the average ntervals wdths for U1, M1 and M, when varous group szes are balanced and unbalanced desgns and are summarzed n the table. Only the results at the nomnal fve-percent level of sgnfcant are presented. For a lnear functon of varances, regardless of balanced or unbalanced desgns, the results n Table 1 show that as groups of observatons are moderate or large, for asymmetrc (skewed) dstrbutons, the M performs substantally better than both the M1 and U1 n terms of holdng the mean coverage closest to the nomnal level, wth narrowest average wdth, for symmetrc non-normal dstrbutons, M and M1 are dentcally and both have coverage close to nomnal. The U1 s clearly the poorest performer for all cases of skewed dstrbutons, as the coverage never reaches 95% and n many cases of symmetrc non-normal dstrbutons ts coverage does not mantan ts nomnal level except when par of samples s tends to nfnty that ts coverage converges to the nomnal level.

7 Interval estmaton for a lnear functon of varances 4915 Table 1 Compares the coverage probabltes () and average wdths () of the U1, M1 and M for a sngle lnear functon of varances, c 1 σ for several dfferent sets of coeffcents from nonnormal dstrbutons. n1 n (U1) (U1) B(1,10) C(1,1) ch(10) C(1, 1) (M1) (M1) (M) (M) (U1) (U1) (M1) (M1) (M) (M) For lnear functon of 3 varances, regardless of balance or unbalance desgns, results (not showed here) state that, for skewed dstrbutons, as groups of observatons are moderate to large, both the M1 and M outperform the u1 but the M1 sometme performs slghtly better than the M n the sense of well control the coverage probabltes to be close enough to the nomnal level and a lttle bt narrower wdths, whle the U1 can better perform when groups of observatons are extremely large, for symmetrc non-normal dstrbutons, the M and M1 are seem to be dentcally and mantanng ther coverage qute well whle the U1 perform poorly,only wth the unform and beta dstrbutons the U1 tends to be most precse n some value of lnear coeffcents, regularly, t can better perform when groups of observatons are large. Results from constructng confdence ntervals for lnear functons of 4 varances n Table show agan that the M s dentcal to the M1 and both also hold ther level well for symmetrc non-normal dstrbutons but for some skewed dstrbutons the M provdes naccurate coverage that exceeds the nomnal level whle the M1 s slghtly below the target level. However, t s also shows that the M and M1 perform comparably well for moderate to large szes. Results from the U1 depart from the nomnal too often, except n some value of lnear coeffcents for a number of symmetrc non-normal dstrbutons (.e., beta (3,3) and unform(0,1)), t performs best, generally, as hgh szes, the U1 performs reasonably well.

8 4916 Srma Suwan and Sa-aat Nwtpong Table compares the coverage probabltes () and average wdths () of the U1, M1 and 4 M for a sngle lnear functon of varances, c 1 σ for several dfferent sets of coeffcents from nonnormal dstrbutons. c(1/4,1/4,1/4,1/4) c(1, 1,1, 1) n1,n,n3,n4 (U1) (U1) (M1) B(1,10) (M1) (M) (M) (U1) (U1) logt(0,1) 10,10,10, ,5,5, ,50,50, ,100,100, ,15,15, ,150,150, ,00,00, ,300,300, ,0,30, ,50,75, ,100,150, ,150,5, ,00,300, ,50,375, ,300,450, ,400,600, (M1) (M1) Cp s(m) (M) In further nvestgaton, as comparson, we determned the performances when samples are collected from normal populaton dstrbutons, n both cases of equal and unequal varances when varous group szes are balanced and unbalanced desgns. Smulaton results are also establshed but not shown here. Fndngs suggest that, the pattern of results were smlar for the equal and unequal varances regardless of balanced or unbalanced desgns, so n most cases, when sample szes are moderate, small to large, the top performng ntervals are the M1 and M, moreover, they both yeld almost dentcal results that gve reasonably smlar coverage and wdth and these are mantaned as sample szes ncreases, wth the U1, n many cases, as sample szes ncrease, generally there s an ncrease n coverage probablty, whle n some lnear coeffcents value, t s the best performer. Although not reported here we also found that, for a wde varety of lnear functons of varances, from k populatons that are not dentcal, the results usually show an extremely large departure from the nomnal level that are not attractve. 4 Conclusons Two new ntervals for a sngle lnear combnaton of varances, the M1 and M whch are generated from the MBBE and the adjusted MBBE of varance respectvely, are proposed here and demonstrate that they both have better coverage probabltes than the U1 that s generated from the unbased varance

9 Interval estmaton for a lnear functon of varances 4917 estmator n most cases nvestgated by both symmetrc and asymmetrc dstrbutons when varous group szes are balanced or unbalanced desgns. As a whole, two possble explanatons for our results are that the three ntervals, the M, M1 and U1 for lnear functon of varances obvously fal to hold ther nomnal coverage, often provdng ntervals whch are poor. In most cases small-sample performance we nvestgated, all the confdence ntervals converge to the target level, as groups of observatons are large. Confdence nterval construct for a sngle lnear combnatons of varances k c 1 σ, from k ndependent populatons, the M performed best n terms of holdng the correct coverage probabltes wth narrowest average wdth n most cases, for asymmetrc dstrbutons regardless of balanced or unbalanced desgns, and dentcal to the M1 for all symmetrc dstrbutons when the group szes are moderate to large. For k > ndependent populatons, when samples are drawn from symmetrc populaton dstrbutons, regardless of balanced or unbalanced desgns, mostly cases for a varety of coeffcent, the M s also dentcal to the M1 but has a lttle bt wder wdths than M1 when group of szes are moderate to large and they both better perform than the U1, except cases n whch at least one negatve term occurs n a varety of coeffcents then they both seem to be nferor n a few dstrbutons. For skewed dstrbutons, n several cases coverage of the M s stll at or above nomnal whle the M1 s at or below nomnal, though they are stll close to nomnal, t s not clear whether the M or M1 perform best snce these two ntervals alternately outperform. In addton, we found a few nstance where the coverage probabltes of U1 was superor to the M and M1 when samples come from Normal (regardless of equal or unequal varances) or symmetrcally dstrbuted populatons (.e., Beta(3,3) and Unform(0,1)) as there s at least one negatve coeffcent occurrng n a varety of coeffcents. To nvestgate the rule for varances combnatons that gve the best performer, that s left for a subsequence paper. Fnally, we thank the work of Donner and Zou [3], and the work of Zou et al.[6]. References [1] D. G. Bonett, Approxmate confdence nterval for standard devaton of nonnormal dstrbutons, Computatonal Statstcs & Data Analyss, 50 (006), [] T. S. Donald and P. Intarapanch, A note on a estmator for the varance that utlzes the kurtoss, The Amercan Statstcan, 44 (1990),

10 4918 Srma Suwan and Sa-aat Nwtpong [3] A. Donner and G. Zou, Closed form confdence ntervals for functons of the normal mean and standard devaton, Statstcal Methods n Medcal Research, 9 (01), [4] R Development Core Team, R: A Language and Envronment for Statstcal Computng. R Foundaton for Statstcal Computng, Venna, Austra, ISBN: , 011. [5] E. Wencheko and H. W. Chpoyera, Estmaton of the varance when kurtoss s known, Statstcs Papers, 50 (009), [6] G. Y. Zou, W. Huang, and X. Zhang, A note on confdence nterval estmaton for a lnear functon of bnomal proportons, Computaton Statstcs & Data Analyss, 53 (009), Receved: July 4, 013