PREDICTION INTERVALS FOR FUTURE SAMPLE MEAN FROM INVERSE GAUSSIAN DISTRIBUTION

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Qatar Uiv. Sci. J. (1991), 11: 19-26 PREDICTION INTERVALS FOR FUTURE SAMPLE MEAN FROM INVERSE GAUSSIAN DISTRIBUTION By MUHAMMAD S. ABU-SALIH ad RAFIQ K. AL-BAITAT Departmet of Statistics, Yarmouk Uiversity, Irbid, Jorda ABSTRACT A radom sample Xt. X 2,, X from Iverse Gaussia distributio I(Jt,A) is observed. O the basis of this observed sample a 113 predictio iterval of the mea Y of a future sample Yt...., Ym from I(Jt,A) has bee costructed whe either or both Jt & ;>,. are ukow. INTRODUCTION A predictio iterval is a iterval which cotais the results of a future sample from a populatio depedig o the results of a past sample from the same populatio with a specified probability. Predictio itervals play a importat role i quality cotrol ad reliability. I statistics, they are beig used i goodess of fit tests, hypothesis testig, classifyig observatios, sample surveys,... etc. (Eglehardt & Bai 1979). This topic has bee discussed by may authors, of whom we metio a few who derived predictio itervals for derived predictio itervals for the future sample mea. Lawless (1972) gave predictio limits for Y i the case of the expoetial distributio. Kamisky ad Nelso (1974) gave predictio iterval for the mea of a future sample usig susbsets of a observed sample whe the samples are from the expoetial distributio. Hah (1975) gave a predictio iterval to cotai the differece betwee the sample meas of two future samples from ormal distributios. Meeker ad Hah (1980) gave predictio iterval for the ratio of the meas of two future samples from ormal distributios. Azzam ad Awad (1981) gave predictio itervals for the differece ad ratio of the meas of two future samples uder the assumptio that the samples are from a

Predictio itervals for future sample mea ormal ad a expoetial distributio. Vee-Mig (1984) costructed predictio itervals for the mea life time of a future sample based o icomplete data take from a 2-parameters expoetial distributio. I this work we derive predictio itervals for the future sample mea from the iverse Gaussia distributio. Chhiekara ad Guttma (1982) used a observed sample from the iverse Gaussia distributio to costruct a predictio iterval to cotai the ext future observatio. The probability desity fuctio (pdf) f(.) of a iverse Gaussia distributio deoted by I(JL,A) is give by f(x,jl,a) (1.1) 0 o.w. X. is shape parameter (Tweedi, 1957). The pdf is skewed uimodal ad is a member of the expoetial family. Iverse Gaussia distributio is used as a life time model (Chhikara & Folks, 1977). It is kow that if X - I(JL,A), the E(X)=JL, ad Var(X)=JL 3 /X.. PREDICTION INTERVAL FOR THE FUTURE SAMPLE MEAN Let XI>..., X be a radom sample from a iverse Gaussia distributio I(JL,A) with desity fuctio f(x,jl,a), as give i (1.1). Let Y 1,..., Y m be a future radom sample from the same populatio with the same parameters JL ad X.. Assume that the two samples are idepedet of oe aother ad that both JL ad X. are ukow parameters. Shuster (1968) showed that (2.1) 20

MUHAMMAD S. ABU-SALIH ad RAFIQ K. AL-BATIAT (2.2) (2.3) m X= I X/, Y I Y/m i=1 i=j It may be see that X ad I ( 11 Xi - 1 I X ) are the maximum likelihood i=1 estimators of JL ad ~ respectively ad the two statistics are idepedet, (Tweedi, 1957), Furthermore, X - l(p., ~) ad ~I ( 1/ xi-11 X ) - X 2-1 (Wasa, ad Roy, 1969). i=1 Clearly ~(X - P-) 2 - x\ i view of (2.1). P-2 x Thus from (2.2) ad (2.3) (2.4) m I ( 11 xi - 1/.x) + I ( 11 Yi- H Y) (2.5) i=1 02 ~(X - p.) 2 d 0 m~(y - p.)z -:...-=--~ a 3 = --'-----'--'-- IL2 X IL2 y are idepedetly distributed as x~ 1 + m _ 2, xi ad xi respectively (Hogg & Craig, 1978).. Now, 0 2 ad 0 3 ca be combied differetly so that the right side of (2.4) equals 0 1 +A+ B. 21

Predictio itervals for future sample mea ad A= ma(x- Y) 2 XY(X +my) (2.6) A[(X +my) -~t( + mh B = -------==----=-- ~A-2 (X + my) (2.7) I view of (2.1), it ca be see that B has x\. Therefore, A, B give i (2.6) ad (2.7) are idepedetly distributed each as x\ (Hogg & Craig, 1978, p. 279). Notice that 0 1 ad A do ot ivolve the parameter IL ad ca be used to obtai 1~ percet predictio itervals for Y whe IL is ukow ad A is kow. I this case we use the statistic A= ma(x- Y) /XY(X +my) for the predictio of Y. (2.8) Sice A~ x\ the 1~% predictio iterval for Y will be obtaied by solvig P[A =::::; x\,~] = 1- ~for Y. The predictio iterval for Y whe A is kow will be give by: (2.9) Whe IL ad A are ukow, cosider the statistic R = ( + m - Ql 2 ) A, so that substitutig for A ad 0 1, we get: - -2 -- - - R = ( + m- 2) (X- Y) /XY(X +my) V (2.10) V = 0 1 /m (2.11) R does ot deped o ay parameter IL or A ad has F distributio with 1 ad (+m-2) degrees of freedom. Hece, R ca be used to costruct predictio itervals for Y whe both IL ad A are ukow. The 1~ predictio iterval for Y will be obtaied by ivertig the iequality R =::::; F 1,+ m _, 13 F 1, +m-, 13 is the 1~% poit of the F 1,+'m-,2 distributio. 22

MUHAMMAD S. ABU-SALIH ad RAFIQ K. AL-BA IT AT Usig (2.10) ad simplifyig we get the 113% predictio iterval for Y to be: [ 1 + (F1,+m-2, 13 ) V + {(VFl,+m-2,13)2 + V(+m)Fl,+m-2,13}]<~. 12 ) X 2(+m-2) 4(+m-2) 2 (+m-2)x Whe J.L is kow, ad A is ukow, cosider the followig statistic Q =I (Xi - i=l J.L) 2 1 xi T = m(y - J.L) 2 YQ It is easy to see that Tis distributed as F 1, Agai the 113% predictio iterval for Y (for J.L kow) will be obtaied by solvig T :!S F 1,, 13 ad is give by [ Q ] _ Q [ 4mJ.L 2 ] lf2 +-F +- --F +F J.L 2m l,, 13 2m Q l,, 13 l,, 13 (2.13) Note that A, R ad T do ot always provide two sided itervals because there is a possibility that the differece i the two terms of (2.9), (2.12) ad (2.13) may be egative. Sice y > 0 it ca be see that oly the oe-sided itervals with oo as the upper limit are admissible ad these are obtaied by restrictig the solutio of a iequality to the positive real lie. Example: Cosider the data give by Chhikara ad Folks (1977) with = 46, jl = x = 3.61 ad A= 1.66, V = -A- 1 = 0.66. Based o these data a 113 percet predictio limits usig (2.9) ad (2.12) were computed for various values of 13. These limits are give i Table 1 ad Table 2 respectively. Table 1 Predictio itervals for varyig A for m 13 A = 1.66 46, X 3.61 (formula (2.9)) A= 2.0 0.9 2.29 0.950 2.12 0.975 1.84 0.990 1.74 6.515 2.38 6.127 7.5 2.22 6.927 10.332 1.94 9.095 11.904 1.85 10.221 23

Predictio itervals for future sample mea A= 2.5 A= 3.0 0.9 0.950 0.975 0.990 2.48 2.32 2.05 1.96 5.747 6.383 8.016 8.814 2.55 5.491 2.40 6.025 2.15 7.349 2.06 7.973 A= 3.5 A= 4.0 Lowet 0.9 0.950 0.975 0.990 2.62 2.47 2.22 2.14 5.304 5.769 6.891 7.407 2.67 5.160 2.53 5.574 229 6.555 2.20 6.997 Commet: As we ca see from the results give above that the legth of the iterval becomes smaller as A icreases, which is a desirable property. Table 2 Predictio itervals for varyig m for 46, X 3.61 (formula 2.12) m = 4 m= 16 0.9 0.950 0.975 0.990 0.46 0.45 0.28 0.21 57.02 138.48 1416.28 0.54 0.41 0.33 0.26 m = 36 m = 76 0.9 0.950 0.975 0.990 0.63 0.49 0.40 0.32 0.74 0.60 0.51 0.42 Commet: It ca be see from the above table that as m icreases the upper limit goes to oo ad the lower limit icreases. If we compare the result of Table 1 ad 2, we ifer that we get shorter predictio itervals whe A is kow, rather tha whe A is ukow. 24

MUHAMMAD S. ABU-SALIH ad RAFIQ K. AL-BATIAT REFERENCES Azzam, M.M. ad A.M. A wad, 1981. Predictio itervals for a future sample mea ad for the differece or ratio of two future sample meas. Dirasat, 8: 21-27. Chbikara, R.S. ad I. Guttma, 1982. Predictio limits for the iverse Gaussia distributio. Techometrics, 24: 319-324. Egelhardt, M. ad L.J. Bai, 1979. Predictio limits ad two-sample problems with complete or cesored Weibull data. Techometrics, 21: 233-237. Hah, G.J. 1975. A simultaeous predictio limit o the mea of future samples from a expoetial distributio. Techometrics, 17: 341-344. Hogg, R. V. ad A. T. Craig, 1978. Itroductio to Mathematical Statistics, Fourth Ed. MacMilla Publishig Co. Ic. New York. Kamisky, K.S. ad P.I. Nelso, 1974. Predictio itervals for the expoetial distributio usig subsets of the data. Techometrics, 16: 57-59. Lawless, J.F. 1972. O predictio itervals for samples from the expoetial distributio ad predictio limits for system survival. Sakhya, SerB, 34: 1-12. Meeker, J.V. ad G.J. Hah, 1980. Predictio iterval for the ratio of ormal distributio samples mea. Techometrics, 22: 357-366. Shuster, J.J. 1968. O the iverse Gaussia distributio fuctio. J. Amer. Statist. Ass., 63: 1514-1516. Tweedie, M.C.K. 1957. Statistical Properties of Iverse Gaussia Distributio I. A. Math. Statist. 28: 362-377. Wasa, M.T. ad L.K. Roy, (1969). Tables of Iverse Gaussia Percetage Poits. Techometrics. 11: 591-604. 25

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