Bayesia Cotrol Charts for the Two-parameter Expoetial Distributio if the Locatio Parameter Ca Take o Ay Value Betwee Mius Iity ad Plus Iity R. va Zyl, A.J. va der Merwe 2 Quitiles Iteratioal, ruaavz@gmail.com 2 Uiversity of the Free State
Abstract By usig data that are the mileages for some military persoel carriers that failed i service give by Grubbs (97 ad Krishamoorthy ad Mathew (2009 a Bayesia procedure is applied to obtai cotrol limits for the locatio ad scale parameters, as well as for a oe-sided upper tolerace limit i the case of the two-parameter expoetial distributio. A compariso betwee the assumptios of < µ < ad 0 < µ < are also made. A advatage of the upper tolerace limit is that it moitors the locatio ad scale parameter at the same time. By usig Jereys' o-iformative prior, the predictive distributios of future maximum likelihood estimators of the locatio ad scale parameters are derived aalytically. The predictive distributios are used to determie the distributio of the ru-legth ad expected ru-legth. This paper illustrates the exibility ad uique features of the Bayesia simulatio method. Keywords: Jereys' prior, two-parameter expoetial, Bayesia procedure, ru-legth, cotrol chart Itroductio I this sectio the same otatio will be used as give i Krishamoorthy ad Mathew (2009 with the exceptio that the locatio parameter ca ow take o values betwee ad, similarly as i some literature, see for example Johso ad Kotz (970. Therefore the two-parameter expoetial distributio has the probability desity fuctio f (x; µ, { } exp (x µ x > µ, < µ <, > 0 where µ is the locatio parameter ad the scale parameter. As before, let X, X 2,..., X be a sample of observatios from the two-parameter expoetial distributio. The maximum likelihood estimators for µ ad are give by ˆµ X ( ad ˆ ( Xi X ( X X( i where X ( is the miimum or the rst order statistic of the sample. It is well kow (see Johso ad Kotz (970; Lawless (982; Krishamoorthy ad Mathew (2009 that ˆµ ad ˆ are idepedetly distributed with (ˆµ µ χ2 2 2 ad ˆ χ2 2 2 2. (. Let ˆµ 0 ad ˆ 0 be observed values of ˆµ ad ˆ the it follows from Equatio. that a geeralized pivotal quatity (GPQ for µ is give by G µ ˆµ 0 χ2 2 ˆ χ 2 0 (.2 2 2 ad a GPQ for is give by 2
G 2ˆ 0 χ 2 2 2 (.3 From a Bayesia perspective it will be show that G µ ad G are actually the posterior distributios of µ ad if the prior p (µ, is used. 2 Bayesia Procedure I this sectio it will be show that the Bayesia procedure is the same as the geeralized variable approach. If a sample of observatios are draw from the two-parameter expoetial distributio, the the likelihood fuctio is give by ( { } L (µ, data exp (x i µ. i As prior the Jereys' prior p (µ, will be used. The joit posterior distributio of µ ad is p (, µ data p (µ, L (µ, data K ( + exp { ( x µ} < µ < x (, 0 < < (2. It ca easily be show that K (ˆ Γ ( where ˆ x x (. The posterior distributio of µ is p (µ data 0 p (, µ data d ( ( (ˆ x µ < µ < x ( (2.2 ad the posterior distributio of is p ( data x( p (, µ data dµ { } exp K 2 ( ˆ where 0 < < (2.3 a Iverse Gamma distributio where 3
(ˆ K 2 Γ (. The coditioal posterior distributio of give µ is give by p ( µ, data p(,µ data p(µ data K 3 ( + exp { ( x µ} where 0 < < (2.4 a Iverse Gamma distributio where K 3 { ( x µ}. Γ ( Also the coditioal posterior distributio of µ give is p (µ, data p(,µ data p( data The followig theorem ca ow easily be proved: exp { ( x( µ } (2.5 where < µ < x ( Theorem 2.. The distributio of the geeralized pivotal quatities G µ ad G deed i Equatios.2 ad.3 are exactly the same as the posterior distributios p (µ data ad p ( data give i Equatios 2.2 ad 2.3. Proof. The proof is give i Appedix A. 3 The Predictive Distributios of Future Sample Locatio ad Scale Maximum Likelihood Estimators, ˆµ f ad ˆ f Cosider a future sample of m observatios from the two-parameter expoetial populatio: X f, X 2f,..., X mf. The future sample mea is deed as Xf m m j X jf. The smallest value i the sample is deoted by ˆµ f ad ˆ f X f ˆµ f. To obtai cotrol charts for ˆµ f ad ˆ f their predictive distributios must rst be derived. If < µ <, the the descriptive statistics, posterior distributios ad predictive distributios will be deoted by a tilde (~. Theorem 3.. The predictive distributio of a future sample locatio maximum likelihood estimator, ˆµ f is give by [ K ( x ˆµ f ] < ˆµ f < x ( f (ˆµ f data [ ] K x ( < ˆµ f < ˆ+m(ˆµ f x ( (3. where K ( m + m 4 (ˆ.
Proof. The proof is give i Appedix B. The reaso why f (ˆµ f data (Equatio 3. diers from f (ˆµ f data is that it is assumed that 0 < µ < { ( } ( which results i a posterior distributio of p (µ data ( ˆ x ( x µ, while for Equatio 3. it is assumed that < µ < ad the posterior distributio for µ is therefore p (µ data ( (ˆ ( x µ. Theorem 3.2. The mea ad variace fo ˆµ f is give by Ẽ (ˆµ f data x m m ˆ (3.2 m ( 2 ad V ar (ˆµ f data 3 ( 2 + (m m 2 2 (ˆ m 2 ( ( 3 ( 2 2 (3.3 Proof. By deletig the term ( x from results from the previous report where 0 < µ < Equatio 3.2 ad Equatio 3.3 follows. Theorem 3.3. The predictive distributio of a future sample scale maximum likelihood estimator, ˆ f is give by f (ˆf data m 2 Γ (m + 2 m (ˆ (ˆf m Γ (m Γ ( ( mˆ f + ˆ m+ 2 0 < ˆ f < (3.4 Proof. The proof is give i Appedix C. Corollary 3.4. ˆ f data ˆ (m m ( F 2m 2,2 2. Proof. The proof is give i Appedix D. Theorem 3.5. The mea ad variace of ˆ f is give by Ẽ (ˆf data (m m ( 2 ˆ (3.5 ad var (ˆf 2 (m data m 2 ( 2 2 ( 3 ( + m 3 ˆ 2 (3.6 Proof. By deletig the term ( x from results from the previous report where 0 < µ <, Equatio 3.5 ad Equatio 3.6 follow. 5
4 Example The followig data is give i Grubbs (97 as well as i Krishamoorthy ad Mathew (2009. The failure mileages give i Table 4. t a two-parameter expoetial distributio. Table 4.: Failure Mileages of 9 Military Carriers 62 200 27 302 393 508 539 629 706 777 884 008 0 82 463 603 984 2355 2880 For this data, the maximum likelihood estimates are ˆµ x ( 62, ˆ i ( xi x ( x x( 835.2 ad 9. As metioed i the itroductory sectio, the aim of this article is to obtai cotrol charts for locatio ad scale maximum likelihood estimates as well as for a oe-sided upper tolerace limit. 4. The Predictive Distributio of ˆµ f ( < µ < By usig Equatio 3. the predictive distributio f (ˆµ f data for m 9 future failure mileage data is illustrated i Figure 4.. Figure 4.: Distributio of ˆµ f, 9, m 9 mea (ˆµ f media (ˆµ f mode (ˆµ f 62, var (ˆµ f 529.2 95% iterval (ˆµ f (.2; 32.8 99.73% iterval (ˆµ f ( 54.85; 477.36 96.08% iterval (ˆµ f ( 2; 326 6
For < µ <, 9, m 9, the predictive distributio p (ˆµ f data is symmetrical. mea (ˆµ f media (ˆµ f mode (ˆµ f 62. Mode (ˆµ f for < µ < is exactly the same as that for 0 < µ <. A further compariso of Figure 4. ad results from previous report shows that var (ˆµ f 529.2 is somewhat larger that var (ˆµ f 3888.7, whe 0 < µ <. Also the predictive itervals are somewhat wider. A dissatisfactory aspect for < µ < is that the 99.73% iterval (ˆµ f ( 54.85; 477.36, i.e., cotais egative values. I Table 4.2 descriptive statistics are give for the ru-legth ad expected ru-legth. Table 4.2: Descriptive Statistics for the Ru-legth ad Expected Ru-legth i the case of ˆµ f ; < µ < ad β 0.039 for 9 ad m 9 Descriptive Statistics f (r data Expected Ru-legth Equal Tail Equal Tail mea 367.84 373.34 media 44.50 69.93 var 6.663 0 7.44 0 8 95% iterval (0; 740.50 (0; 724.64 A compariso of Table 4.2 with results from previous report shows that if < µ < the expected (mea ru-legth 370 if β 0.0392 while this is the case if β 0.0258 for 0 < µ <. Also the 95% itervals are somewhat shorter for < µ <. 4.2 A Compariso of the Predictive Distributios for ˆ f I Figure 4.2 comparisos are made betwee f (ˆf data m 2 Γ (m + 2 m (ˆ (ˆf m Γ (m Γ ( ( mˆ f + ˆ m+ 2 0 < ˆ f < ad f (ˆf data m m Γ (m + 2 Γ (m Γ ( { ( ˆ ( } m 2 (ˆf x ( m+ 2 ( m+ 2 mˆ f + ˆ mˆ f + x 0 < ˆ f <. As metioed f (ˆf data deotes the predictive desity fuctio if < µ < ad f (ˆf data deotes the predictive desity fuctio if 0 < µ <. 7
Figure 4.2: f (ˆf data, 9, m 9 I Table 4.3 descriptive statistics are give for f (ˆf data ad f (ˆf. Table 4.3: Descriptive Statistics of f (ˆf data ad f (ˆf data Descriptive Statistics f (ˆf data f (ˆf data Mea (ˆf 884.34 876.98 Media (ˆf 835.2 829. Mode (ˆf 747.3 743.3 V ar (ˆf 95037 999 (430; 622 (428; 60 95% Equal tail Iterval (ˆf 95% HP D Iterval (ˆf Legth92 Legth73 (370.8; 500.5 (369.2; 482.6 Legth29.7 Legth3.4 Als, the exact meas ad variaces are give by: (m Ẽ (ˆf data m ( 2 ˆ 884.34, V ar (ˆf 2 (m m 2 ( 2 2 ( 3 ( + m 3 ˆ 2 95042, 8
(m E (ˆf data m ( 2 KL 976.98, { } V ar (ˆf data 2 (m KM m ( 2 m m ( K 2 L 2 999 where { ( K ( ˆ L 2 { ( 2 ˆ ( }, x ( } 2 x ad M 3 { ( 3 ˆ ( } 3. x From Figure 4.2 ad Table 4.3 it ca be see that f (ˆf data ad f (ˆf data are for all practical purposes the same. Also the exact meas ad variaces are very much the same as the umerical values. It therefore seems that whether the assumptio is that < µ < or that 0 < µ < does ot play a big role i the predictio of ˆ f. I Table 4.4 comparisos are made betwee the ru-legths ad expected ru-legths i the case of ˆ f for < µ < ad 0 < µ <. Table 4.4: Descriptive Statistics for the Ru-legth ad Expected Ru-legths i the case of ˆ f, < µ <, 0 < µ < ad β 0.08 < µ < 0 < µ < Descriptive Statistics f (r data Expected Ru-Legth f (r data Expected Ru-Legth Equal Tail Equal Tail Equal Tail Equal Tail Mea 369.29 370.42 375.25 375.25 Media 22.8 229.28 27.4 238.38 Variace 3.884 0 5.2572 0 5 3.955 0 5.2698 0 5 95% Iterval (0; 586 (7.423; 089.7 (0; 602.5 (7.809; 096.7 From Table 4.4 it ca be see that the correspodig statistics for < µ < ad 0 < µ < are for all practical purposes the same. So with respect to ˆ f it does ot really matter whether it is assumed that µis positive or ot. 9
5 Phase I Cotrol Chart for the Scale Parameter i the Case of the Two-parameter Expoetial Distributio Statistical quality cotrol is implemeted i two phases. I Phase I the primary iterest is to assess process statbility. Phase I is the so-called retrospective phase ad Phase II the prospective or moitorig phase. The costructio of Phase I cotrol charts should be cosidered as a multiple testig problem. The distributio of a set of depedet variables (ratios of chi-square radom variables will therefore be used to calculate the cotrol limits so that the false alarm probability (FAP is ot larger tha F AP 0 0.05. To obtai cotrol limits i Phase I, more tha oe sample is eeded. Therefore i the example that follows there will be m 5 samples for a subgroup each of size 0. Example 5. The data i Table 5. are simulated data obtaid from the followig two-parameter expoetial distributio: f (x ij ;, µ { exp x } ij µ i, i, 2,..., m, j, 2,..., ; x ij > µ i 8; µ i 2i; m 5; 0. Table 5.: Simulated Data for the Two-parameter Expoetial Distributio µ i 2 4 6 8 0 3.6393 8.7809 9.3759 0.7846 6.5907 2.796 4.2388 32.6582 35.578 7.7079 8.5094 4.3502 7.3084 8.272 2.376 2.749 9.7827 6.5463 32.6032.8333 5.6664 5.7823 9.002 26.6535 23.486 20.699 9.628 8.293 9.5539 5.706 2.2267 0.9065 8.3750 0.927 6.4669 6.8282 4.7042 3.4873 7.883 3.498 2.3474 5.8636 9.379 8.4085 2.747 2.2859 4.3308 20.200 34.9469 2.266 From the simulated data i Table 5. we have ˆ i X i X (,i [ 5.4780 4.5974 5.907 2.087 34.023 ], ad m ˆ i 3.470 i ˆ m ˆ i 6.2940. m i 0
It is well kow that ˆ i 2 χ2 2 2 2 Y i ad therefore m i ˆ i 2 m i Y i. Let where Z ˆ i m i ˆ i 2 Y i 2 m i Y i Y i χ 2 2 2. Y i m i Y i i, 2,..., m For further details see also Huma, Chakraborti, ad Smit (200. To obtai a lower cotrol limit for the data i Table 5., the distributio of Z mi mi (Z, Z 2,..., Z m must be obtaied. Figure 5.: Distributio of Z mi mi (Z, Z 2,..., Z m, 00 000 simulatios The distributio of Z mi obtaied from 00 000 simulatios is illustrated i Figure 5.. The value Z 0.05 0.0844 is calculated such that the FAP is at a level of 0.05. The lower cotrol limit is the determied as LCL Z 0.05 m i ˆ i (0.0844(3.470 2.656. Sice ˆ i > 2.656 (i, 2,..., m it ca be cocluded that the scale parameter is uder statistical cotrol.
6 Lower Cotrol Limit for the Scale Parameter i Phase II I the rst part of this sectio, the lower cotrol limit i a Phase II settig will be derived usig the Bayesia predictive distributio. The followig theorem ca easily be proved: Theorem 6.. For the two-parameter expoetial distributio ( { f (x ij ;, µ i exp } (x ij µ i i, 2,..., m, j, 2,...,, ad x ij > µ i the posterior distributio of the parameter give the data is give by ( m( ˆ p ( data Γ (m ( ( ( m( + exp ˆ > 0 a Iverse Gamma Distributio. Proof. The proof is give i Appedix E. Theorem 6.2. Let ˆ f be the maximum likelihood estimator of the scale parameter i a future sample of observatios, the the predictive distributio fo ˆ f is f (ˆf data 2 (ˆf Γ [m ( + ] Γ ( Γ [m ( ] (ˆf ˆ ˆf m( + > 0 + which meas that ˆ f data ˆ m F 2( ;2m( where m ˆ ˆ i. i Proof. The proof is give i Appedix F. At β 0.0027 the lower cotrol limit is obtaied as.8847 for m 5 ad 0. ˆ m F 2( ;2m( (0.0027 3.470 5 (0.29945 Assumig that the process remais stable, the predictive distributio for ˆ f ca als be used to derive the distributio of the ru-legth, that is the umber of samples util the cotrol chart sigals for the rst time. The resultig regio of size β usig the predictive distributio for the determiatio of the ru-leght is deed as ˆ β R(β f (ˆf data dˆ f 2
where R (β (0;.8847 is the lower oe-sided cotrol iterval. Give ad a stable process, the distributio of the ru-legth r is Geometric with parameter ˆ ψ ( f (ˆf dˆ f where f (ˆ is the distributio of a future sample scale parameter estimator give. R(β The value of the parameter is however ukow ad its ucertaity is described by the posterior distributio p ( data. The followig theorem ca also be proved. Theorem 6.3. For give the parameter of the Geometric distributio is ( ψ ( ψ χ 2 2m( for give χ 2 2m( which meas that the parameter is oly depedet o χ2 2m( ad ot o. Proof. The proof is give i Appedix G. As metioed, by simulatig from p ( data the probability desity fuctio of f (ˆf as well as the parameter ψ ( ca be obtaied. This must be doe for each future sample. Therefore, by simulatig a large umber of values from the posterior distributio a large umber of ψ ( values ca be obtaied. A large umber of Geometric ad ru-legth distributios with dieret parameter values (ψ (, ψ ( 2,..., ψ ( l will therefore be available. The ucoditioal ru-legth distributio is obtaied by usig the Rao- Blackweel method, i.e., the average of the coditioal ru-legth distributios. I Table 6. results for the ru-legth at β 0.0027 for 0 ad dieret values for m are preseted for the lower cotrol limit of the scale parameter estimator. 3
4 Table 6.: Two Parameter Expoetial Ru Legths Results m mea (ARL media (ARL Mea (PDF Media (PDF Oe-sided Two-sided Low High Low High 0 5 040.87866 56.904434 06.07825 365 20.668063 3285.604745 90.5340623 4875.2548 0 6 859.2946037 528.04343 850.6496099 340 32.63677 2633.7792 03.84474 3585.574629 0 7 752.7840665 502.7053642 747.964747 330 37.365507 263.362738 07.504466 293.68755 0 8 692.3723 478.8328 688.5627995 320 47.7899272 904.50695 2.760763 2547.589774 0 9 644.233737 467.3792855 640.6357323 30 49.98895 707.55249 9.5874038 298.55465 0 0 66.400894 456.27994 63.095372 305 54.965304 58.5262 28.5788067 2094.95052 0 582.54467 445.48059 579.3980237 295 59.475642 466.358 33.3858464 874.78944 0 2 563.84757 445.48059 560.857002 295 65.74253 340.987775 39.7090749 68.4247 0 3 542.293566 440.999559 539.396828 290 70.6404787 264.44683 45.034248 534.26048 0 4 523.2584573 429.8629367 520.42609 285 70.6404787 93.03076 45.034248 466.358 0 5 54.4424077 429.8629367 5.680866 285 77.4499555 42.597024 49.98895 38.3827 0 6 498.708368 424.804032 496.040877 285 79.20395 064.049573 50.60839 283.08557 0 7 493.4525335 49.86607 490.7979663 280 79.20395 064.049573 54.965304 264.44683 0 8 49.9807528 424.804032 489.3787352 285 88.298034 09.99956 64.476967 228.23456 0 9 479.4053832 44.8994965 476.898625 280 84.5942927 978.005367 59.475642 75.93545 0 20 470.638992 40.055567 467.6270667 275 88.298034 95.3034035 64.476967 0.378304 0 50 408.680438 386.808266 406.4862335 265 235.674776 654.3843849 24.737467 726.2252763 0 00 389.048886 377.9609703 386.9577637 260 267.625879 548.056023 248.3896595 583.470484 0 500 374.375597 373.6276458 372.3573508 255 38.8260483 440.999559 308.3784626 450.837436 0 000 372.979505 373.6276458 370.9756458 255 333.430277 44.8994965 326.0292806 424.804032 0 5000 37.9360097 373.6276458 369.9394557 255 352.838786 39.329358 348.850358 395.96345 0 0000 37.7307984 373.6276458 369.7352037 255 356.882629 386.808266 356.882629 386.808266 mea(arl ad media(arl refer to results obtaied from the expected ru-legth while mea(pdf ad media(pdf refer to results obtaied from the probability desity fuctio of the ru-legth.
From Table 6. it ca be see that as the umber of samples icrease (larger m the mea ad media ru-legths coverge to the expected ru-legth of 370. Further dee ψ ( l l i ψ ( i. From Mezefricke (2002 it is kow ( that as l, ψ ( β 0.0027 ad the harmoic mea of the ucoditioal ru-legth will be 370. Therefore β 0.0027 it does ot matter how small m ad is, the harmoic mea of the ru-legth will always be β if l. I the case of the simulated example the mea ru-legth is 040.88 ad the media ru-legth 56.90. The reaso for these large values is the ucertaity i the parameter estimate because of the small sample size ad umber of samples ( 0 ad m 5. β however ca easily be adjusted to get a mea ru-legth of 370. 7 A Compariso of the Predictive Distributios ad Cotrol Charts for a Oe-sided Upper Tolerace Limit, U f A future sample tolerace limit is deed as U f ˆµ f k 2 ˆf where ˆµ f > µ ad ˆ f > 0. Also ( m f (ˆµ f µ, { exp m } (ˆµ f µ ˆµ f > µ which meas that f (U f µ,, ˆ ( m f { exp m [ U f (µ k ]} 2 ˆf U f > µ k 2 ˆf (7. A compariso will be made betwee f (U f data ad f (U f data. The dierece i the simulatio procedure for these two desity fuctios is that i the case of f (U f data it is assumed that 0 < µ < { ( } ( which results i a posterior distributio of p (µ data ( ˆ x ( x µ while for f (U f data it is assumed that < µ < ad the posterior distributio for µ is the p (µ data ( (ˆ ( x µ. I the followig gure comparisos are made betwee f (U f data ad f (U f data. 5
Figure 7.: Predictive Desities of f (U f data ad f (U f data The descriptive statistics obtaied from Figure 7. are preseted i Table 7.. Table 7.: Descriptive Statistics of ad f (U f data ad f (U f data Descriptive Statistics f (U f data f (Uf data Mea (U f 3394.7 3406.6 Media (U f 32.5 3225.8 Mode (Ûf 2900 2907 V ar (Ûf.237 0 6.2694 0 6 95% Equal-tail Iterval (736.5; 6027 (738; 606 99.73% Equal-tail Iterval (249.05; 7973 (249.9; 8620 Also the exact meas ad variaces for f (U f data are E (U f data x + KL (ah 3394.8 ad V ar (U f data 2 } {J m 2 + H2 KM { + ( ah 2 KM ( ( 2 K2 L 2}.2439 0 6 6
where a m (, H k 2 (m, J + k 2 2 (m ad K, L ad M deed as before. The exact meas ad variaces for f (U f data are ad Ẽ (U f data x + 2 (ah ˆ 345.0 { } V 2 ar (U f data (J ( 2 ( 3 m 2 + H2 + ( ah 2 2 (ˆ.29 0 6. 2 Ẽ (U f data ad V ar (U f data are derived from E (U f data ad V ar (U f data by deletig the term i K, L ad M. ( x It seems that the predictive itervals for f (U f data are somewhat wider tha i the case of f (U f data. I Table 7.2 comparisos are made betwee the ru-legths ad expected ru-legths i the case of U f for < µ < ad 0 < µ <. Table 7.2: Descriptive Statistics for the Ru-legths ad Expected Ru-legths i the Case of U f ; < µ < ; 0 < µ < ad β 0.08 < µ < 0 < µ < Descriptive Statistics f (r data Expected Ru-Legth f (r data Expected Ru-Legth Equal Tail Equal Tail Equal Tail Equal Tail Mea 444.95 444.95 48.68 49.68 Media 36.5 258.2 32. 248.03 Variace 7.6243 0 5 2.8273 0 5 5.8236 0 5 2.035 0 5 95% Iterval (0; 892.4 (0; 803.6 It is clear from Table 0. that the correspodig statistics do ot dier much. The mea, media ad variace ad 95% iterval are however somewhat larger for < µ <. 8 Coclusio This paper develops a Bayesia cotrol chart for moitorig the scale parameter, locatio parameter ad upper tolerace limit of a two-parameter expoetial distributio. I the Bayesia approach prior kowledge about the ukow parameters is formally icorporated ito the process of iferece by assigig a prior distributio to the parameters. The iformatio cotaied i the prior is combied with the likelihood fuctio to obtai the posterior distributio. By usig the posterior distributio the predictive distributios of ˆµ f, ˆ f ad U f ca be obtaied. The theory ad results described i this paper have bee applied to the failure mileages for military carriers aalyzed by Grubbs (97 ad Krishamoorthy ad Mathew (2009. The example illustrates the exibility ad uique features of the Bayesia simulatio method for obtaiig posterior distributios ad ru-legths for ˆµ f, ˆ f ad U f. 7
Refereces Grubbs, F., 97. Approximate ducial bouds o reliability for the two parameter egative expoetial distributio. Techometrics 3, 873876. Huma, S., Chakraborti, S., Smit, C., 200. Shewart-type cotrol charts for variatio i Phase I data aalysis. Computatioal Statistics ad Data Aalysis 54, 863874. Johso, N., Kotz, S., 970. Distributios i statistics : cotiuous uivariate distributios. New York ; Chichester, etc. : Wiley, icludes bibliographical refereces. Krishamoorthy, K., Mathew, T., 2009. Statistical Tolerace Regios: Theory, Applicatios ad Computatio. Wiley Series i Probability ad Statistics. Lawless, J., 982. Statistical models ad methods for lifetime data. Statistics i Medicie (3. Mezefricke, U., 2002. O the evaluatio of cotrol chart limits based o predictive distributios. Commuicatios i Statistics - Theory ad Methods 3(8, 423440. 8
Mathematical Appedices A Proof of Theorem 2. (a The posterior distributio p ( data is the same as the distributio of the pivotal quatity G Proof: 2ˆ. χ 2 2 2 Let Z χ 2 2 2 f (z 2 2 Γ( z 2 exp { 2 z} We are iterested i the distributio of 2ˆ Z. Therefore Z 2ˆ From this it follows that ad dz 2ˆ d Z. ( f (G f ( 2 Γ( (ˆ Γ( ( exp { 2 { 2ˆ 2ˆ exp 2 ˆ ˆ } p ( data } See Equatio 2.3. (b The posterior distributio p (µ data is the same as the distributio of the pivotal quatity G µ ˆµ χ2 2 ˆ. χ 2 2 2 Proof: Let F g (f χ 2 2 /2 χ 2 2 2 /(2 2 F 2,2 2 ( + f where 0 < f < We are iterested i the distributio of µ ˆµ 0 df (. dµ Therefore ˆ g (µ { + ˆ (ˆµ 0 µ} ˆ ( ˆ ( p (µ data 2ˆ ( 2 2F which meas that F (ˆµ µ ad ˆ x µ where < µ < ˆµ See Equatio 2.2. (c The posterior distributio of p(µ, data is the same as the distributio of the pivotal quatity G µ ˆµ χ2 2 2 (see Equatio.. 9
Proof: Let Z χ 2 2 the g ( z 2 exp { 2 z}. Let µ ˆµ z 2, the z 2 Therefore (ˆµ µ ad d z 2. dµ g (µ exp { (ˆµ µ} < µ < ˆµ p (µ, data See Equatio 2.5. B Proof of Theorem 3. As before ( m f (ˆµ f µ, { exp m } (ˆµ f µ ˆµ f > µ ad therefore f (ˆµ f µ, data ˆ 0 f (ˆµ f µ, p ( µ, data d. Sice it follows that p ( µ, data f (ˆµ f µ, data { ( x µ} Γ ( ( + exp { } ( x µ + ( x µ m [m (ˆµ f µ + ( x µ] + ˆµ f > µ. For < µ < x (, ( p (µ data ( (ˆ x µ ad f (ˆµ f, µ data f (ˆµf µ, data p (µ data. + m( (ˆ [m(ˆµ f µ+( x µ] + Therefore 20
f (ˆµ f data ˆµf f (ˆµ f, µ data dµ < ˆµ f < x ( x( f (ˆµ f, µ data dµ x ( < ˆµ < [ K ( x ˆµ f ] < ˆµ f < x ( K [ ˆ+m(ˆµ f x ( ] x ( < ˆµ f < where K ( m ( + m (ˆ. C Proof of Theorem 3.3 From Equatio. it follows that ˆ f χ2 2m 2 2m which meas that m 2 ( ( m m (ˆf exp f (ˆf m ˆ f Γ (m 0 < ˆ f < (C. The posterior distributio of (Equatio 2.3 is ( ˆ ( ( p ( data exp Γ ( 0 < <. Therefore f (ˆf data f (ˆf p ( data d 0 m m (ˆ Γ(m Γ( m 2 ( { ( } m+ (ˆf 0 exp mˆ f + ˆ d Γ(m+ 2m m (ˆ (ˆ f m 2 Γ(m Γ( (mˆ f +ˆ m+ 2 0 < ˆ f <. D Proof of Corollary 3.4 ˆ f χ2 2m 2 2m ad data ˆ 2 χ 2 2 2. 2
Therefore ˆ f data χ2 2m 2 2m 2 χ 2 2 2 ˆ, ˆ 2m 2 2 2 2m 2 ˆ f F 2m 2;2 2 ad ˆ f ˆ (m m ( F 2(m ;2(. E Proof of Theorem 6. Let ˆ m ˆ i. As metioed i Sectio 6 (see also Krishamoorthy ad Mathew (2009 that it is well kow that i ˆ i 2 χ2 2( which meas that Therefore f (ˆ ˆ 2 χ2 2m(. m( ( ( m( (ˆ exp Γ [m ( ] ˆ ( L ˆ i.e, the likelihood fuctio. As before we will use as prior p (. The posterior distributio ( ( p ˆ p ( data L ˆ p ( (ˆ m( Γ[m( ] ( m( + exp ( ˆ A Iverse Gamma distributio. 22
F Proof of Theorem 6.2 ˆ f 2 χ2 2(. Therefore f (ˆf data f (ˆf p ( data d 0 0 ( ( (ˆ f 2 exp ˆ f Γ( (ˆ m( Γ[m( ] ( m( + exp ( ( (ˆ f 2 (ˆ m( Γ( Γ[m( ] Γ[m( + ](ˆ m( Γ( Γ[m( ] ˆ d ( { m( + 0 exp [ˆf ˆ]} + d (ˆ f 2 (ˆ f +ˆ m( + ˆf > 0 From this it follows that where ˆ f data ˆ m F 2( ;2m( ˆ m ˆ i. i G Proof of Theorem 6.3 For give ψ ( p (ˆf ˆ m F 2( ;2m( (β ( p ( p 2 χ2 2( ˆ 2ˆ χ 2 2m( χ 2 2( 2 m F 2( ;2m( (β give ˆ m F 2( ;2m( (β ( p χ 2 2( χ2 2m( m F 2( ;2m( (β ( ψ χ 2 2m( give χ 2 2m( give χ 2 2m( 23