Bayesia Cotrol Charts for the Two-parameter Expoetial Distributio 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 Krishamoorthy Mathew (2009 a Bayesia procedure is applied to obtai cotrol limits for the locatio scale parameters, as well as for a oe-sided upper tolerace limit i the case of the two-parameter expoetial distributio. A advatage of the upper tolerace limit is that it moitors the locatio scale parameter at the same time. By usig Jereys' o-iformative prior, the predictive distributios of future maximum likelihood estimators of the locatio scale parameters are derived aalytically. The predictive distributios are used to determie the distributio of the ru-legth expected ru-legth. This paper illustrates the exibility uique features of the Bayesia simulatio method. Keywords: Jereys' prior, two-parameter expoetial, Bayesia procedure, ru-legth, cotrol chart Itroductio The two-parameter expoetial distributio plays a importat role i egieerig, life testig medical scieces. I these studies the data are positively skewed, the expoetial distributio is as importat as the ormal distributio is i samplig theory agricultural statistics. Researchers have studied various aspects of estimatio iferece for the two-parameter expoetial distributio usig either the frequetist approach or the Bayesia procedure. However, while parameter estimatio hypothesis testig related to the two-parameter expoetial distributio are well documeted i the literature, the research o cotrol charts has received little attetio. Ramalhoto Morais (999 developed a cotrol chart for moitorig the scale parameter while Sürücü Sazak (2009 preseted a cotrol chart scheme i which momets are used. Mukherjee, McCracke, Chakraborti (204 o the other h proposed several cotrol charts moitorig schemes for the locatio the scale parameters of the two-parameter expoetial distributio. I this paper cotrol charts for the locatio scale parameters as well as for a oe-sided upper tolerace limit will be developed by derivig their predictive distributios usig a Bayesia procedure. Bayarri García-Doato (2005 give the followig reasos for recommedig a Bayesia aalysis: Cotrol charts are based o future observatios Bayesia methods are very atural for predictio. Ucertaity i the estimatio of the ukow parameters is adequately hled. Implemetatio with complicated models i a sequetial sceario poses o methodological diculty, the umerical diculties are easily hled via Mote Carlo methods; Objective Bayesia aalysis is possible without itroductio of exteral iformatio other tha the model, but ay kid of prior iformatio ca be icorporated ito the aalysis, if desired. Krishamoorthy Mathew (2009 Hah Meeker (99 deed a tolerace iterval as a iterval that is costructed i such a way that it will cotai a specied proportio or more of the populatio with a certai degree of codece. The proportio is also called the cotet of the tolerace iterval. As opposed to codece itervals that give iformatio o ukow populatio parameters, a oe-sided upper tolerace limit for example provides iformatio about a quatile of the populatio. 2
2 Prelimiary Statistical Results I this sectio the same otatio will be used as give i Krishamoorthy Mathew (2009. The two-parameter expoetial distributio has the probability desity fuctio f (x; µ, { exp (x µ x > µ, µ > 0, > 0 µ is the locatio parameter the scale parameter. Let X, X 2,..., X be a sample of observatios from the two-parameter expoetial distributio. The maximum likelihood estimators for µ are give by ˆµ X ( ˆ ( Xi X ( X X( i X ( is the miimum or the rst order statistic of the sample. It is well kow (see Johso Kotz (970; Lawless (982; Krishamoorthy Mathew (2009 that ˆµ ˆ are idepedetly distributed with (ˆµ µ χ2 2 2 ˆ χ2 2 2 2. (2. 3 Bayesia Procedure 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 followig theorems ca ow be proved. Theorem 3.. The joit posterior distributio of µ is p (, µ data p (µ, L (µ, data ( K + { exp ( µ 0 < µ < x (, 0 < < (3. K Γ ( { ( ˆ (. 3
Proof. The proof is give i Appedix A. Theorem 3.2. The posterior distributio of µ is p (µ data 0 p (, µ data d { ( ( ( ˆ ( µ 0 < µ < x (. (3.2 Proof. The proof is give i Appedix B. Theorem 3.3. The posterior distributio of is p ( data 0 p (, µ data dµ ( K { exp ( ˆ exp ( 0 < <. (3.3 Proof. The proof follows easily from the proof of Theorem 3.. Theorem 3.4. The coditioal posterior distributio of µ give is p (µ, data p(,µ data p( data K 2 exp ( µ (3.4 K 2 { exp ( x(. Proof. The proof follows easily from the proof of Theorem 3.. Theorem 3.5. The coditioal posterior distributio of give µ is p ( µ, data p(,µ data p(µ data ( K + { 3 exp ( µ 0 < < (3.5 K 3 { ( µ. Γ ( Proof. The proof is give i Appedix C. 4
4 The Predictive Distributios of Future Sample Locatio Scale Maximum Likelihood Estimators, ˆµ f 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 ˆ f X f ˆµ f. To obtai cotrol charts for ˆµ f ˆ f their predictive distributios must rst be derived. The followig theorems ca ow be proved. Theorem 4.. The predictive distributio of a future sample locatio maximum likelihood estimator, ˆµ f, is give by {[ K f (ˆµ f data {[ K ( ˆµ f ] [ m(ˆµ f x (+ˆ mˆµ f + ] [ ] mˆµ f + ] 0 < µ f < x ( x ( < ˆµ f < (4. K ( m { (. ( ( + m ˆ Proof. The proof is give i Appedix D. Theorem 4.2. The mea of ˆµ f is give by E (ˆµ f data KL ( a the variace by { 3 V ar (ˆµ f data m 2 ( 2 + ( a2 ( 2 KM ( a 2 K2 L 2 a m (, { ( K ( ˆ L ( 2 {( 2 ˆ (, ( 2 M ( 3 {( 3 ˆ ( 3. 5
Proof. The proof is give i Appedix E. Theorem 4.3. The predictive distributio of a future sample scale maximum likelihood estimator, ˆ f, is give by f (ˆf data { m m Γ(m+ 2 ( ( Γ(m Γ( ˆm 2 f { ( m+ 2 ( mˆ f +ˆ mˆ f + m+ 2 ˆ f > 0. (4.2 Proof. The proof is give i Appedix F. Theorem 4.4. The mea variace of ˆ f is give by (m E (ˆf data m ( KL variace { V ar (ˆf data 2 (m KM (m m ( 2 m ( K 2 L 2. Proof. The proof is give i Appedix G. 5 Example The followig data is give i Grubbs (97 as well as i Krishamoorthy Mathew (2009. The failure mileages give i Table 5. t a two-parameter expoetial distributio. Table 5.: 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( 835.2 9. As metioed i the itroductory sectio, the aim of this article is to obtai cotrol charts for locatio scale maximum likelihood estimates as well as for a oe-sided upper tolerace limit. Therefore i the ext sectio a cotrol chart for a future locatio maximum likelihood estimate will be developed. 6 Cotrol Chart for ˆµ f It is well kow that statistical quality cotrol is actually implemeted i two phases. I Phase I the primary iterest is to assess process stability. The practitioer must therefore be sure that the process is i statistical cotrol before cotrol limits ca be determied for olie moitorig of the process i Phase II. 6
By usig the predictive distributio (deed i Equatio 4. a Bayesia procedure will be developed i Phase II to obtai a cotrol chart for ˆµ f. Assumig that the process remais stable, the predictive distributio ca be used to derive the distributio of the ru-legth average ru-legth. For the example give i Table 5. (failure mileage data the predictive distributio, f (ˆµ f data for m 9 future data is illustrated i Figure 6.. Figure 6.: Distributio of ˆµ f, 9, m 9 Mea (ˆµ f 68.78, Media (ˆµ f 63.9, Mode (ˆµ f 62, V ar (ˆµ f 3888.7 95% Iterval (ˆµ f (55.8; 37.45 99.73% Iterval (ˆµ f (3.527; 489.52 From Figure 6. it follows that for a 99.73% two-sided cotrol chart the lower cotrol limit is LCL 3.527 the upper cotrol limit is UCL 489.52. Let R (β represets those values of ˆµ f that are smaller tha LCL larger tha UCL. The ru-legth is deed as the umber of future ˆµ f values (r util the cotrol chart sigals for the rst time (Note that r does ot iclude the ˆµ f value whe the cotrol chart sigals. Give µ a stable Phase I process, the distributio of the ru-legth r is geometric with parameter ˆ ψ (µ, f (ˆµ f µ, dˆµ f R(β f (ˆµ f µ, m { exp m (ˆµ f µ ˆµ f > µ i.e., the distributio of ˆµ f give that of µ are kow. See also Equatio 2.. The values of µ are however ukow the ucertaity of these parameter values are described by their joit posterior distributio p (, µ data give i Equatio 3.. By simulatig µ from p (, µ data p ( µ, data p (µ data the probability desity fuctio of f (ˆµ f µ, as well as the parameter ψ (µ, ca be obtaied. This must be doe for each future sample. I other words, for each future sample µ must rst be simulated from p (, µ data the ψ (µ, 7
calculated. Therefore, by simulatig all possible combiatios of µ from their joit posterior distributio a large umber of ψ (µ, values ca be obtaied. Also, a large umber of geometric distributios, i.e., a large umber of ru-legth distributios each with a dieret parameter value (ψ (µ,, ψ (µ 2, 2,..., ψ (µ m, m ca be obtaied. As metioed the ru-legth r for give µ is geometrically distributed with mea E (r µ, ψ (µ, ψ (µ, variace V ar (r µ, ψ (µ, ψ 2 (µ,. The ucoditioal momets, E (r data, E ( r 2 data V ar (r data ca therefore be obtaied by simulatio or umerical itegratio. For further details refer to Mezefricke (2002, 2007, 200a,b. The mea of the predictive distributio of the ru-legth for the 99.73% two-sided cotrol limits is E (r data 37526, much larger tha the 370 that oe would have expected if β 0.0027. The reaso for this large average ru-legth is the small sample size large variatio i the data. The media rulegth 450. Dee ψ (µ, m m i ψ (µ i, i. From Mezefricke (2002 it follows that if m, the ψ (µ, β the harmoic mea of r β. For β 0.0027 the harmoic mea would therefore be 370. I Table 6. the average ru-legth for dieret values of β are give. Table 6.: β Values Correspodig Average Ru-legth β 0.0027 0.003 0.005 0.007 0.009 0.0 0.05 0.02 0.025 0.0258 0.03 E (r data 37526 3260 023 4655 2807 233 987.6 594.2 399.7 369.67 280. I Figure 6.2 the distributio of the ru-legth for 9, m 9 β 0.0258 is illustrated i Figure 6.3 the histogram of the expected ru-legth is give. Figure 6.2: Ru-legth, 9, m 9, β 0.0258 E (r data 369.67, Media(r data 7.22, V ar (r data 6.5275 0 6 95% iterval (r data (0; 242.2 8
Figure 6.3: Expected Ru-legth, 9, m 9, β 0.0258 Mea 36.4, Media.93, V ar 3.757 0 6, Mea (ψ 0.026 95% iterval (0; 94.3 7 Cotrol Chart for ˆ f I this sectio a cotrol chart for ˆ f, a future scale maximum likelihood estimator, will be developed. The predictive distributio f (ˆf data give i Equatio 4.2 is displayed i Figure 7. for the example previously give m 9. Figure 7.: Predictive Distributio of ˆ f, 9, m 9 Mea (ˆf 876.73, Media (ˆf 829.23, Mode (ˆf 743.34 9
For a 99.73% two-sided cotrol chart the lower cotrol limit is LCL 297.5 the upper cotrol limit is UCL 2278. R (β represets those values of ˆf that are smaller tha LCL larger tha UCL. Give a stable Phase I process, the distributio of the ru-legth is geometric with parameter ˆ ψ ( f (ˆf dˆ f R(β f (ˆf is deed i Equatio 2.. As before the value of is ukow, but ca be simulated from the posterior distributio p ( data give i Equatio 3.3 or equivaletly by rst simulatig µ from p (µ data the from p ( µ, data. By simulatig the probability desity fuctio f (ˆf Equatio 3.3 as well as the parameter ψ ( ca be obtaied. As metioed i Sectio 6, this must be doe for each future sample. The mea of the predictive distributio of the ru-legth for the 99.73% two-sided cotrol limits is E (r data 888.6. As i the case of ˆµ f, this is much larger tha the 370 that oe would have expected if β 0.0027. I Table 7. the average ru-legth versus probabilities β are give. Table 7.: β Values Correspodig Average Ru-legth β 0.0027 0.003 0.005 0.007 0.009 0.0 0.05 0.08 0.02 0.025 E (r data 0939.2 9029.8 357.7 93 238.2 00.7 52. 372.4 3.2 2.8 I Figure 7.2 the distributio of the ru-legth for 9, m 9 β 0.08 is illustrated i Figure 7.3 the histogram of the expected ru-legth is give. Figure 7.2: Ru Legth, 9, m 9, β 0.08 E (r data 372.4, Media (r data 27.4, V ar (r data 3.955 0 5 95% iterval (r data (0; 602.5 0
Figure 7.3: Expected Ru-legth, 9, m 9, β 0.08 Mea 372.4, Media 238.38, V ar.2698 0 5, Mea (ψ 0.077 95% iterval (7.809; 096.7 8 A Bayesia Cotrol Chart for a Oe-sided Upper Tolerace Limit As metioed i the itroductio, a codece iterval for a quatile is called a tolerace iterval. A oe-sided upper tolerace limit is therefore a quatile of a quatile. It ca easily be show that the p quatile of a two-parameter expoetial distributio is give by q p µ l ( p. By replacig the parameters by their geeralized pivotal quatities (GPQs, Krishamoorthy Mathew (2009 showed that a GPQ for q p ca be obtaied as G qp [ ] χ 2 ˆµ 2 + 2 l ( p ˆ. χ 2 2 2 Let E p;α deotes the α quatile of E p χ2 2 +2 l( p χ 2 2 2, the ˆµ E p;α ˆ ˆµ k2 ˆ (8. is a α upper codece limit for q p, which meas that (p, α is a upper tolerace limit for the expoetial (µ, distributio. A advatage of the upper tolerace limit is that it moitors the locatio scale parameters of the two-parameter expoetial distributio at the same time. It is show i Roy Mathew (2005 Krishamoorthy Mathew (2009 that the upper lower tolerace limits are actually exact, which meas that they have the correct frequetist coverage probabilities.
9 The Predictive Distributio of a Future Sample Upper Tolerace Limit From Equatio 8. it follows that a future sample upper tolerace limit is deed as U f ˆµ f k 2 ˆf ˆµ f > µ ˆ f > 0. From Equatio 2. it follows that ( 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 (9. From Equatio 9. it ca be see that the derivatio of the ucoditioal predictive desity fuctio f (U f data will be quite complicated. A approximatio of the desity fuctio ca however be obtaied by usig the followig Mote Carlo simulatio procedure:. Simulate µ from p (, µ data. This ca be achieved by rst simulatig µ from p (µ data deed i Equatio 3.2 the from p ( µ, data deed i Equatio 3.5. 2. For give, simulate ˆ f from χ2 2m 2 2m. 3. Substitute the simulated µ,, ˆ f values i Equatio 9. draw the expoetial distributio. Repeat this procedure l times obtai the average of the l simulated expoetial desity fuctios (Rao-Blackwell method to obtai the ucoditioal predictive desity f (U f data. Although (as metioed the derivatio of the exact ucoditioal predictive desity will be quite complicated the exact momets ca be derived aalytically. The followig theorem ca be proved: Theorem 9.. The exact mea variace of U f give by ˆµ f k 2 ˆf, a future sample tolerace limit, is E (U f data + KL (ah (9.2 V ar (U f data 2 {J m 2 + H2 KM { + ( ah 2 KM ( ( 2 K2 L 2 (9.3 H { k 2 (m J + k 2 2 (m. K, L, M a are deed i Theorem 4.2. 2
Proof. The proof is give i Appedix H. For the failure mileage data give i Table 5., k 2 3.6784 if m 9 by usig 0, 000, 000 Mote Carlo simulatios as described i (, (2 (3, the ucoditioal predictive desity fuctio ca be obtaied is illustrated i Figure 9.. Figure 9. is therefore the distributio of a upper future tolerace limit for the mileages of the ext 9 military persoel carriers that will fail i service. Figure 9.: Predictive Desity of f (U f data Mea (U f 3394.7, Mode (U f 2900, Media (U f 32.5, V ar (U f.237 0 6 95% iterval (U f (736.5; 6027 99.73% iterval (U f (249.05, 7973 As i the previous sectios the predictive distributio ca be used to derive the ru-legth average ru-legth. From Figure (9. it follows that for a 99.73% two-sided cotrol chart the lower cotrol limit is LCL 249.05 the upper cotrol limit is UCL 7973. R (β therefore represets those values of U f that are smaller tha LCL larger tha UCL. As before, the ru-legth is deed as the umber of future U f values (r util the cotrol chart sigals for the rst time. Give µ the distributio of the ru-legth r is geometric with parameter ˆ ψ (µ, f (U f µ, du f R (β f (U f µ, is the distributio of a future U give that µ are kow. As metioed before the values of µ are however ukow the ucertaity of these parameter values are described by their joit posterior distributio p (µ, data. By simulatig µ from p (µ, data, the probability desity fuctio f (U f µ, ca be obtaied from Equatios 2. 9. i the followig way. I. f ( U f µ,, χ 2 2m 2 ( m exp { m [ ( U f µ k χ 2 2m 2 2 2m ]. II. The ext step is to simulate l 00000 χ 2 2m 2 values to obtaie l expoetial desity fuctios for give µ. 3
III. By averagig the l desity fuctios (Rao-Blackwell method f (U f µ, ca be obtaied also ψ (µ,. This must be doe for each future sample. I other words, for each future sample µ must be simulated from p (µ, data the the steps described i (I., (II. (III.. The mea of the predictive distributio of the ru-legth for the 99.73% two-sided cotrol limits is E (r data.709 0, much larger tha the 370 that would have bee expected for β 0.0027. I Table 9. the average ru-legths versus probabilities β are give. Table 9.: β Values Correspodig Average Ru-legth β 0.007 0.009 0.0 0.05 0.08 0.02 0.025 E (r data 0000 240.2 020.7 530. 374.2 300.2 208.8 I Figure 9.2 the distributio of the ru-legth for 9, m 9 β 0.08 is illustrated Figure 9.3 the histogram of the expected ru-legth is give. Figure 9.2: Distributio of Ru-legth whe β 0.08 E (r data 374.2, Media (r data 32., V ar (r data 5.8236 0 5 95% iterval (r data (0; 803.6 4
Figure 9.3: Expected Ru-legth whe β 0.08 Mea 374.2, Media 248.03, V ar 2.035 0 5 95% iterval (3.554; 463.2 0 Coclusio This paper develops a Bayesia cotrol chart for moitorig the scale parameter, locatio parameter 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 U f ca be obtaied. The theory results described i this paper have bee applied to the failure mileages for military carriers aalyzed by Grubbs (97 Krishamoorthy Mathew (2009. The example illustrates the exibility uique features of the Bayesia simulatio method for obtaiig posterior distributios ru-legths for ˆµ f, ˆ f U f. Refereces Bayarri, M., García-Doato, G., 2005. A bayesia sequetial look at u-cotrol charts. Techometrics 47(2, 425. Grubbs, F., 97. Approximate ducial bouds o reliability for the two parameter egative expoetial distributio. Techometrics 3, 873876. Hah, G., Meeker, W., 99. Statistical Itervals: A Guide for Practitioers. Joh Wiley & Sos, Ic. 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 Computatio. Wiley Series i Probability Statistics. Lawless, J., 982. Statistical models methods for lifetime data. Statistics i Medicie (3. 5
Mezefricke, U., 2002. O the evaluatio of cotrol chart limits based o predictive distributios. Commuicatios i Statistics - Theory Methods 3(8, 423440. Mezefricke, U., 2007. Cotrol chart for the geeralized variace based o its predictive distributio. Commuicatios i Statistics - Theory Methods 36(5, 03038. Mezefricke, U., 200a. Cotrol chart for the variace the coeciet of variatio based o their predictive distributio. Commuicatios i Statistics - Theory Methods 39(6, 2930294. Mezefricke, U., 200b. Multivariate expoetially weighted movig average chart for a mea based o its predictive distributio. Commuicatios i Statistics - Theory Methods 39(6, 29422960. Mukherjee, A., McCracke, A., Chakraborti, S., 204. Cotrol charts for simultaeous moitorig of parameters of a shifted expoetial distributio. Joural of Quality Techology, Accepted. Ramalhoto, M., Morais, M., 999. Shewart cotrol charts for the scale parameter of a weibull cotrol variable with xed variable samplig itervals. Joural of Applied Statistics 26, 2960. Roy, A., Mathew, T., 2005. A geeralized codece limit for the reliability fuctio of a two-parameter expoetial distributio. Joural of Statistical Plaig Iferece 28, 50957. Sürücü, B., Sazak, H., 2009. Moitorig reliability for a three-parameter weibull distributio. Reliability Egieerig System Safety 94 (2, 503508. 6
Mathematical Appedices A Proof of Theorem 3. K x( ( + { 0 0 exp ( µ dµd ( + { 0 exp [ x ( exp ( ] µ 0 dµ d Sice ˆ x( 0 exp ( µ dµ { exp ( x( it follows that K ( { 0 exp ˆ d ( 0 { exp d ( ( ˆ Γ ( ( ( Γ ( { ( ( ( Γ ( ˆ. Sice K ( K the theorem follows. B Proof of Theorem 3.2 p (µ data 0 p (µ, data d ( K + { 0 exp ( µ d { K ( µ Γ (. By substitutig K the result follows. C Proof of Theorem 3.5 Sice K 3 ( + { 0 exp ( µ d Γ( {( µ, the result follows. 7
D Proof of Theorem 4. From Equatio 2. it follows that ˆµ f µ, χ2 2 2m + µ ˆµ f > µ. Therefore ( m f (ˆµ f µ, { exp m (ˆµ f µ ˆµ f > µ. Further ˆ ˆ f (ˆµ data f (ˆµ µ, p ( µ, data p (µ data ddµ p ( µ, data { ( µ Γ ( ( + exp { ( µ 0 < < { ( p (µ data ( ˆ ( ( µ 0 < µ < x (. Now f (ˆµ f µ, data 0 f (ˆµ f µ, p ( µ, data d m{( µ Γ( ( +2 { 0 exp [m (ˆµ f µ + ( µ] d + ( µ m [m(ˆµ f µ+( µ] + ˆµ f > 0 f (ˆµ f data ˆµf f (ˆµ 0 f, µ data dµ 0 < ˆµ f < x ( x( f (ˆµ 0 f, µ data dµ x ( < ˆµ f < f (ˆµ f, µ data + ( m { ( {(mˆµ ( ˆ f + µ ( + m (. Therefore f (ˆµ f data K {[ K {[ ( ˆµ f ] [ m(ˆµ f x (+ˆ mˆµ f + ] [ ] mˆµ f + 0 < ˆµ f < x ( ] x ( < ˆµ f < K ( m ( + m { ( ˆ (. 8
E Proof of Theorem 4.2 Expected Value of ˆµ f It follows from Equatio 2. that Therefore ˆµ f χ2 2 2m + µ. E (ˆµ f µ, m + µ. From Equatio 3. it follows that p ( µ, data { ( µ Γ ( ( + exp { ( µ which meas that E ( µ, data ( µ ( therefore E (ˆµ f µ, data ( µ m( + µ ( a µ + a (E. a m (. Also from Equatio 3.2 it follows that p (µ data K ( µ 0 < µ < x ( { ( K ( ˆ (. Now E {( µ data E (µ data E {( µ data + K ( 2 { ( 2 ˆ ( 2 KL which meas that E (µ data KL +. (E.2 Substitute Equatio E.2 i E. the result follows as E (ˆµ f data KL ( a. 9
Variace of ˆµ f From Equatio 2. it also follows that Further Sice V ar (ˆµ f µ, data 2 4 4m 2 2 m 2. V ar (ˆµ f µ, data E µ {V ar (ˆµ f µ, + V ar µ {E (ˆµ f µ,. E ( 2 µ, data { ( µ2 ( ( 2 { ( µ2 V ar ( µ, data ( 2 ( 2 it follows that V ar (ˆµ f µ, data m 2 {( µ 2 ( ( 2 + m 2 {( µ 2 ( 2 ( 2 3 ( µ 2 m 2 ( 2 ( 2. Also Sice V ar (ˆµ f data E µ {V ar (ˆµ f µ, data + V ar µ {E (ˆµ f µ, data. M E ( µ 2 KM ( 3 {( 3 ˆ ( 3 it follows that Further E µ {V ar (ˆµ f µ, data KM. m 2 ( 2 ( 2 V ar µ {E (ˆµ f µ, data ( a 2 V ar (µ data ( V ar (µ data E µ {µ KL 2 3 E µ { ( µ 2 2 ( µ KL + K 2 L 2 KM 2 KL KL + K2 L 2 KM K 2 L 2. Therefore V ar µ {E (ˆµ f µ, data ( a 2 { KM K2 L 2 V ar (ˆµ f data { 3 m 2 ( 2 + ( a2 ( 2 KM ( a 2 K2 L 2. 20
F Proof of Theorem 4.3 From Equatio 2. it follows that ˆ f χ2 2m 2 2m ˆf > 0. Therefore m 2 ( ( m m (ˆf exp f (ˆf m ˆ f Γ (m ˆ f > 0. The posterior distributio of is ( { ( p ( data K exp ˆ exp ( K Γ ( { ( ( (see Theorem 3.3. The ucoditioal predictive desity fuctio of ˆ f is therefore give by f (ˆf data f (ˆf p ( data d 0 m m ˆ ( Γ(m f K { [ ( ] [ ( ] m+ 0 exp mˆ f + ˆ exp mˆ f + d m m Γ(m+ 2 Γ(m Γ( m+ 2 ( { ( mˆ f +ˆ { ( ˆ ( m 2 (ˆf m+ 2 mˆ f + ˆ f > 0. G Proof of Theorem 4.4 Expected Value of ˆ f From Equatio 2. it follows that Therefore ˆ f µ, χ2 2m 2 2m. E (ˆf µ, (m m (m V ar (ˆf µ, m 2 2. 2
By usig p ( µ, data (give i Equatio 3.5 it follows that E ( µ, data ( µ ( therefore E (ˆf µ, data (m ( µ m (. Sice p (µ data K ( µ 0 < µ < x ( it follows that E ( µ KL (m E (ˆf µ, data m ( KL. Variace of ˆ f V ar (ˆf µ, data [ ] E µ {V ar (ˆf µ, + V ar µ E (ˆf µ, (m m 2 {( µ 2 ( ( 2 + (m 2 m 2 {( µ 2 ( 2 ( 2 (m 2 m 2 ( 2 ( 2 (m + 2 ( µ2. Further { { V ar (ˆf data E µ V ar (ˆf µ, data + V ar µ E (ˆf µ, data E µ {V ar (ˆf data (m 2 m 2 ( 2 (m + 2 KM. ( 2 Also (m 2 V ar µ {E (ˆf 2 µ, data m 2 2 V ar (µ data ( therefore V ar µ {E (ˆf µ, data { ( (m 2 2 m E µ KL 2 2 ( 2 (m 2 2 E {( µ 2 2 KL ( µ + K 2 L 2 m 2 ( 2 { (m 2 2 m 2 ( KM 2 KL KL + K2 L 2 2 { (m 2 2 m 2 ( KM K2 L 2. 2 22
From this it follows that V ar (ˆf data (m 2 (m + 2 KM { + (m 2 2 m 2 ( 2 ( 2 m 2 ( KM K2 L 2 2 2 (m m( { KM ( 2 (m m( K 2 L 2. H Proof of Theorem 9. Proof of E (U f data From Equatio 9. it follows that f (U f µ,, ˆ ( m f { exp m [ U f (µ k ] 2 ˆf U f > µ k 2 ˆf which meas that E (U f µ,, ˆ f m + µ k 2 ˆf. Sice it follows that E (U f µ, µ + m ˆ f, µ χ2 2m 2 2m { k 2 (m µ + m H. Also sice the posterior distributio p ( µ, data { ( µ Γ ( ( + exp { ( µ 0 < < it follows that therefore E ( µ, data ( µ E (U f µ, data ah + µ ( ah Further a { ( p (µ data ( ˆ m (. ( ( µ 0 < µ < x ( therefore E (µ data KL which meas that E (U f data + KL (ah. 23
Proof of V ar (U f data From Equatio 9. it follows that V ar (U f µ,, ˆ ( 2 f. m Now V ar (U f µ, V {E (U arˆf f µ,, ˆ { f + V ar (U Eˆf f µ,, ˆ f V arˆf ( k2 2 V ar m + µ k 2 ˆf + Eˆf { ( m ( χ 2 2m 2 2m + ( 2 m ( 2 m { k2 2 (m + ( 2 m J. 2 Further V ar (U f µ, data E µ {V ar (U f µ, + V ar µ {E (U f µ, E µ { ( m 2 J + V ar µ { µ + m H J m 2 E ( 2 µ, data + H2 m 2 V ar ( µ, data. Sice it follows that Fially E ( 2 µ, data { ( µ2 ( ( 2 { ( µ2 V ar ( µ, data ( 2 ( 2 V ar (U f µ, data E µ {V ar (U f µ, data 2 ( µ 2 {J m 2 + H2. ( ( 2 2 {J m 2 + H2 E ( µ 2. ( ( 2 From p (µ data it follows that E ( µ 2 KM E µ {V ar (U f µ, data 2 {J m 2 + H2 KM. ( ( 2 24
Also V ar µ {E (U f µ, data V ar {ah + µ ( ah ( ah 2 V ar (µ data ( ah 2 ( KM K2 L 2. Therefore V ar (U f data 2 {J m 2 + H2 KM + ( ah 2 { KM ( ( 2 K2 L 2. 25