On multivariate folded normal distribution

Transcription

1 Sankhyā : The Indan Journal o Stattc 03, Volume 75-B, Part, pp. -5 c 03, Indan Stattcal Inttute On multvarate olded normal dtrbuton Ah Kumar Chakraborty and Moutuh Chatterjee Indan Stattcal Inttute, Kolkata, Inda Abtract Folded normal dtrbuton are when we try to nd out the dtrbuton o abolute value o a uncton o a normal varable. The properte and ue o unvarate and bvarate olded normal dtrbuton have been tuded by varou reearcher. We tudy here the properte o multvarate olded normal dtrbuton and ndcate ome area o applcaton. AMS 000 ubject clacaton. Prmary 6E5; Secondary 6H0. Keyword and phrae. Folded normal dtrbuton, multvarate dtrbuton, multvarate olded normal dtrbuton, proce capablty ndce. Introducton Normal dtrbuton ha been wdely ued a the underlyng dtrbuton or many qualty charactertc ued n varou ndutre. However Johnon 963 ha ponted out that, n many tuaton, even the underlyng dtrbuton normal, whle collectng data, or example, on derence and devaton n meaurement, oten the algebrac gn o the data rretrevably lot. The reultng oberved varable no more ollow a normal dtrbuton- rather t ollow a olded normal dtrbuton ee Johnon, 963 and Kng, 988. Ln 004 ued olded normal dtrbuton to tudy the magntude o devaton o an automoble trut algnment. Leone, Nelon and Nottngham 96 have mentoned ome more applcaton o olded normal dtrbuton pecally when meaurng traghtne and latne o any object. The pd o unvarate olded normal dtrbuton, a propoed by Leone et al. 96, gven by X z h Z z+h Z z { σ exp } { z μ + exp } z + μ, z > 0, π σ σ

2 A.K. Chakraborty and M. Chatterjee where, X ollow unvarate olded normal UFN dtrbuton wth mean μ and varance σ and Z Nμ, σ wthh Z. beng t pd. Here, μ σ π exp μ σ + μ μ, σ { σ μ + σ σ π exp μ σ + μ μ } σ and. the cd o unvarate tandard normal dtrbuton. The ubcrpt ued to dtnguh the mean and varance o a olded normal dtrbuton rom that o normal dtrbuton. Elandt 96 ormulated a general expreon or the r th moment o unvarate olded normal dtrbuton. The author alo propoed two method o etmatng the parameter μ and σ o the parent normal dtrbuton, vz., one baed on the rt and econd raw and central moment o olded normal dtrbuton and the other baed on t thrd and ourth raw and central moment. Many nteretng reult o the theory o tattcal qualty control owe ther development rom the olded normal dtrbuton. Kng 988 made a thorough reearch regardng the poble tuaton where a olded dtrbuton, epecally a olded normal dtrbuton, may are. He put empha on ome practcal conequence encountered durng proce capablty analy n many common ndutral procee. In act, Ln 004 ha ponted out that olded normal proce data common n mechancal ndutre. Johnon 963 dcued the ue o CUSUM control chart when the underlyng varable ollow olded normal dtrbuton whle, Lao 00 ha propoed economc tolerance degn or the olded normal data n manuacturng ndutre. Unvarate olded normal dtrbuton ha alo ound ueul applcaton n Ln 004 and Ln 005 whle tudyng the properte o ome unvarate proce capablty ndce PCI. Vannman 995 propoed a upertructure o unvarate PCI vz. C p u, v, gven by d u μ M C p u, v 3 σ + vμ T, u 0,v 0 where, USL and LSL are repectvely the upper and lower peccaton lmt o a proce, d USL LSL/, M USL+ LSL/, μ, σ and T are the mean, varance and the target o the proce and u and v are the calar contant that can take any non-negatve nteger value. However, Taam, Subbaah and Lddy 993 ponted out that n mot o the practcal tuaton, the manuacturng procee cont o more than

3 On multvarate olded normal dtrbuton 3 one nterdependent qualty charactertc. A a reult, calculaton o proce capablty ndce or ndvdual component may yeld mleadng reult. A number o multvarate proce capablty ndce MPCI are developed to meaure the capablty o a proce havng multple varable to control. See Taam et al. 993, Wang and Chen 998, Polanky 00, Krman and Polanky 009 and the reerence there n or urther detal on multvarate proce capablty ndce MPCI. The necety o contructng the multvarate olded normal dtrbuton wa elt by Chatterjee and Chakraborty 03 whle tudyng the properte o a upertructure o MPCI, analogou to C p u, v, gven by C G u, v d ud Σ d ud 3 +vμ T Σ μ T where, USL and LSL are, repectvely, the upper and lower peccaton lmt o the th qualty charactertc, or p, D μ M, μ M,, μ p M p, d USL LSL /, USL LSL /,, USL p LSL p /, T T,T,,T p and M M,M,,M p wth M USL + LSL /, p. Here, T the target value and M the nomnal value or the th charactertc o the tem. p denote the total number o qualty charactertc under conderaton, Σ varance covarance matrx o the vector X o the p qualty charactertc X,X,,X p and μ the mean vector o X. Here t aumed that X N p μ, Σ and u and v are the calar contant that can take any nonnegatve nteger value. Note that here we ue bold aced letter to denote vector or the remanng part o the artcle. The pd o a bvarate olded normal BVFN dtrbuton wa developed by Parak and Panareto 00. Suppoe that Z Z,Z σ N, Σ or μ,μ and Σ σ σ σ, then, Z, Z ollow BVFN wth mean vector and dperon matrx Σ, where the upercrpt denote the dmenon o Z. Thepdo Z, Z canbe derved a: Z, Z z,z h Z,Z u, v, or z,z > 0, u z, z v z, z where h Z,Z.,. denote the pd o bvarate normalbvn dtrbuton wth mean vector and varance-covarance matrx Σ. However, Parak and Panareto 00 have derved the expreon o the mg or the bvarate olded tandard normal dtrbuton, aumng

4 4 A.K. Chakraborty and M. Chatterjee 0 and Σ I ung Tall 96 ormula or mg o the truncated mult-normal dtrbuton, where only the cae o tandard multvarate normal dtrbuton condered. It to be noted that whle dervng the mg, Parak and Panerato 00 have decompoed the exponent o the correpondng bvarate normal dtrbuton and a a reult, whle generalzng the expreon or p 3, one ha to undergo dcult computatonal procedure. In the preent paper, we have developed an expreon or multvarate olded normal dtrbuton, ollowng Chakraborty and Chatterjee 00 and ound out t mean vector, dperon matrx and the mg. Etmaton procedure or the parameter can be preented a a eparate artcle. In the ollowng ecton, a ew notaton ued throughout the text, are preented. In Secton 3, the pd o the propoed multvarate olded normal dtrbuton and the orm o the mean vector, dperon matrx and mg are developed. Th ollowed by concluon n Secton 4. Notaton. Let Sp { :,,..., p, wth ±, p}.. dag,,..., p Λ p, 3. For any Sp, let W p Z > 0, p and Σ p Cholek actorzaton method. 4. Let B p Λ p, Σ p Λ p Σ p Λ p Z,..., p Z p N p B p B p, where B p, Σ p wth obtaned ung denote the th rowothep p matrx B p, p, and. denote the pd o unvarate tandard normal dtrbuton. Then we dene the ollowng: Q p j I p B p p j j p j B p j B p { B p B p,.. B p },.

5 On multvarate olded normal dtrbuton 5 I p k B p p j,, k B p B p k μp j, k..3 It can be noted that nce B p and k o Q p j, Ip and I p k ap p matrx, the ubcrpt, j can aume any o the value p wth condton mentoned above. We can then dene I,p a I,p I p j, where p, j p. 3 Multvarate olded normal dtrbuton The pd o the multvarate ay, p-varate olded normal dtrbuton MVFN can be wrtten a: p z,z,...,z p,,..., p Sp,,..., p Sp h p z, z,..., p z p h p Λ p z p or each z > 0, 3. where z z. p z p dag,,..., p z z. z p Λp z p. 3. For Z p N p, Σ p, we then have W p wth Λ p and Σ p Λ p Σ p Λ p. Λ p Z p N p, Σ p We rt obtan the mean vector, dperon matrx and the mg o the bvarate olded normal dtrbuton and then generalze that to get the ame or MVFN dtrbuton. It may be noted that the pd o the BVFN dtrbuton ha been recontructed here a mentoned n Secton. Accordngly, t mean vector, dperon matrx and mg are derved on the ba o the new orm o pd.

6 6 A.K. Chakraborty and M. Chatterjee 3.. Mean vector. Suppoe the mean vector o bvarate olded normal dtrbuton gven by. Then the expreon or μ can be obtaned a ollow. From 3. and 3. t can be een that E Z ; Z N, Σ, S π Σ exp w w 0 w 0 Σ w w w dw dw 3.3 Now nce Σ a varance-covarance matrx and hence potve dente, we can expre Σ a Σ B B, S. It may be noted that n general B not unque. However, we retrct our attenton unque Tong, to the ubcla o all lower trangular matrce, then B 990. Let u now conder the tranormaton W Y : where Y y y W B Y, S, 3.4. Hence rom 3.3 and 3.4, we get EW π Σ exp w π exp π exp y B y y w w 0 w 0 μ dy y B μ y y dy Σ w y B μ y B μ dw + B y

7 On multvarate olded normal dtrbuton 7 + B π exp + B y B μ y y B dy B μ B μ y B μ B μ B μ y y 3.5 From 3.5 the expreon or mean o the bvarate olded normal dtrbuton can be obtaned a { B, S + B B μ B μ B μ B μ. 3.6 Thu generalzng 3.6 or p-varate cae one can obtan mean vector o MVFN dtrbuton a: p B p Sp + Sp B p B p μp B p j B p p. p j p μp B p. p B p B p

8 8 A.K. Chakraborty and M. Chatterjee where Q p Sp p B p + Sp B p Q p, 3.7 ap vector whoe j th element Q p j dened n.. It can be ealy vered that or p, the expreon n 3.7 reduce to the ame expreon or the mean o unvarate olded normal dtrbuton a gven by Leone et al Dperon matrx. Suppoe the dperon matrx o a BVFN dtrbuton gven by Σ. Then the expreon or Σ can be obtaned a ollow: Σ E X X : Now, EW W X Z, Z N, Σ EX X μ, S π exp Σ w EW W w 0 w 0 μ, ay. 3.8 w w Σ w dw π + B y B μ y exp y B μ y y +B y dy, 3.9 by ung the tranormaton 3.4. Hence rom 3.9 we have EW W π y B μ + B Y + B Y Y B y B μ + Y B

9 On multvarate olded normal dtrbuton 9 exp y dy dy 4 I, ay, 3.0 where I μ π y B μ y B μ I exp B B π y dy dy B 3. y y y B μ B μ exp y dy dy B μ B μ μ 3. Smlar to 3., t can be hown that, 3 I I 4 B π B { B μ B μ y I I I I y y y B μ B μ exp B, 3.3 } y dy dy B B, 3.4

10 0 A.K. Chakraborty and M. Chatterjee where I π { B B y y exp y B } μ μ B y μ μ, dy dy I B μ B μ B B μ μ, I π y y y exp y y dy dy B I. Ung 3. to 3.4, we can obtan the expreon or EW W rom 3.0. A uch rom 3.8, the expreon or dperon matrx o bvarate olded normal dtrbuton can be obtaned a: Σ S S + S EW W W Λ B B B μ B μ Z μ B μ B μ μ

11 On multvarate olded normal dtrbuton + B μ B μ B S B μ B μ +, S B I I I I B μ. 3.5 We are now n a poton to generalze 3.5 or multvarate p-varate, ay cae. Wth the help o the matrx I,p I p j,,j p, whch dened n. and.3, the dperon matrx o p-varate olded normal dtrbuton can be obtaned a: Σ p Sp { p + B p Q p μ p B p + + B p I,p B p Q p } B p μp, where,,..., p. It may be noted that here alo the unvarate analogue o th varancecovarance matrx the ame a that o the expreon gven by Leone et al Moment generatng uncton mg. Suppoe the mg o a BVFN dtrbuton denoted by M X t. Then the expreon or M X t canbe obtaned a ollow: M X t X Eet X Z, Z N, Σ Ee t W S S π Σ exp w w 0 w 0 exp t w Σ w dw

12 A.K. Chakraborty and M. Chatterjee Ung the tranormaton W Y a gven n 3.4, we have M X t S π y y exp t + B y exp t π + t B S y y exp y dy dy B t y y t B y + t B B t dy dy 3.6 Now, let t B t B t B B t B t t, t where B th row o B,,. Thu rom 3.6 mg o BVFN can be obtaned a M X t exp y π t dy dy S y y { exp t + } t Σ t B t +B S 3.7

13 On multvarate olded normal dtrbuton 3 It worth mentonng here that, or bvarate olded tandard normal dtrbuton,.e. or 0, Σ ρ and Σ ρ ρ, ρ the expreon n 3.7 can be mpled a M X t e t Σ t B t +e t Σ t B t e T b ; R+e T b ; R, 3.8 where, T /t Σ t, T /t Σ t, b Rt or R t a the cae may be, b ; R the cd o the bvarate tandard normal dtrbuton wth correlaton matrx R Σ and b ; R that wth correlaton matrx R Σ. The mg o bvarate olded tandard normal dtrbuton, propoed by Parak and Panareto 00, matche wth the expreon gven n 3.8. Thu ther expreon a pecal cae o M X t. In act, the aumpton o 0 would al to dcrmnate a olded normal dtrbuton rom a hal normal dtrbuton a ha been dcued by Leone et al. 96. We now generalze 3.7 or multvarate p-varate, ay cae. Thu mg o MVFN can be obtaned a M p X t,,..., p Sp { exp t p B p + t Σ p t t +B p } 3.9 It nteretng to note that or p,m p X t n 3.9 gve the mean and varance o the unvarate olded normal dtrbuton a obtaned rom Leone et al Concluon Unvarate olded normal dtrbuton wa developed by Leone et al. 96. In the preent paper, we have developed t multvarate counterpart. Although, Parak and Panareto 00 had already propoed bvarate olded normal dtrbuton, they contructed the mg o only the bvarate olded tandard normal dtrbuton. On the contrary, we have contructed a more general orm o the dtrbuton or p -varate cae wth a

14 4 A.K. Chakraborty and M. Chatterjee general orm o the mean vector and the dperon matrx o the correpondng p-varate normal dtrbuton. We have alo derved the expreon o the mean vector, dperon matrx and the mg o the MVFN dtrbuton. It can be hown that the correpondng expreon obtaned by Leone et al. 96 and Parak and Panareto 00 can be derved a pecal cae o our. We have alo made a bre dcuon on the poble applcaton o MVFN dtrbuton n multvarate proce capablty analy whch one o the core area o the theory o tattcal qualty control. However, we have not dealt wth the etmaton procedure o the parameter nvolved n MVFN whch requred or explorng the dtrbuton urther. Gven the complcated orm o the dtrbuton tel, uch etmaton wll dentely be challengng yet nteretng. Acknowledgement. The author acknowledge ome very ueul dcuon wth Pro. Shbda Bandapadhyay, Appled Stattc Unt, Indan Stattcal Inttute, Kolkata and the anonymou reeree or ther valuable comment to make the paper better. Reerence chakraborty, a.k. and chatterjee, m. 00. On Multvarate Folded Normal Dtrbuton. Techncal Report No. SQCOR , SQC & OR Unt. Indan Stattcal Inttute, Kolkata. chatterjee, m. and chakraborty, a.k. 03. Some Properte O C Gu, v. In Proceedng o Internatonal Conerence On Qualty and Relablty Engneerng, Bangalore, Inda. To be publhed. elandt, r.c. 96. The olded normal dtrbuton: two method o etmatng parameter rom moment. Technometrc, 3, johnon, n.l Cumulatve um control chart or the olded normal dtrbuton. Technometrc, 5, kng, j.r When a normal varable not a normal varable? Qual. Eng.,, krman,. and polanky, a.m Multvarate proce capablty va lowner orderng. Lnear Algebra Appl., 430, leone,.c., nelon, l.. and nottngham, r.b. 96. The olded normal dtrbuton, Technometrc, 3, lao, m.y. 00. Economc tolerance degn or olded normal data. Int. J. Prod. Re., 8, ln, h.c The meaurement o a proce capablty or olded normal proce data. Int. J. Adv. Manu. Technol., 4, 3 8. ln, p.c Applcaton o the generalzed olded-normal dtrbuton to the proce capablty meaure, Int. J. Adv. Manu. Technol., 6, polanky, a.m. 00. A mooth nonparametrc approach to multvarate proce capablty, Technometrc, 43, 99.

15 On multvarate olded normal dtrbuton 5 parak,. and panareto, j. 00. On ome bvarate extenon o the olded normal and the olded T dtrbuton. J. Appl. Statt. Sc., 0, tall, g.m. 96. The moment generatng uncton o the truncated mult-normal dtrbuton. J. R. Stat. Soc. Ser. B Stat. Methodol., 3, 3 9. taam, w., ubbaah, p. and lddy, j.w A note on multvarate capablty ndce. J. Appl. Stat., 0, tong, y.l The multvarate normal dtrbuton. Sprnger-Verlag, New York. vannman, k A uned approach to capablty ndce. Statt. Snca, 5, wang,.k. and chen, j.c Capablty ndex ung prncpal component analy. Qual. Eng.,, 7. Ah Kumar Chakraborty and Moutuh Chatterjee SQC & OR Unt, Indan Stattcal Inttute Kolkata, Inda E-mal: akchakraborty.@gmal.com tuh.tat@gmal.com Paper receved: 4 Aprl 0; reved: 7 March 03.