APPENDIX A DERIVATION OF JOINT FAILURE DENSITIES

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1 APPENDIX A DERIVAION OF JOIN FAILRE DENSIIES I his Appedi we prese he derivaio o he eample ailre models as show i Chaper 3. Assme ha he ime ad se o ailre are relaed by he cio g ad he sochasic are o his relaioship ca be represeed by reaig oe or wo o he parameers o g as radom variables. We cosider he ollowig or eample orms: i g ii g iii g ad iv g e e. I each case we irodce radomess io he cio by reaig he parameer as a radom variable havig disribio. sig ad he rasormaio o variables we cosrc he margial probabiliy disribio o sage. ha is d g. A. d Oce he margial disribio o sage is obaied we he cosrc he joi ailre desiy sig he codiioig relaio i.e.. A. 7

2 73 I Eq. A. he codiioal desiy is obaied by sig he well-kow relaioship bewee a desiy ad is hazard cio: { } { } g g d z z d z z ep ep. A.3 We assme ha he codiioal bivariae hazard cio o age give sage may be saed as: z A.4 so ha he deiiios o he cios ad g deermie he codiioal hazard ad limaely he bivariae lie disribio. I here i order o ocs o he cios g we assme ha ad are simple liear cios i.e. ad. hs we assme z. A.5 der his modelig orma we may obai he bivariae lie disribios correspodig o orms i ii iii ad iv respecively as ollows: ep A.6 Eq ep A.7 Eq. 3.8

3 74 ep A.8 Eq. 3.9 ad ep. A.9 Eq. 3. A. Derivaio o Eq. A.6 For case i g solvig or yields: ad d d so:. A. he codiioal bivariae hazard cio o age give sage ca be cosrced as: z A. Sbsiig Eq. A. io Eq.A.3 we obai d ep

4 75 ep A. Sbsiig Eqs. A. ad A. io Eq. A. we derive ep. A. Derivaio o Eq. A.7 For case ii g solvig or yields: ad d d so:. A.3 he codiioal bivariae hazard cio o age give sage ca be cosrced as: z A.4 Sbsiig Eq. A.4 io Eq.A.3 we obai

5 76 d ep ep A.5 Sbsiig Eqs. A.3 ad A.5 io Eq. A. we derive ep. A.3 Derivaio o Eq. 3.9 For case ii g solvig or yields: ad d d so:. A.6 he codiioal bivariae hazard cio o age give sage ca be cosrced as: z A.7 Sbsiig Eq. A.7 io Eq.A.3 we obai

6 77 d ep ep A.8 Sbsiig Eqs. A.6 ad A.8 io Eq. A. we derive ep. A.4 Derivaio o Eq. 3. For case iv g e e solvig or yields: ad d d so:. A.9 he codiioal bivariae hazard cio o age give sage ca be cosrced as: z ep ep. A. Sbsiig Eq. A. io Eq.A.3 we obai

7 78 d ep ep ep d ep Le he d ep ep ep ep A. Sbsiig Eqs. A. ad A.9 io Eq. A. we derive

8 79 ep.

9 APPENDIX B HE BIVARIAE DIRAC DELA FNCION I his Appedi we irodce he bivariae Dirac Dela cio ha is sed i Chaper 6 o model he logeviy o he sigle-i sysem i he PM cycle o he ARPM model. We begi wih he discssio o he ivariae Dirac Dela cio ad is properies. he we eed he ivariae cio o wo dimesios ad prese he resls or he bivarie Dirac Dela cio. B. ivariae Dirac Dela Fcio Cosider he ollowig cio: ε ε ε > F ε B. oherwise. Figre B.. shows he plo o F ε. Geomerically as ε we have he ollowig iegral: F ε /ε. he F ε d. B. Le δ be he limiig cio o F ε as ε. δ is called ivariae Dirac Dela or i Implse cio or F ε see Spiegel [965]. 8

10 F ε /ε ε Figre B.. Plo o F ε. We desigae he ivariae Dirac Dela cio as δ ha has he ollowig properies: δ d B.3 i δ or ay coios cio φ. B.4 ii φ d φ Noe ha δ has a siglariy o iiie vale a ad is eqal o zero a all oher vales o. Propery i gives he Dirac Dela cio he characerisic o a probabiliy desiy cio. Propery ii helps i akig Laplace rasorm o he Dirac Dela cio. he Laplace rasorm o δ is give by 8

11 L s v { } δ s s δ e d e B.5 B. Bivariae Dirac Dela Fcio We ow cosrc he bivariae Dirac Dela cio wih aalogical properies as i ivariae case. Cosider he ollowig cio: ε ε ε ε > F ε B.6 oherwise. Figre B.. shows he plo o F ε. Geomerically as ε F ε /ε. he we have he ollowig iegral: F ε dd. B.7 Le δ be he limiig cio o as ε. δ is called bivariae Dirac F ε Dela or i Implse cio or. We desigae he ivariae Dirac Dela cio as δ ha has he ollowig properies: F ε he bivariae Dirac Dela cio or his PM cycle δ ollowig properies: has he i δ dd B.8 ii δ φ dd φ or ay coios cio φ. B.9 8

12 F ε /ε ε /ε ε Figre B.. Plo o F ε. Propery i gives he bivariae Dirac Dela cio he characerisic o a bivariae probabiliy desiy cio. Propery ii helps i akig Laplace rasorm o he bivariae Dirac Dela cio. he Laplace rasorm o δ is L { } δ s v s v s v δ e dd e. B. 83

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Appendix. The solution begins with Eq. (2.15) from the text, which we repeat here for 1, (A.1)

Appendix. The solution begins with Eq. (2.15) from the text, which we repeat here for 1, (A.1) Aenix Aenix A: The equaion o he sock rice. The soluion egins wih Eq..5 rom he ex, which we reea here or convenience as Eq.A.: [ [ E E X, A. c α where X u ε, α γ, an c α y AR. Take execaions o Eq. A. as