America Joural of Theoretical ad Applied tatistics 6; 5(-): -6 Published olie November 8 5 (http://wwwsciecepublishiggroupcom/j/ajtas) doi: 648/jajtass65 IN: 36-8999 (Prit); IN: 36-96 (Olie) Tolerace imits o Order tatistics i Future amples Comig from the Two-Parameter Expoetial Distributio Nicholas A Nechval * Kostati N Nechval Departmet of Mathematics Baltic Iteratioal Academy Riga atvia Departmet of Applied Mathematics Trasport ad Telecommuicatio Istitute Riga atvia Email address: echval@juilv (N A Nechval) osta@tsilv (K N Nechval) To cite this article: Nicholas A Nechval Kostati N Nechval Tolerace imits o Order tatistics i Future amples Comig from the Two-Parameter Expoetial Distributio America Joural of Theoretical ad Applied tatistics pecial Issue: Novel Ideas for Efficiet Optimizatio of tatistical Decisios ad Predictive Ifereces uder Parametric certaity of derlyig Models with Applicatios Vol 5 No - 6 pp -6 doi: 648/jajtass65 Abstract: This paper presets a iovative approach to costructig lower ad upper tolerace limits o order statistics i future samples Attetio is restricted to ivariat families of distributios uder parametric ucertaity The approach used here emphasizes pivotal quatities relevat for obtaiig tolerace factors ad is applicable wheever the statistical problem is ivariat uder a group of trasformatios that acts trasitively o the parameter space It does ot require the costructio of ay tables ad is applicable whether the past data are complete or Type II cesored The proposed approach requires a quatile of the F distributio ad is coceptually simple ad easy to use For illustratio the two-parameter expoetial distributio is cosidered A practical example is give Keywords: Order tatistics F Distributio ower Tolerace imit pper Tolerace imit Itroductio tatistical tolerace limits are aother tool for maig statistical iferece o a uow populatio As opposed to a cofidece limit that provides iformatio cocerig a uow populatio parameter a tolerace limit provides iformatio o the etire populatio; to be specific oe-sided tolerace limit is expected to capture a certai proportio or more of the populatio with a give cofidece level For example a upper tolerace limit for a uivariate populatio is such that with a give cofidece level a specified proportio or more of the populatio will fall below the limit A lower tolerace limit satisfies similar coditios It is ofte desirable to have statistical tolerace limits available for the distributios used to describe time-to-failure data i reliability problems For example oe might wish to ow if at least a certai proportio say of a maufactured product will operate at least T hours This questio ca ot usually be aswered exactly but it may be possible to determie a lower tolerace limit ( ) based o a prelimiary radom sample ( ) such that oe ca say with a certai cofidece that at least % of the product will operate loger tha ( ) The reliability statemets ca be made based o ( ) or decisios ca be reached by comparig ( ) to T Tolerace limits of the type metioed above are cosidered i this paper That is if f (x) deotes the desity fuctio of the paret populatio uder cosideratio ad if is ay statistic obtaied from the prelimiary radom sample ( ) of that populatio the () is a lower probability tolerace limit for proportio if Pr f ( xdx ) () ( ) ad () is a upper probability tolerace limit for proportio if ( ) Pr f ( xdx ) () is the parameter (i geeral vector) The commo distributios used i life testig problems are the ormal expoetial Weibull ad gamma distributios [] Tolerace limits for the ormal distributio have bee
Nicholas A Nechval ad Kostati N Nechval: Tolerace imits o Order tatistics i Future amples Comig from the Two-Parameter Expoetial Distributio cosidered i [] [3] [4] ad others Tolerace limits ejoy a fairly rich history i the literature ad have a very importat role i egieerig ad maufacturig applicatios Patel [5] provides a review (which was fairly comprehesive at the time of publicatio) of tolerace itervals for may distributios as well as a discussio of their relatio with cofidece itervals for percetiles ad predictio itervals Dusmore [6] ad Guether Patil ad ppuluri [7] both discuss -parameter expoetial tolerace itervals ad the estimatio procedure i greater detail Egelhardt ad Bai [8] discuss how to modify the formulas whe dealig with type II cesored data Guether [9] ad Hah ad Meeer [] discuss how oe-sided tolerace limits ca be used to obtai approximate two-sided tolerace itervals by applyig Boferroi's iequality I cotrast to other statistical limits commoly used for statistical iferece the tolerace limits (especially for the order statistics) are used relatively rarely Oe reaso is that the theoretical cocept ad computatioal complexity of the tolerace limits is sigificatly more difficult tha that of the stadard cofidece ad predictio limits Thus it becomes ecessary to use the iovative approaches which will allow oe to costruct tolerace limits o future order statistics for may populatios I this paper the iovative approach to costructig lower ad upper tolerace limits o order statistics i future samples is proposed For illustratio the two-parameter expoetial distributio is cosidered Mathematical Prelimiaries Probability Distributio Fuctio of Order tatistic Theorem If there is a radom sample of m ordered observatios Y Y m from a ow distributio (cotiuous or discrete) with desity fuctio f (y) distributio fuctio F (y) the the probability distributio fuctio of the th order statistic Y { m} is give by m m j PY ( y m) [ F ( y )] [ F ( y )] j j mj f ( xdx ) ( m + ) (3) F ( y ) F ( y ) ( m + ) [( m + ) + ]/ ( m + ) x + x > is the probability desity fuctio of a F distributio with (m+) ad degrees of freedom Proof uppose a evet occurs with probability p per trial It is well-ow that the probability P of its occurrig or more times i m trials is termed a cumulative biomial probability ad is related to the icomplete beta fuctio I x (a b) as follows: (4) m m j mj P p ( p) I ( m + ) p j (5) j It follows from (5) that m m j P { Y y m} [ F ( y )] [ F ( y )] j j I ( m + ) ( ) F y F( y ) ( m + ) u ( u) du Β( m + ) ( m + )/ ( m ) + ( m + ) Β ( m + )/ F ( y ) ( m + ) + u mj u du u ( m + ) ( m + ) u ( m + )/ ( m ) + ( m + ) Β F ( y ) F ( y) ( m + ) x [( m + ) + ]/ ( m + ) x + dx ( m + )/ (6) f ( x) + ( m + ) Β ( m ) This eds the proof Corollary u x u ( m + ) (7) ( m + ) ( m + ) x ( m + )/ m j PY ( > y m) [ F ( y )] [ F ( y )] j j mj
America Joural of Theoretical ad Applied tatistics 6; 5(-): -6 3 F ( y ) F ( y ) ( m + ) f ( m + ) ( xdx ) Corollary If y m; is the quatile of order for the distributio of Y we have from (3) that y m; is the solutio of ( m + ) ; (8) F ( y ) m ; (9) + ( m + ) q ; is the quatile of order for the F distributio with (m+) ad degrees of freedom Two-Parameter Expoetial Distributio The two-parameter expoetial distributio is a widely used ad widely ow distributio It is characterized by the desity fuctio is the pivotal quatity with the desity fuctio gw w w Γ( r ) r ( ) exp( ) w 3 Tolerace imits for Order tatistic 3 over Tolerace imit (8) Theorem et r be the first r ordered observatios from the prelimiary sample of size from a two-parameter expoetial distributio defied by the desity fuctio () The a lower oe-sided -cotet tolerace limit at level () (o the th order statistic Y from a set of m future ordered observatios Y Y m also from the distributio () ) which satisfies ( PY m ) Pr ( > ) (9) yµ f ( y) exp y µ σ σ () (µσ) σ (scale parameter) ad µ (shift parameter) are uow The distributio fuctio of the two-parameter expoetial distributio is yµ F ( y) exp σ () The sufficiet statistic for the parameter based o the r ( ) smallest observatios ( r ) i a radom sample of size from the two-parameter expoetial distributio () is r mi( ) + ( r ) () r i r i x ( µ ) ~ h ( x ) exp x µ σ σ V (3) µ (4) σ is the pivotal quatity with the desity fuctio ( ) hv ( ) exp v v (5) s r ~ g ( s ) s exp s σ r Γ( r ) σ σ (6) is give by r l( ) + if l r l( ) if < l ( m + ) q( m + ) ; ( m + ) q + ( m + ) ; Proof It follows from (8) () ad (9) that ( PY > m ) Pr ( ) F ( ) F ( ) ( m + ) Pr f ( xdx ) ( m + ) F ( ) Pr q ( m + ) ; F ( ) ( m + ) Pr F ( ) ( m ) q + + ( m + ) ; () () W (7) σ µ ( m ) q + ( m + ) ; Pr exp σ ( m ) q + + ( m + ) ;
4 Nicholas A Nechval ad Kostati N Nechval: Tolerace imits o Order tatistics i Future amples Comig from the Two-Parameter Expoetial Distributio µ ( m ) q + m Pr l ( ) ( + ) ; σ m q + + ( m + ) ; ( l ) PrV W l w hvdv ( ) () r l + if l r l if < l (8) is the lower tolerace factor (3) ( m + ) q( m + ) ; ( m + ) q + ( m + ) ; (9) ( m + ) q( m + ) ; ( m + ) q + ( m + ) ; It follows from (8) (9) ad () that l w arg hvgwdvdw ( ) ( ) (4) Proof It follows from (3) () ad (7) that ( PY m ) Pr ( ) F ( ) F( ) ( m + ) Pr f ( xdx ) ( m + ) /( r) (5) Taig ito accout (3) ad (5) we have () This completes the proof Corollary If m the r l( ) + if l r l( ) if < l The result similar to that of (6) ca be foud i [7 8] 3 pper Tolerace imit (6) Theorem 3 et r be the first r ordered observatios from the prelimiary sample of size from a two-parameter expoetial distributio defied by the desity fuctio () The a lower oe-sided -cotet tolerace limit at level () (o the th order statistic Y from a set of m future ordered observatios Y Y m also from the distributio () ) which satisfies is give by ( PY m ) Pr ( ) (7) F ( ) F ( ) ( m + ) Pr f ( xdx ) ( m + ) F ( ) Pr q ( m + ) ; F ( ) ( m + ) Pr F ( ) ( m ) q + + ( m + ) ; µ ( m ) q + ( m + ) ; Pr exp σ ( m ) q + + ( m + ) ; µ ( m ) q + m Pr l ( ) ( + ) ; σ m q + + ( m + ) ; ( V W l ) Pr is the upper tolerace factor hvdv ( ) (3) l w (3)
America Joural of Theoretical ad Applied tatistics 6; 5(-): -6 5 ( m + ) q( m + ) ; ( m + ) q + It follows from (8) (7) ad (3) that ( m + ) ; arg hvgwdvdw ( ) ( ) wl /( r) (3) (33) Goodess-of-fit testig It is assumed that x µ ~ f ( x) exp x µ i σ σ (35) the parameters µ ad σ are uow Thus for this example we have that r 5 m 3l 5 95 95 8 ( ) 66 (36) i i It ca be show that the Taig ito accout (3) ad (33) we have (8) This completes the proof Corollary 3 If m the r ( ) l + if l( ) r ( ) l if < l( ) (34) The result similar to that of (34) ca by foud i [7] Remar It will be oted that a upper tolerace limit may be obtaied from a lower tolerace limit by replacig by by 4 Practical Example A idustrial firm has the policy to replace a certai device used at several locatios i its plat at the ed of 4-moth itervals It does t wat too may of these items to fail before beig replaced hipmets of a lot of devices are made to each of three firms Each firm selects a radom sample of l 5 items ad accepts his shipmet if o failures occur before a specified lifetime has accumulated I order to fid this specified lifetime the maufacturer wishes to tae a radom sample of size 5 ad to calculate the lower oe-sided simultaeous tolerace limit () which is expected to capture a certai proportio 95 or more of the populatio of selected items (m3l) with a give cofidece level 95 This tolerace limit is such that oe ca say with a certai cofidece that at least % of the product selected for testig by firms will operate loger tha () The resultig lifetimes (rouded off to the earest moth) of the iitial sample of size 5 from a populatio of the aforemetioed devices are give i Table Table The resultig lifetimes of the iitial sample of size 5 Observatios (i terms of moth itervals) 3 4 5 6 7 8 8 9 4 7 5 9 3 4 5 9 3 35 4 47 54 6 j j+ ( i )( ) + i i j i () j () j+ ( i )( ) + i i i (37) are iid () rv s (Nechval et al []) We assess the statistical sigificace of departures from the two-parameter expoetial model () by performig the Kolmogorov-mirov goodess-of-fit test We use the K statistic (Muller et al []) The rejectio regio for the α5 level of sigificace is {K K ;α } The percetage poits for K ;α were give by Muller et al [] For this example K 8 < K 3;α5 36 (38) Thus there is ot evidece to rule out the two-parameter expoetial model Now the lower oe-sided simultaeous -cotet tolerace limit at the cofidece level () (o the order statistic Y from a set of m future ordered observatios Y Y m ) ca be obtaied from () ice l( ) 5< 9 (39) l ( m + ) q( m + ) ; 989848 ( m + ) q + it follows from () that ( m + ) ; r ( ) 4 (4) (4) tatistical iferece Thus the maufacturer has 95% assurace that o failures will occur i the proportio 95 or more of the populatio of selected items before 4 moth itervals
6 Nicholas A Nechval ad Kostati N Nechval: Tolerace imits o Order tatistics i Future amples Comig from the Two-Parameter Expoetial Distributio 5 Coclusio Tolerace limits ejoy a fairly rich history i the literature ad have a very importat role i egieerig ad maufacturig applicatios I cotrast to other statistical limits commoly used for statistical iferece the tolerace limits (especially for the order statistics) are used relatively rarely Oe reaso is that the theoretical cocept ad computatioal complexity of the tolerace limits is sigificatly more difficult tha that of the stadard cofidece ad predictio limits Thus it becomes ecessary to use the iovative approaches which will allow oe to costruct tolerace limits o future order statistics for may populatios Refereces [] V Medehall A bibliography o life testig ad related topics Biometria vol V pp 5 543 958 [] I Guttma O the power of optimum tolerace regios whe samplig from ormal distributios Aals of Mathematical tatistics vol VIII pp 773 778 957 [3] A Wald ad J Wolfowitz Tolerace limits for a ormal distributio Aals of Mathematical tatistics vol VII pp 8 5 946 [4] W A Wallis Tolerace itervals for liear regressio i ecod Bereley ymposium o Mathematical tatistics ad Probability iversity of Califoria Press Bereley 95 pp 43 5 [5] J K Patel Tolerace limits: a review Commuicatios i tatistics: Theory ad Methodology vol 5 pp 79 76 986 [6] I R Dusmore ome approximatios for tolerace factors for the two parameter expoetial distributio Techometrics vol pp 37 38 978 [7] W C Guether A Patil ad V R R ppuluri Oe-sided -cotet tolerace factors for the two parameter expoetial distributio Techometrics vol 8 pp 333 34 976 [8] M Egelhardt ad J Bai Tolerace limits ad cofidece limits o reliability for the two-parameter expoetial distributio Techometrics vol pp 37 39 978 [9] W C Guether Tolerace itervals for uivariate distributios Naval Research ogistics Quarterly vol 9 pp 39 333 97 [] G J Hah ad W Q Meeer tatistical Itervals: A Guide for Practitioers New Yor: Joh Wiley & os 99 [] N A Nechval ad K N Nechval Characterizatio theorems for selectig the type of uderlyig distributio i Proceedigs of the 7th Vilius Coferece o Probability Theory ad d Europea Meetig of tatisticias Vilius: TEV 998 pp 35 353 [] P H Muller P Neuma ad R torm Tables of Mathematical tatistics eipzig: VEB Fachbuchverlag 979