Technical Support Document Bias of the Minimum Statistic

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1 Tchical Support Documt Bias o th Miimum Stattic Itroductio Th papr pla how to driv th bias o th miimum stattic i a radom sampl o siz rom dtributios with a shit paramtr (also kow as thrshold paramtr. Ths dtributios iclud th thr-paramtr Wibull logormal log-logtic gamma ad th two-paramtr potial dtributios. Dtributio o th Miimum Stattic Lt b a radom sampl o siz draw rom a dtributio with probability dsity uctio (PDF dotd by ( ad a cumulativ dtributio uctio (CDF dotd by F (. Th mi{ } has th PDF ad th CDF Th pctd valu o ( ( ( F( F ( ( F( ] ( ( F( Th pctd valu or pctatio ca b asily calculatd or som modls such as th potial modl. Howvr i gral th plicit prssio o th pctatio uavailabl ad vry compl to calculat umrically. Th complity o th calculatios grows potially with th sampl siz. Wh th calculatios o th plicit prssio bcom too uwildy such as or th thr-paramtr logormal ad gamma dtributios w us a approimatio mthod. For th thr-paramtr Wibull (two-paramtr potial a spcial cas ad th log-logtic dtributios w prss th pctatio i trms o wll-kow uctios such as th gamma ad bta uctios. Thr-Paramtr Wibull ad Two-Paramtr Epotial Modls To simpliy ad stramli th tchical drivatios or ths modls w prst th mai rsults irst as thorms. Thorm 3. Lt b a radom sampl o siz draw rom th Wibull dtributio with shap paramtr / ad scal paramtr (Wibull (. Th th miimum dtributd as Wibull (. d Kowldgbas ID 2357: Pag

2 Th PDF ad CDF o th Wibull ( dtributio ar ( ( ad rspctivly. Thus th PDF o which ca b rwritt as / Thror Wibull ( /. Tchical Support Documt Bias o th Miimum Stattic F( ( / / ( / ( / ( ( p / / ( / / Thorm 3.2 I a radom sampl o siz draw rom th thr-paramtr Wibull dtributio with thrshold paramtr shap paramtr ad scal paramtr (Wibull ( th / dtributd as Wibull ( /. Th E ] ( / ad th bias o as a / stimator o th thrshold paramtr ( /. Bcaus a radom sampl rom th thr-paramtr Wibull ( a radom sampl rom Wibull (. By Thorm 3. th miimum o th radom sampl / Wibull (. By diitio o thr-paramtr dtributios dtributd as Wibull ( / / /. Thus ] ( / as rquird. Corollary 3. I a radom sampl o siz draw rom th two-paramtr potial dtributio with thrshold paramtr ad scal paramtr (Epotial ( th dtributd as Epotial ( /. Thus E ] / ad th bias o as a stimator o /. Th rsult ollows dirctly rom Thorm 3.2 by sttig th shap paramtr o th Wibull dtributio to uity. Kowldgbas ID 2357: Pag 2

3 Thr-Paramtr Log-Logtic Modl Tchical Support Documt Bias o th Miimum Stattic Thorm 4. Lt b a radom sampl o siz draw rom th log-logtic dtributio with paramtrs ad (log-logtic (. Th th pctd valu o th miimum E ] p( ( ( / ( p( Bta (. Th PDF ad CDF o th log-logtic ( ar ad rspctivly whr ad Thus th PDF o ad log( ( log( F( ( 2 ( log( log( ( log( log( ] d Usig th trasormatio y (log( / th abov pctatio bcoms ( y dy ] p( y ( y By itgratig by part (lt u p( y ad dv ( y( ( y dy th pctatio bcoms Bcaus ( y y y ( ] p( y ( y dy ] y Applyig th trasormatio /( yilds p( y dy y ( ] ( which by diitio th bta uctio Bta ( providd. dy Kowldgbas ID 2357: Pag 3

4 Tchical Support Documt Bias o th Miimum Stattic Thorm 4.2 Lt b a radom sampl o siz draw rom th thr-paramtr log-logtic dtributio with paramtrs ad (Log-logtic (. Th th pctd valu o miimum E ] p( ( ( / ( p( Bta ( ad th bias o as a stimator o p( ( ( / ( p( Bta ( providd. Bcaus a radom sampl rom th thr-paramtr log-logtic ( a radom sampl rom log-logtic (. By Thorm 4. th miimum o th radom sampl has th pctatio p( ( ( / ( p( Bta ( providd. Thr-Paramtr Logormal ad Gamma Modls For ths two modls w caot obtai plicit prssios or th irst momt o. W ca obtai rcursiv prssios but th calculatios ar trmly log ad compl or larg sampls. For th raso w us approimatios o th bias o as a stimator o th thrshold paramtr. Approimatios ad bouds or th momts o ordr stattics ca b oud i Ordr Stattics by H.E. David. I a radom sampl rom a dtributio with CDF F ( th or suicitly larg r ( r ] F ( r Applyig th approimatio to th logormal ad th gamma dtributio yilds th ollowig rsults: Thorm 5. Lt b a radom sampl o siz draw rom th thr-paramtr logormal (. Th or suicitly larg E ] p p whr ( th ivrs CDF o th stadard ormal dtributio. Thror a approimat bias o as a stimator o th thrshold paramtr p p Th ivrs CDF o th thr- paramtr logormal dtributio ca b writt as F ( p ( Thorm 5. ollows immdiatly rom th rsult. Kowldgbas ID 2357: Pag 4

5 Tchical Support Documt Bias o th Miimum Stattic Thorm 5.2 Lt b a radom sampl o siz draw rom th thr-paramtr gamma dtributio with thrshold paramtr shap paramtr ad scal paramtr (gamma (. Th or suicitly larg ] I ; I ; whr ( th ivrs o th icomplt gamma uctio did as I a y y dy I ( a ( Thror a approimat bias o as a stimator o th thrshold paramtr I ; I ; Th ivrs CDF o th thr- paramtr gamma dtributio ca b writt as F ( ; Thorm 5.2 ollows immdiatly rom th rsult. Not: W obtai similar approimatios or th Wibull ad log-logtic dtributios. For th thr-paramtr Wibull dtributio a approimat pctd valu o E ] I Th rsult ollows rom th ivrs CDF o th thr-paramtr Wibull dtributio / F ( log( ad th act that or larg by Lockhart ad Stphs. 2 log( / For th thr-paramtr log-logtic modl th approimatio yilds E ( ] p log( /( /. A approimat bias thus / as rportd Th rsult ollows rom th ivrs CDF o thr-paramtr log-logtic dtributio: F ( p log Rrcs H. A. David (98. Ordr Stattics. Joh Wily & Sos Ic 7. 2 R. A. Lockhart ad M. A. Stphs (994. Estimatio ad Tsts o Fit or th Thr-Paramtr Wibull Dtributio J R Statt Soc B 56( Kowldgbas ID 2357: Pag 5