ON THE SCALE PARAMETER OF EXPONENTIAL DISTRIBUTION

Transcription

1 Review of the Air Force Academy No. (34)/7 ON THE SCALE PARAMETER OF EXPONENTIAL DISTRIBUTION Aca Ileaa LUPAŞ Military Techical Academy, Bucharet, Romaia DOI:.96/ Abtract: Eoetial ditributio i oe of the widely ued cotiuou ditributio i variou field for tatitical alicatio. I thi aer we tudy the eact ad aymtotical ditributio of the cale arameter for thi ditributio. We will alo defie the cofidece iterval for the tudied arameter a well a the fied legth cofidece iterval.. INTRODUCTION Eoetial ditributio i ued i variou tatitical alicatio. Therefore, we ofte ecouter eoetial ditributio i alicatio uch a: life table, reliability tudie, etreme value aalyi ad other. I the followig aer, we focu our attetio o the eact ad aymtotical reartitio of the eoetial ditributio cale arameter etimator.. SCALE PARAMETER ESTIMATOR OF THE EXPONENTIAL DISTRIBUTION We will coider the radom variable X with the followig cumulative ditributio fuctio: F( ; ) e (, ) () where i a ukow cale arameter Uig the relatiohi betwee M ( X ) ; ( X ) ; ( X ), we obtai ( X ) a theoretical variatio coefficiet. Thi i a ueful idicator, eecially if M( X) you have obervatioal data which eem to be eoetial ad with variatio coefficiet of the electio cloed to. If we coider,,... a a art of a oulatio that follow a eoetial ditributio, the by uig the maimum likelihood etimatio method we obtai the followig etimate ˆ i () i 9

2 ˆ Sice M O the Scale Parameter of Eoetial Ditributio, it follow that ˆ i a ubiaed etimator for. Similarly, becaue D ( ˆ ) ad lim, we obtai that ˆ i a abolutely correct etimator. The efficiecy of the etimator ˆ ca be calculated uig the followig formula: e ( ˆ ) l f (, ) f (, ) d D( ˆ ) (3) where f ( ; ) e (, ) i the robability deity fuctio of the l f ( ; ) eoetial ditributio. After the calculatio we obtai f ( ; ) d ad thu e ( ˆ ) which imlie that ˆ i alo a efficiet etimator. Takig ito accout the reroducibility roerty of the eoetial ditributio we ca calculate the eact ditributio of the radom variable ˆ. A um X X... X, coitig of radomly elected variable, all eoetially ditribuited, ha a ditributio deity fuctio equal to f( ) e. ( ) (4) Thi lead u to the cumulative ditributio fuctio u ˆ u ( ) H ( ) P( ) P( X... X ) e du F ( ) (5) from which we obtai that the ditributio deity fuctio of the etimator ˆ i equal to h( ) f( ) e ( ) From the above we obtai that ˆ D h (6) M ( ˆ ) h ( ) d ad ( ) ( ) ( ) which i coitet with reviouly kow reult. By uig the cetral limit theorem we ca tate that the radom variable X follow a ormal ditributio with the arameter ad, which mea that X N;.

3 Review of the Air Force Academy No. (34)/7 Baed o thi we ca calculate the aymtotical cumulative ditributio fuctio for the etimator ˆ, G ˆ ( ) lim P( ) lim P( X ). (7) Thi how that the etimator ˆ ha the ame aymtotical ditributio a X ad a ditributio deity fuctio equal to: ( ) g( ) e (8) Uig the aymtotical ditributio of the etimator ˆ we ca determie the cofidece iterval for the cale arameter. To do thi we eed to coider the reduced ( ˆ ) ormal radom variable Z ad the igificace level. By defiitio we will get P ( Z Z ) (9) where Z i the rewritte uch a quartile of the tadard ormal ditributio. Equatio (9) ca be ( ˆ ) P( Z Z ). () Z We may aume that becaue we deal with aymtotical ditributio ad thu i ufficietly high. After carryig out the calculatio we obtai for the arameter the followig cofidece iterval which deed o the maimum likelihood etimatio ˆ ˆ ˆ Z Z. () 3. SET LENGTH CONFIDENCE INTERVALS FOR THE SCALE PARAMETER Uig a imilar method to that which Stei rooed for the double electio method we fid a et legth cofidece iterval for the arameter. Let be a ytem of ideedet electio, each with a volume, take from a oulatio which ha the ame eoetial ditributio a the radom variable X

4 O the Scale Parameter of Eoetial Ditributio,...,...,..., () Thee electio allow u to obtai ideedet ad idetical aiged etimator ˆ i N ; i,...,. (3) Let ˆ ˆ i ad i ˆ ˆ ˆ ( i ) i. Becaue ˆ i the electio mea for the radom variable ˆ i ; i,..., we kow that ˆ N( ; ). Net let u coider a ecod ytem with m ideedet electio, each of volume, take from a oulatio which ha the ame eoetial ditributio a the radom variable X...,..., ; ;,..., m; m; (4) Thee ew electio allow u to calculate m ideedet ad idetically aiged etimator ˆ j N( ; ) j,..., m. (5) Let ˆ be the calculated average of both electio ytem: ˆ m ˆ i (6) m i It i kow that ˆ N ;. Note that for m= we obtai ˆ ˆ. Due to the ( m ) fact that ˆ N( ; ) ad i the variace of radom variable ˆ coidered i equatio, we deduce the followig equatio ˆ ( ) (7)

5 Review of the Air Force Academy No. (34)/7 Alo the radom variable ˆ matche with the reduced radom variable through the followig equatio: ( m ) ( ˆ ) N(,) (8) The radom variable from equatio (7) ad (8) are ideedet ad thu we ca coider the followig ratio ( m ) ( ˆ ) ( m ) ( ˆ ) t ˆ ˆ (9) where t i a Studet radom variable with - degree of freedom. For the igificace level we have the followig equatio: P m ( ˆ ) t ; ˆ () from which we deduce the followig cofidece iterval for ˆt ; ˆt ; ˆ ˆ m m () The legth of thi iterval ca be eaily calculated ad i equal to t l m ˆ ;. () The legth of thi iterval mut ot be greater tha the coidered legth. We will tart with m= ad comare the legth l with. If l the the iterval ˆ ˆ ; i a cofidece iterval for of ize, that ha a cofidece coefficiet greater or at leat equal to, build oly with the hel of the firt electio ytem. If l, we eed to carry out the ecod electio ytem, where m i equal with the t mallet iteger for which ˆ ;. I thi cae ˆ ˆ ; will be the m cofidece iterval for of ize, build o both electio ytem ad havig a cofidece coefficiet greater the -. 3

6 O the Scale Parameter of Eoetial Ditributio REFERENCES [] Gheorghe, F.C., Obreja, G. Tete tatitice etru elecţii de volum mic ( 5) Ed. Uiv. Piteşti 999. [] Margoli,H.B., Maurer,W. Tet of the Kolmogorov Smirov tye for eoetial data with ukow cale ad related roblem. Biometrica (976), 63, g