ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 Estimation o f the Parameter o f an Exponential Distribution When Applying Maximum Likelihood and Probability Plot Methods Using Simulation A.M. Hamad Department of Mathematical,College of Education Ibn Al -Haitham, Unive rsity of Baghdad Received in: 4September2011, Accepte d in: 18October2011 Abstract Exp onential Dist ribution is p robably the most imp ortant distribution in reliability work. In this p ap er, estimating the scale p arameter of an exp onential distribution was p rop osed through out emp loying maximu m likelihood estimator and p robability p lot methods for different samp les size. M ean square error was implemented as an indicator of p erformance for assumed several values of the p arameter and comp uter simulation h as been carr ied out to analysis t he obtained results. Key words: maximu m likelihood estimators; probability p lot methods; exp onential distribution. Introduction Exp onential distribution is one of the most imp ortant distributions which can be used in many p laces such as in the statist ics, engineering, p hy sics, chemist ry and others [1]. It is good to use with reliability because value of failure is constant since it has one p arameter[2]. Exp onential gives distribution of time between indep endent events occurring at a constant rate-equivalently, and it is a sp ecial case of both weibull and gamma distributions [2]. An imp ortant p rop erty of this distribution is that its memory is less so it is used in any sy stem in the life and to solve p roblems of surviv al theory and analy sis live table and it is called lif e distribution [3]. Exp onential distribution has the density function below:- tetf  )( t, λ>0 ---------------------------------(1) Where  is scale parameter and a continuous random variable X is said to have an exp onential distribution with rate p arameter λ as shown in figure(1). Not ed that This distribution is valuable and has t he following advantages see; [4] (1) A single and easily estimated p arameter (2) Is mathematically tractable (3) Has fairly wide app licability Anot her p rop erties of exp onential distribution are list ed in table (1) From table (1) note that tetF  1);( ------------------------------------ (2) ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 If T~exp (λ) and U represent a uniform random varib le from[0,1] then:-     0 1 )(uf )1ln( 1 ut   tey 1 t, λ>o Q te dx dy J   1( ) ( )g t f u t J      ( ) tg t e    1 ln(1 )t y     --------------------------------------- (3) Maximum Likelihood Estimator In estimating unknown p arameters the most p op ular method is the M aximum-likelihood estimator (M LE). One imp ortant reason is that the M LE is asy mptotically optimal in that it approximates the minimum variance unbiased (M VU) estimator for lar ge data r ecords [4].M aximum Likelihood Estimator (M LE) represents a very gener al method of p oint estimation which is ap p licable whether the regu larity condition are or are not satisfied [5]. Consider est imation of λ when :- Let nTTT ,,, 21  be a random samp le of size n≥2 from exp (λ ) then the loglikehood function L(λ ) is J.P.F of .,,, 21 nttt  Hence ),(,),(),(),,,,( 2121  nn tftftftttf   ---- ------------------------------- (4) = 1 2 t t tne e e             K K K = 1 n t in ie      By taking ),,,,()( 21  ntttfL  --------------------------------- (5) ln ( ) ln 1 nnL ti i       And ln ( ) 1 nL n ti i          By equating 0 )(ln     L 0 1 nn ti i      0,1 If u Ot herwise ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 We get an estimator of λ which is denoted by   1 n n ti i      ---------------------------------- (6) Probability Plotting Method (P.P.E) To estimate the p arameter, one can also use the graphical method called p robability p lotting method using the data obtained [6]. The following transformations have b een employed: Probability Plotting M odel: BAF  C.D.F: F (t) From table (1) tetF 1)( tetF  )(1 ttF  )](1ln[ -----------------------------------(7) )(ln)](1ln[ tftF  ttF )(ln -----------------------------------(8) Taking the natural logarithim of the sided for equation(8): )ln()](ln[ln ttF  ---------------------------------(9) Hence ytF )](ln[ln &   A ln and xt )ln( Hence equation[9] will p roduce linear from: iii BxAy   ---------------------------------(10) )ln(ln tyi    --------------------------------(11) Now, in added error term to equation (10), the iii eyy   ; e~E(t) Using estimater,we obtan est imater the Exp onential p arameter λ frome equation (11): ))ln(exp ( tyi    ------------------------------(12) Now, p arameter estimaterfor the other distribution, p robability p lotting model can b e written as in equation (12) above. ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 Simulation & Empirical Work One of the most imp ortant app lication of comp uter science is comp uter simulation [7]. Simulation can be used to show the eventual real effects of alternative conditions and courses of action. Simulation is also used when the real sy st em cannot be engaged, because it may not be accessible, or it may be dangerous or unaccep table to engage, or it is being designed but not y et built, or it may simply not exist [8].simulation app roaches offer great op p ortunities for working out p robabilities, cofidence intervals and similar concepts [9].This analysis may be done, sometimes, through analytical or numerical methods, but the model may be too comp lex to be dealt with. Essentially , simulation p rocess consists of building a comp uter model that describes the behavior of a sy st em and exp erimenting with this comp uter model to reach conclusions that sup p ort decisions [10]. Sometimes, it is not feasible or p ossible, to build a p rototy p e, y et we may obtain a mathematical model describing, through equations and constraints, the essential behavior of the sy st em. In such extreme cases, we may use simulation to replicate real world st udies that cannot be done, simulation exercises may encounter st atist ical p itfalls that degrade their p erformance, or fail to take advantage of the op p ortunities st atist ics can p rovide for controlling simulation error and p roducing st atist ically reliable results [10]. In order to mak e the b est estimation of p arameter of it can exp onentail distribution for (M LE)and (P.P.E). We make a simulation p rototype provide assumption of many cases whh icbe existed in real word and use the basic st ep p rocess in any simulation exp eriment once we have estimated the corresp onding simulation model. Algorithms steps :- (I) Frist step:- Sp ecified the assumed valu es by choosing d ifferent sample sizes of exp onential distrbution, such as samp le size (n=20) and samp le size (n=50) and samle size (n=100) Then choosing the valu es of assump tion p arameter λ in each several contrast s and choosing for the in itial values of t he parameter (scale) it as shown in Table (2):- (II) S econd ste p :- Generation of data which include: - Generated the random data which was taken from the uniform distribution in the interval [0,1] using Excel,and SP SS,soft ware comp uter p ackage. - The generation of errors for all d ata and in method the random errors have been generated using the standard exp onential distribution. (III) Third ste p :- This step contains t he following :- - Using the same value of t  &  iy for amethods (M LE),(P.P.E)and app ly ing the equation 1 ln(1 )t y      & )ln(ln tyi    as mentioned in (3),(11). - Finding the time(t) by using the equation t t ei i    where i=1,2,……….,n - The value of   of exp onential distribution can be determined according to the estimators of (M LE)and(P.P.E) in equation (6),(12). (IV) Fourth ste p: smoothing the obtained values ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 - In this st ep the iteration of data will b e repeated 500 times t o generate a new d ifferent error, so we obtain100 value of t  for each contrast . Then the mean of each case will be calculated to find the est imated t . (V) Fifth ste p :- In this st ep the followin g comp arison indicator will b e employed to make a comp orison between different values of λ and different methods for (M LE) and Probability Plotting estimation (P.P.E). Conclusions & Future Work As a consequence for p ractical work and taking the mean square error as the indicator of p reference between the different est imator methods, the following r esults are obtained:- (1) S ample size (n=20) For the assumed contrast p arameters (λ= 0.5) the M LHE and P.P.E estimators was given the best results. (2) S ample size (n=50) For the assumed contrast p arameters (λ= 0.5) the M LHE and P.P.E estimators was given the best results. (3) S ample size (n=100) For the assumed contrast p arameters (λ=0.5) the M LHE and P.P.E estimators was given the best results. (4) The best results from different samp le sizes (20, 50, 100) is samp le size n=100 for the assumed contrast p arameters (λ=0.5, 1, 2) the M LHE estimator method was given the best results. (5) The best results from different sample sizes (20, 50, 100) is sample size n=50 for the assumed contrast p arameters (λ=0.5, 1, 2) the P.P.E method was given the best results. The results of simulation for differ ent sample sizes (n=20, 50, and100) are list ed in the table (3) Re ference 1. Green. J. r & M arge Rison, D. (1978), Statistical treatment of Exp erimental data, Amst erdam: Nort h-Holland Publ ishing Co mpany; New York: Elsevier/Nort h-Holland, In c. x + 382 p p . 2. Jap ar Abd Modhe, (1999), Same of valuables Reliability Estimators for Exp onential Dist ribution by Using Shrink Estimators", Ibn AL-Haitham Education College, Baghd ad University , M.SC thesis. 3. Felle.W. R, (1971), Introduction to Probability Theory and Its App lications, II, (2nd edition), Wiley . Section I.3, ISBN 0-471-25709-5 . 4. Quan Din g and Steven Kay , (2011), M aximum Likelihood Estimator under a M issp ecified M odel with High Si gnal-to-Noise Ratio, ie transactions on si gn al p rocessing, Journal: IEEE Transactions on Sign al Processin g. 59, no. 8, 1053587X Pages: 4012-4016 5. Hogg, (1986), Introduction to M athematical Statist ics edition, NewYork : The M acmillan Company, x+415pp . 6. Rajini ,V. (2010), Prediction of Life Time of Poly meric Insulators: A Statist ical App roach, Iranian Journal of Electrical and Comp uter Engineering, 9(1) winter-sp ring 1682-0053. ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 7. Banks, and J. Carson, (2001). "Discrete-Event Sy st em Simulation" Prentice Hall. P.3. ISBN 0-13-088702-1. 8. Banks .Sokolowski, J.A., , C.M . (2009). Princip les of M odeling and Simu lation. Hoboken, NJ: Wiley. p . 6. ISBN 978-0-470-28943-3 . 9. M ichael Wood, (2005),”The Role of Simulation Ap p roaches in Statist ics”, University of Port smouth, U.K., Journal of. Statist ics Education.13, no.3. 10. Insua.D.R. and etal,(2005), Simulation in Indust rial Statistics, Statistical and App lied M athematical Sciences Inst itut e, Technical Report, Research Trian gle Park,NC 27709- 4006,PO Box 14006, available at www.samsi.info. Table (1):S ome Propertie s of an Expone nti al Distribution Where (Scdf) is standard cumu lative density function. 1  Mean 1 2  Variance ln 2 0.693    Median ln 4 ln 3   First quartile ln 4  Thi rd quartile ( ) 1 (1 ) t R t e     Survival function          )1(1)(1 )( )( te te tF tf th Hazard function 0 1y , ln(1 )y    1 ( )F y  t etF   1);( (S cdf) F(t) ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 Table (2):Assumed contrast paramete r Table (3) Table (3):Esti mation of S cale Parameter of Expone nti al Distributi on For (MLH) an d (P.P.E ) 0.5 1 2 Assume d Paramet er Esti mator Indicator S ample λ   M SE M LH M SE P.P.E 0.5 0.327733 0.162901 . 02128338 1 0.550218 1.08878 . 08668342 20 2 0.616011 10.11944 . 34332615 0.5 0.285936 0.093745 . 02059832 1 0.602633 0.327592 . 08300531 50 2 0.717962 3.364929 . 32840239 0.5 0.344761 0.0245 . 02084406 1 0.537857 0.21746 . 08338210 100 2 0.65473 1.829821 .33505150 ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 Fig.(1): The Probaability Density Function for Expone nti al Distributi on ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 سي بتطبیق طریقتي دالة االمكان االعظم ودالة الرسم ألتخمین معلمة التوزیع ا باستخدام المحاكاة البیاني االحتمالي األء ماجد حمد ابن الھیثم ،جامعة بغداد -كلیة التربیة ، قسم الریاضیات 2011تشرین االول 18: ، قبل البحث في 2011ایلول 4:استلم البحث في الخالصة والرسم ، في هذا البحث تم تخمین معلمة القیاس للتوزیع االسي من خالل تطبیق طریقتي دالة االمكان االعظم أ "ااستخدم مؤشر. للمعلمة عدیدة وألحجام وعینات مختلفة مع تولیفات افتراضیةالبیاني االحتمالي معدل مربعات الخط .كمؤشر ألفضل اداء باستخدام تقنیة المحاكاة الحاسوبیة و تحلیل القیم والنتائج المستحصلة .وزیع االسي تتقدیر الجوار االعظم ،طریقة الرسم االحتمالي، ال: المفتاحیةالكلمات ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012