مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 Estimation of the Parameter of 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, University of Baghdad Received in: 4September2011, Accepted in: 18October2011 Abstract Exponential Distribution is probably the most important distribution in reliability work. In this paper, estimating the scale parameter of an exponential distribution was proposed through out employing maximum likelihood estimator and probability plot methods for different samples size. Mean square error was implemented as an indicator of performance for assumed several values of the parameter and computer simulation has been carried out to analysis the obtained results. Key words: maximum likelihood estimators; probability plot methods; exponential distribution. Introduction Exponential distribution is one of the most important distributions which can be used in many places such as in the statistics, engineering, physics, chemistry and others [1]. It is good to use with reliability because value of failure is constant since it has one parameter[2]. Exponential gives distribution of time between independent events occurring at a constant rate-equivalently, and it is a special case of both weibull and gamma distributions [2]. An important property of this distribution is that its memory is less so it is used in any system in the life and to solve problems of survival theory and analysis live table and it is called life distribution [3]. Exponential 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 exponential distribution with rate parameter λ as shown in figure(1). Noted that This distribution is valuable and has the following advantages see; [4] (1) A single and easily estimated parameter (2) Is mathematically tractable (3) Has fairly wide applicability Another properties of exponential distribution are listed in table (1) From table (1) note that tetF  1);( ------------------------------------ (2) مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 If T~exp(λ) and U represent a uniform random varible 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 parameters the most popular method is the Maximum-likelihood estimator (MLE). One important reason is that the MLE is asymptotically optimal in that it approximates the minimum variance unbiased (MVU) estimator for large data records [4].Maximum Likelihood Estimator (MLE) represents a very general method of point estimation which is applicable whether the regularity condition are or are not satisfied [5]. Consider estimation of λ when :- Let nTTT ,,, 21  be a random sample 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 tin ie      By taking ),,,,()( 21  ntttfL  --------------------------------- (5) ln ( ) ln 1 nnL tii       And ln ( ) 1 nL n tii          By equating 0 )(ln     L 0 1 nn tii      0,1 If u Otherwise مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 We get an estimator of λ which is denoted by  1 n n tii      ---------------------------------- (6) Probability Plotting Method (P.P.E) To estimate the parameter, one can also use the graphical method called probability plotting method using the data obtained [6]. The following transformations have been employed: Probability Plotting Model: 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 produce 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 estimater the Exponential parameter λ frome equation (11): ))ln(exp( tyi    ------------------------------(12) Now, parameter estimaterfor the other distribution, probability plotting model can be written as in equation (12) above. مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 Simulation & Empirical Work One of the most important application of computer science is computer 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 system cannot be engaged, because it may not be accessible, or it may be dangerous or unacceptable to engage, or it is being designed but not yet built, or it may simply not exist [8].simulation approaches offer great opportunities for working out probabilities, cofidence intervals and similar concepts [9].This analysis may be done, sometimes, through analytical or numerical methods, but the model may be too complex to be dealt with. Essentially, simulation process consists of building a computer model that describes the behavior of a system and experimenting with this computer model to reach conclusions that support decisions [10]. Sometimes, it is not feasible or possible, to build a prototype, yet we may obtain a mathematical model describing, through equations and constraints, the essential behavior of the system. In such extreme cases, we may use simulation to replicate real world studies that cannot be done, simulation exercises may encounter statistical pitfalls that degrade their performance, or fail to take advantage of the opportunities statistics can provide for controlling simulation error and producing statistically reliable results [10]. In order to make the best estimation of parameter of it can exponentail distribution for (MLE)and (P.P.E). We make a simulation prototype provide assumption of many cases whh icbe existed in real word and use the basic step process in any simulation experiment once we have estimated the corresponding simulation model. Algorithms steps :- (I) Frist step:- Specified the assumed values by choosing different sample sizes of exponential distrbution, such as sample size (n=20) and sample size (n=50) and samle size (n=100) Then choosing the values of assumption parameter λ in each several contrasts and choosing for the initial values of the parameter (scale) it as shown in Table (2):- (II) Second step :- Generation of data which include: - Generated the random data which was taken from the uniform distribution in the interval [0,1] using Excel,and SPSS,software computer package. - The generation of errors for all data and in method the random errors have been generated using the standard exponential distribution. (III) Third step :- This step contains the following :- - Using the same value of t  &  iy for amethods (MLE),(P.P.E)and applying 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 exponential distribution can be determined according to the estimators of (MLE)and(P.P.E) in equation (6),(12). (IV) Fourth step: smoothing the obtained values مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 - In this step the iteration of data will be repeated 500 times to 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 estimated t . (V) Fifth step :- In this step the following comparison indicator will be employed to make a comporison between different values of λ and different methods for (MLE) and Probability Plotting estimation (P.P.E). Conclusions & Future Work As a consequence for practical work and taking the mean square error as the indicator of preference between the different estimator methods, the following results are obtained:- (1) Sample size (n=20) For the assumed contrast parameters (λ= 0.5) the MLHE and P.P.E estimators was given the best results. (2) Sample size (n=50) For the assumed contrast parameters (λ= 0.5) the MLHE and P.P.E estimators was given the best results. (3) Sample size (n=100) For the assumed contrast parameters (λ=0.5) the MLHE and P.P.E estimators was given the best results. (4) The best results from different sample sizes (20, 50, 100) is sample size n=100 for the assumed contrast parameters (λ=0.5, 1, 2) the MLHE 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 parameters (λ=0.5, 1, 2) the P.P.E method was given the best results. The results of simulation for different sample sizes (n=20, 50, and100) are listed in the table (3) Reference 1. Green. J. r & Marge Rison, D. (1978), Statistical treatment of Experimental data, Amsterdam: North-Holland Publishing Company; New York: Elsevier/North-Holland, Inc. x + 382 pp. 2. Japar Abd Modhe, (1999), Same of valuables Reliability Estimators for Exponential Distribution by Using Shrink Estimators", Ibn AL-Haitham Education College, Baghdad University , M.SC thesis. 3. Felle.W. R, (1971), Introduction to Probability Theory and Its Applications, II, (2nd edition), Wiley. Section I.3, ISBN 0-471-25709-5 . 4. Quan Ding and Steven Kay, (2011), Maximum Likelihood Estimator under a Misspecified Model with High Signal-to-Noise Ratio, ie transactions on signal processing, Journal: IEEE Transactions on Signal Processing. 59, no. 8, 1053587X Pages: 4012-4016 5. Hogg, (1986), Introduction to Mathematical Statistics edition, NewYork: The Macmillan Company, x+415pp. 6. Rajini ,V. (2010), Prediction of Life Time of Polymeric Insulators: A Statistical Approach, Iranian Journal of Electrical and Computer Engineering, 9(1) winter-spring 1682-0053. مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 7. Banks, and J. Carson, (2001). "Discrete-Event System Simulation" Prentice Hall. P.3. ISBN 0-13-088702-1. 8. Banks .Sokolowski, J.A., , C.M. (2009). Principles of Modeling and Simulation. Hoboken, NJ: Wiley. p. 6. ISBN 978-0-470-28943-3 . 9. Michael Wood, (2005),”The Role of Simulation Approaches in Statistics”, University of Portsmouth, U.K., Journal of. Statistics Education.13, no.3. 10. Insua.D.R. and etal,(2005), Simulation in Industrial Statistics, Statistical and Applied Mathematical Sciences Institute, Technical Report, Research Triangle Park,NC 27709- 4006,PO Box 14006, available at www.samsi.info. Table (1):Some Properties of an Exponential Distribution Where (Scdf) is standard cumulative density function. 1  Mean 1 2  Variance ln 2 0.693    Median ln 4 ln 3   First quartile ln 4  Third quartile ( ) 1 (1 )tR t e    Survival function          )1(1)(1 )( )( te te tF tf th Hazard function 0 1y , ln(1 )y    1( )F y tetF  1);( (Scdf) F(t) مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 Table (2):Assumed contrast parameter Table (3) Table (3):Estimation of Scale Parameter of Exponential Distribution For (MLH) and (P.P.E) 0.5 1 2 Assumed Paramet er Estimator Indicator Sample λ   MSE MLH MSE 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 مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 Fig.(1): The Probaability Density Function for Exponential Distribution مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for 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 معدل مربعات الخطأ "ااستخدم مؤشر. للمعلمة عدیدة وألحجام وعینات مختلفة مع تولیفات افتراضیةالبیاني االحتمالي .كمؤشر ألفضل اداء باستخدام تقنیة المحاكاة الحاسوبیة و تحلیل القیم والنتائج المستحصلة .وزیع االسي تتقدیر الجوار االعظم ،طریقة الرسم االحتمالي، ال: المفتاحیةالكلمات مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012