This paper deals with constructing a new generated Marshal – Olkin Uniform family distribution which include finding the probability density function

Ameer, Hayder

Hamza,  Zainab

Hassan, Dhwyia

English

Iraqi Journal for Computers and Informatics

 55 All rights reserved©UOITC                                                                                                                                www.uoitc.edu.iq  Comparing Three Different Algorithms to Estimate Parameters of new Generated Marshal – Olkin Uniform Distribution  Prof. Dhwyia S. Hassan1/ Zainab F. Hamza2  1&2Department of Business Information Technology, College of Business Administration of Informatics, University of Information Technology &Communications, Iraq    Hayder A. Ameer3 MSc Operation Research   Abstract— This paper deals with constructing a new generated Marshal – Olkin Uniform family distribution which include finding the probability density function ( ), cumulative distribution function ( ), this new compound  is necessary to represent the time to failure data of complex system, so we insist to extend Marshal – Olkin distribution to another family called Marshal – Olkin Weibull, we deriving the ( ),  ( ), and reliability function, then applying simulation procedure taking different sample size (  and different set of initial values of ( ), each experiment repeated ( ), all results explained in tables.  Index Terms—MOEU(Marshall-Olkin extended uniform , MLE,Maximum likelihood estimator,MOM,Moment estimators,REM,Regresion estimators I. INTRODUCTION  The  of Marshal – Olkin extended uniform (MOEU) is defined by;                       (1)            Where; ( ) is the shape parameter and ( ) is the scale parameter, while the corresponding cumulative distribution function is;  (16)                       (2)     The reliability function of (MOEU) is;                      (3) For the  in equation (1) we observe that the shape of this ,  depend on parameter ( ), when [ ], the  is decreasing function ( ) with;                                         (4) But when ( ), the  is an increasing function on ( ) with (4). Also we can find hazard rate of the random variable ( ), which has  in (1);                                                       (5) The  moment about origin of this distribution [MOEU( )] is;                                                                                                    According to this formula, the mean and variance of  with [MOEU( )] is;                                                       (7)                                          (8) Also we can show that the coefficient of variation is;                                          (9) Which depend on shape parameter ( ), and the coefficient of Skewness is;                                                        Where;                                                   We can prove that;                                                                        (10)   II. ESTIMATION OF THE PARAMETERS This section deals with introducing some methods to estimate two parameters ( ), then applying simulation procedure to compare the results, these methods are; II.1 MAXIMUM LIKELIHOOD METHOD The estimators by this method obtained from maximizing;  56 All rights reserved©UOITC                                                                                                                                www.uoitc.edu.iq                                                 If we consider [ ], then;  Which is an implicit function can be solved numerically.  II.2 MOMENT ESTIMATOR  Let , then the moment estimators of ( ) obtained from equating sample moment [ ] with population moment ( ) defined in equation (6), we use coefficient of variation ( );                                                       (14) Which is a function of shape parameter ( ). Therefore equating ( ) with sample ( );                                                This yield ( ) and then from  [ ], we can find ( ):                                                         (15)  II.3 MOMENT ESTIMATOR This method used to estimate ( ) for [ ] is regression estimator which explained as follows; Let ( ) be a random sample from [ ], defined in equation (1), then;                                                                          Let                                          simple linear regression model, therefore;                                                                               (16) And                                                                                (16)       III. Application The application has been done through simulation procedure to compare between the estimators, for the values of scale parameter ( ) and shape parameter ( ) with ( ), the comparison for scale parameter and shape parameter was done by MSE.  3 3.5 4  1 1.5 2 Table (1):     n Method      20 MLE 1.031278 0.113673 2.851806 0.039166 Moment 1.068291 0.276983 2.996657 0.001013 Regression 1.031082 0.11297 2.847869 0.040374 BEST                        MOM                      MOM  40 MLE 1.00589 0.059886 2.937422 0.007572 Moment 1.019005 0.106815 2.999201 0.00036 Regression 1.005839 0.059516 2.933136 0.008074 BEST                                REG                      MOM  80 MLE 1.068839 0.080575 2.949082 0.005078 Moment 1.107214 0.134356 3.000366 0.000264 Regression 1.068609 0.080022 2.944403 0.00561 BEST                               REG                   MOM  100 MLE 0.992154 0.026968 2.971291 0.001815 Moment 0.984331 0.036687 2.999123 5.07E-05 Regression 0.992206 0.026804 2.966234 0.002137 BEST                     REG                MOM Table (2):               n Method      20 MLE 1.100314 0.238803 3.32512 0.059525 Moment 1.229697 0.510707 3.502232 2.10E-03 Regression 1.099771 0.236671 3.318926 0.061781 BEST                       REG                       MOM  40 MLE 1.066186 0.062155 3.443084 0.00695 Moment 1.084501 0.083461 3.499059 1.49E-04 Regression 1.065977 0.061891 3.438127 0.007579 BEST                       REG                       MOM  80 MLE 1.012072 0.034551 3.457482 0.003955 Moment 1.00564 0.069503 3.500705 1.36E-04 Regression 1.012036 0.034326 3.451485 0.004553 BEST                       REG                       MOM  100 MLE 0.996889 0.024841 3.465704 0.002261 Moment 1.005863 0.045934 3.500502 9.70E-05 Regression 0.996893 0.024648 3.459205 0.002801 BEST                       REG                       MOM  57 All rights reserved©UOITC                                                                                                                                www.uoitc.edu.iq   Table (3):    Prepare Your PAPER n Method      20 MLE 1.042908 0.138847 3.811649 0.063222 Moment 1.156049 0.287677 4.007926 1.98E-03 Regression 1.042368 0.137414 3.804282 0.065918 BEST                       REG                       MOM  40 MLE 1.029217 0.078406 3.911941 0.014818 Moment 1.043457 0.159298 4.002348 5.13E-04 Regression 1.029042 0.07787 3.904754 0.01623 BEST                       REG                       MOM  80 MLE 1.034246 0.034645 3.958564 0.004069 Moment 1.054223 0.060447 3.999109 1.51E-04 Regression 1.034102 0.034334 3.950485 0.004775 BEST                       REG                       MOM  100 MLE 1.029207 0.041988 3.969557 0.001767 Moment 1.006764 0.050583 3.999158 9.19E-05 Regression 1.029003 0.041595 3.961514 0.002371 BEST                       REG                       MOM Table (4):     n Method      20 MLE 1.603319 0.470726 2.898675 0.020577 Moment 1.677129 1.051682 3.000486 1.98E-03 Regression 1.600803 0.465825 2.895817 0.021229 BEST                       REG                       MOM  40 MLE 1.650908 0.240887 2.946342 0.005478 Moment 1.768623 0.535315 3.008162 6.91E-04 Regression 1.648319 0.238308 2.943197 0.005813 BEST                       REG                       MOM  80 MLE 1.519302 0.10133 2.972898 0.001614 Moment 1.544359 0.182681 3.001927 3.03E-04 Regression 1.51761 0.100462 2.970086 0.001766 BEST                       REG                       MOM  100 MLE 1.510354 0.064359 2.981114 0.00074 Moment 1.546363 0.168409 3.002502 2.96E-04 Regression 1.50853 0.063907 2.977822 0.000893 BEST                       REG                       MOM Table (5):     n Method      20 MLE 1.662229 0.482339 3.395761 0.01763 Moment 1.75725 0.731479 3.500527 2.40E-03 Regression 1.659424 0.477065 3.391316 0.018693 BEST                       REG                       MOM  40 MLE 1.50306 0.184362 3.436737 0.007136 Moment 1.556405 0.355055 3.502946 8.79E-04 Regression 1.500762 0.182108 3.43237 0.007696 BEST                       REG                       MOM  80 MLE 1.531236 0.065936 3.47274 0.001643 Moment 1.565229 0.134749 3.501977 4.93E-04 Regression 1.529203 0.065282 3.468705 0.001913 BEST                       REG                       MOM  100 MLE 1.50272 0.050678 3.476985 0.000941 Moment 1.533148 0.159062 3.502192 3.75E-04 Regression 1.500694 0.05031 3.472647 0.001158 BEST                       REG                       MOM   Table (6):     n Method      20 MLE 1.602384 0.368624 3.869891 0.032305 Moment 1.825823 1.436017 4.015709 3.21E-03 Regression 1.598837 0.362828 3.863818 0.034078 BEST                       REG                       MOM  40 MLE 1.551459 0.185092 3.93784 0.007377 Moment 1.605304 0.440727 4.002972 1.27E-03 Regression 1.548679 0.183124 3.932044 0.00812 BEST                       REG                       MOM  80 MLE 1.549335 0.101966 3.970651 0.001636 Moment 1.619259 0.251243 4.005788 7.38E-04 Regression 1.546586 0.100513 3.964991 0.002042 BEST                       REG                       MOM  100 MLE 1.511176 0.076053 3.971001 0.001802 Moment 1.52876 0.151724 4.002049 3.62E-04 Regression 1.508754 0.075092 3.965391 0.002171 BEST                       REG                       MOM    Table (7):     n Method      20 MLE 2.203633 0.877015 2.936145 0.008075 Moment 2.510654 2.760199 3.012835 2.48E-03 Regression 2.199787 0.869804 2.93423 0.00837 BEST                       REG                       MOM  40 MLE 2.141178 0.345645 2.9709 0.00167 Moment 2.313537 1.099686 3.006987 1.55E-03 Regression 2.136453 0.341346 2.968616 0.001801 BEST                       REG                       MOM  80 MLE 2.034213 0.137878 2.982707 0.000516 Moment 2.139557 0.554344 3.004001 9.57E-04 Regression 2.030249 0.136308 2.980478 0.000607 BEST                       REG                       MOM  100 MLE 1.982068 0.133995 2.985069 0.000551 Moment 1.969524 0.263091 2.99837 6.01E-04 Regression 1.978385 0.132882 2.98279 0.000631 BEST                       REG                       MOM Table (8):     n Method      20 MLE 2.213872 0.653475 3.898859 0.01965 Moment 2.798112 6.377826 4.022145 7.83E-03 Regression 2.207895 0.644301 3.895201 0.020476 BEST                       REG                       MOM  40 MLE 2.109576 0.409909 3.942162 0.007259 Moment 2.266288 0.997309 4.010185 3.08E-03 Regression 2.103151 0.401116 3.938412 0.007763 BEST                       REG                       MOM  80 MLE 2.038608 0.150793 3.977039 0.001124 Moment 2.114785 0.365654 4.004357 1.43E-03 Regression 2.033227 0.148042 3.973238 0.001312 BEST                       REG                       MOM  100 MLE 2.081594 0.142793 3.984962 0.00041 Moment 2.095828 0.26858 3.999847 9.30E-04 Regression 2.076102 0.140141 3.981166 0.000536 BEST                       REG                       MOM  58 All rights reserved©UOITC                                                                                                                                www.uoitc.edu.iq   IV. CONCLUSION From table one the  best estimator is  α ( where MSE(α )=0.026804) REG and the best estimator for θ is  ,since  MSE(θ )=0.002137 also from table two , we find  is best and   is best , where MSE( )=0.024648 MSE( =0.002801 Also from table three, we find best estimator is  and  , similarly from table four we find best estimator  for α is  and for θ is   V. RECOMMENDATION Since  this constructed model represent  a new  generated  Marshal – Olkin uniform family , and  it have two parameters(α) shape parameter , θ is scale parameter so we use simulation taking sample size (n=20,40,80,100) and each experiment  is Repeated 1000 times to estimate these two parameters , and the results  are compared by statistical measure  (Mean Square error ) and the results is explained in tables  . REFERENCES [1] [1] Al _ Athari, Faris, M. (2011), "Parameter Estimation for the double Pareto Distribution", Journal of mathematics and statistics, 7(4):289 – 294. 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Comparing Three Different Algorithms to Estimate Parameters of new Generated Marshal – Olkin Uniform Distribution

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Comparing Three Different Algorithms to Estimate Parameters of new Generated Marshal – Olkin Uniform Distribution

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