ISSN 168-80 Journal of Statistics Volume 0, 01. pp. 1-10 Comparison of Least Square Estimators with Rank Regression Estimators of Weibull Distribution - A Simulation Study Abstract Chandika Rama Mohan 1, Anantasetty Vasudeva Rao and Gollapudi Venkata Sita Rama Anjaneyulu We estimated the parameters of two-parameter Weibull Distribution with the Least Square method from an optimally constructed grouped sample and compared the efficiencies of these estimators with the parameters estimated with Rank Regression method from an ungrouped (complete) sample, though the estimators from both the methods are in closed form, we resorted to Monte-Carlo simulation for computing Bias, Variance and Mean Square Error. The Least Square Estimation method for optimally grouped sample will give an efficient shape parameter than the Rank Regression Estimation method for complete sample. A numerical example is presented to illustrate the methods proposed here. Keywords Weibull parameters, Asymptotically optimal grouped sample, Asymptotically relative efficiency, Least Square method, Rank Regression method. 1. Introduction It is well known that the two-parameter Weibull model is a widely used model in life testing experiments because it has increasing, decreasing or constant failure rate according to its shape parameter being greater than or less than or equal to one, respectively. 1 Lecturer in Statistics, ASRAM Medical College, Eluru, Andhra Pradesh, India. Department. of Statistics, Acharya Nagarjuna University, Nagarjuna Nagar-10, Guntur, Andhra Pradesh, India. Department of Statistics, Acharya Nagarjuna University, Nagarjuna Nagar-10, Guntur, Andhra Pradesh, India.
Chandika Rama Mohan, Anantasetty Vasudeva Rao and Gollapudi Venkata Sita Rama Anjaneyulu The statistical inference for the Weibull Distribution under censoring schemes has been investigated by several authors such as Herd (1960), Johnson (196), Mann (1969), Cohen (197), Cacciari and Montanari (1987), Wong (199) and Viveros and Balakrishna (199). In two-parameter Weibull Distribution, Vasudeva Rao et al. (199) studied the problem of asymptotically optimum grouping for Maximum Likelihood Estimation (MLE) of the parameters for grouped samples and they constructed optimal interval lengths of an equi-class grouped sample. In the same way, Srinivas (199) studied the problem of optimum grouping for Maximum Likelihood Estimation of the parameters in the case of un-equi-class grouped samples and constructed the class limits of an optimal un-equi-class grouped sample. He tabulated the optimal class limits of a grouped sample for k=(1) 10, 1 groups which are presented in Table 1. Shuo-Jye (00), Balakrishnan et al. (00) and Kantam et al. (00) constructed the optimum group limits for unequi-spaced grouped sample with k=(1) 1 groups for Maximum Likelihood Estimation in scaled Log-logistic Distribution. Further, they constructed the optimum interval lengths for equi-spaced grouped sample with k=(1) 10, 1 and 0 groups. The Rank Regression Estimation method is quite good for functions that can be linearized. Olteann and Freeman (009) have studied in their technical report, the Median Rank Regression and Maximum Likelihood Estimation techniques for a two-parameter Weibull Distribution. Srinivas (199) had constructed the optimal group limits of a grouped sample from two-parameter Weibull Distribution. But in practice the use of the optimal group limits constructed by him requires a prior knowledge or guess value of the parameters and hence the optimal group limits constructed by him have a limited use in practical applications. In this context, we develop a practical procedure to construct an optimally grouped sample even when there is no prior knowledge or guess values of the parameters for constructing this optimal grouped sample. The research topic can be considered and discussed under the following headings: Least Squares Estimation of the parameters from the optimally constructed grouped sample. Rank Regression Estimation of the parameters from ungrouped sample. Comparison of the efficiencies of Least Squares Estimators obtained from the optimally constructed grouped sample with Rank Regression Estimators obtained from ungrouped (complete) sample using Monte Carlo simulation based on 1000 samples of sizes.
Comparison of Least Square Estimators with Rank Regression Estimators of Weibull Distribution - A Simulation Study. Least Squares Estimation of the Parameters from the Optimally Constructed Grouped Sample The p.d.f. of two-parameter Weibull Distribution is given by ( ) ( ) ( ) * ( ) + (1.1) where b and c are respectively the scale and shape parameters. Its cumulative distribution function is ( ) * ( ) + (1.) To construct an optimally grouped sample when there is no a priori knowledge or guess values of the parameters suppose N be the number of sampling units put under a life-testing experiment which assumes the Weibull model (1.1) and suppose the experimenter wishes to obtain the grouped life-time data with k classes. Then from Table 1 optimal group limits y i s, i=1,, k-1. can be used to compute the expected number of sampling units to be failed in the time interval (t i-1, t i ) for i=1,, k and is given by f i = Np i for i =1,.k (1.) where ( ) ( ) f i is expected number of failures in the i th interval Y i s are optimal group limits obtained from Table 1 K is number of groups N is total frequency and f i s may be rounded to the nearest integers so that N=f 1 + f +..+f k. Here, it may be noted that the experimenter has to observe the random time instants t 1, t,, t k-1 so as the optimum pre-fixed number of units f i to be failed in the time interval (t i-1, t i ) for i=1,,, k taking t o =0 and t k =. In other words, he/she has to record a random time instant after failure of first f 1 sampling units, but before the failure of (f 1 +1) th sampling unit and to record a random time instant after failure of first f 1 + f sampling units, but before the failure of (f 1 + f +1) th sampling unit and so on. Further, it may be noted that it is difficult to record all exact failure times of the individual units but it is not so difficult to note a random time
Chandika Rama Mohan, Anantasetty Vasudeva Rao and Gollapudi Venkata Sita Rama Anjaneyulu instant between the failure times of two consecutive sampling units, t 1, t,, t k-1, the group limits of the optimally constructed grouped sample using the procedure explained above, are the observed values of the true asymptotic optimal group limits x 1, x,, x k-1 whereas their expected (estimated) values are given by ( ) (1.) where and are obtained by using the principle of Least Squares method that is by minimizing the Error Sum of Squares and y i s are asymptotically optimal group limits are taken from Table 1 as fixed values. (1.) where i is the difference between the log values of the observed and estimated values of the optimal group limits. According to the principle of Least Squares method the estimates of b and c may be obtained by regressing log t i s on log(y i s) that is by fitting the model (that is the power curve x=by 1/c ). ( ) (1.6) and are given by ( ) (1.7) * { ( }+ (1.8) The rationale for applying Least Squares method is that for a given k, y i s are fixed values and can be borrowed from Table 1 whereas t i s are random values and are obtained as observations from the procedure explained as above.. Rank Regression Estimation of the Parameters from Ungrouped (Complete) Sample Rank Regression Estimates of b and c from a complete sample x 1, x,, x N drawn from the above Weibull population (1.1) is that to bring the above function (1.) into a linear form.
Comparison of Least Square Estimators with Rank Regression Estimators of Weibull Distribution - A Simulation Study xi Ln Ln( 1 F( xi)) cln clnb clnxi (.1) b Now denote y Ln Ln( 1 F( xi)) and Ln x i =T i This results in the following linear equations yi ( clnb) ct i (.) where i = 1, N T i = Ln b+ (1/c) Y i (.) Now let us denote By simplification, we have (.) (.) In the Rank Regression method for parameter estimation we replace the unknown F x ) with median rank, namely ( i ˆ i 0. F( x i ) ; i = 1,,----N N 0. Among Y i and T i values we may treat Y i as independent variable (since it is fixed) and T i can be treated as dependent variable (since it is a random variable). But in the above (.) equation, the fixed variable Y is expressed as a function of the dependent variable T which is not reasonable form for applying Least Square method. In view of this important assumption, we rewrite equation (.) as (.).. Comparison of Least Square Estimators Obtained from Optimal Grouped Sample with Rank Regression Estimates Obtained from Ungrouped Samples The Least Square Estimates of and obtained from an optimal grouped sample are compared with the Rank Regression Estimates and obtained from
6 Chandika Rama Mohan, Anantasetty Vasudeva Rao and Gollapudi Venkata Sita Rama Anjaneyulu ungrouped sample. Though the estimates from both the methods in closed form, it is very difficult to obtain the bias and variance of estimates mathematically, since the estimates are in non-linear form. Hence, we have resorted to Manto Carlo simulation to compute bias, variance and Mean Square Error of and as well as and. We simulate these values based on 1000 samples of size N=0, 100, 10, 00, 0, 00 generated from standard Weibull Distribution with shape parameter C= to (1). Here to assess the performances of bˆ and ĉ, we compare these estimates with the corresponding rank regression estimates obtained from ungrouped sample. Results from the simulation results: The Least Square Estimators are less biased when compared to the Rank Regression Estimators, in particularly the scale parameter. Least Square Estimator of scale parameter is having slightly larger variance than Rank Regression Estimator of scale parameter; on the other hand, as Least Square Estimator of shape parameter is having smaller variance than Rank Regression Estimator of shape parameter The trace of the variance-covariance matrix of Least Square Estimators is relatively small than the trace of the variance- covariance matrix of Rank Regression Estimators. If our interest is to find an efficient estimator for shape parameter c, Least Square method may be applied instead of Rank Regression Estimation method.. Conclusion The Least Square Estimators are obtained from an optimal grouped sample whereas Rank Regression Estimators are obtained from complete sample. Thus Least Square method applied for an optimal grouped sample is yielding more efficient estimators than the Rank Regression method applied to a complete sample, especially when sample is large. This is an interesting application of Least Square method.
Comparison of Least Square Estimators with Rank Regression Estimators of Weibull Distribution - A Simulation Study An illustration A random sample of 00 observations is generated from a two-parameter Weibull Distribution with the parameters b=100 and c=. using MINITAB package and the ordered sample is given below: 0.9 1.11 16.8 18.16.09 6.197 8.70 9.819 9.91 0. 1.16 1.61 1.7.610.66.17 6.066 7.0 7.9 7.7 7.777 7.897 0.76 1.96 1.7.91.9.1 6.18 6.8 7.9 7.696 7.7 8.800 9.1 9.18 9.1 9.777 9.9 0.687 1.8 1.8 1.6.0.99.87.6..608 6.1 6.91 8. 9.06 9.71 9.78 61.0 61.76 6.60 6.917 6.97 66.976 67.090 67.1 67.89 67.7 67.79 68.69 69.0 69.78 70.09 70.6 70.7 71.1 71.67 7.1 7.90 7.10 7.9 7.11 7.160 7.768 7.881 7.7 76. 77.7 77.91 77.686 78.098 78.87 78.96 78.77 79.868 80.16 80.9 80.969 81.61 8.1 8.87 8.08 8. 8.1 8.81 8.99 8.9 8.08 8.779 86.087 86.098 86.1 86.890 86.97 87.6 88.16 88. 88.79 90.08 90.7 90.96 91.9 91.76 9.69 9.069 9.160 9.1 9.79 96.0 96.1 96.60 96.677 97.6 98.18 99.76 100.71 100.898 101.70 101.8 10.81 10.89 10. 10.910 106.60 108.11 108.17 110. 111. 111.6 111.891 111.998 11.96 11.66 11.898 11.01 11.0 11.169 11.1 116.08 119.97 119.98 10.98 11. 11.7 1.616 1.77 1.09 1.81 16.09 18.60 18.787 10.107 10.17 10.7 1.11 1.706 1.0 1.86 1. 16. 17.96 18.8 18.7 18.17 19. 1.87 1.70 16.090 16.11 11.6 1.79 1.70 1.79 17.108 160.668 160.711 16.69 16.699 168.1 168.96 17.68 181. 07.16 The computation of and from the optimal grouped sample of k=6 and size N=00 using (1.8), (1.7) and the computation of and from an ungrouped sample of size N=00 using (.) and (.) is carried out using MS-Excel and computation Table is presented in Tables and, respectively. 7
8 Chandika Rama Mohan, Anantasetty Vasudeva Rao and Gollapudi Venkata Sita Rama Anjaneyulu Table 1: Asymptotically optimal group limits X i in the form Y i= (X i /B) C, ( I=1,,,K-1) of Weibull scale (b) and shape(c) parameters from a grouped sample with k groups k y 1 y y y y y 6 y 7 y 8 y 9 y 10 y 11 y 1 y 1 y 1 0.7.6067 0.10 1.979.1 0.10 0.1 1.99.860 6 0.077 0.69 1.6.7.09 7 0.00 0.19 0.67 1.719.99.79 8 0.07 0.179 0.8 1.190.0.8.0 9 0.07 0.169 0. 0.78 1.60.71.767.7 10 0.01 0.0987 0.6 0.77 1.180 1.99.97.10.87 1 0.007 0.0 0.091 0.187 0.9 0.7 0.98 1..011.67..16. 7.00 Table : Computation of and from the optimal grouped sample using Least-Squares method t y Ln(t) ln(y) ln^(y) ln(t)ln(y).9 0.077.6 -.61 6.60-9.076 67.1 0.69.06-1.0081 1.016 -.06 108.1 1.68.68 0.0 0.018 0.97 16.1.7.98 0.98 0.897.709 176.8.096.17 1.88.016 7.6790.9-0.96 10.710 0.088 Optimal Grouped Sample Complete Sample Estimates using Estimates using Parameter Least Squares Rank Regression c (shape).7.6 b (scale) 98.90 97.98 TR RR =Trace of the variance covariance matrix of Rank Regression Estimators = V ( ) + V ( ) T LS = Trace of the variance covariance matrix of Least Square Estimators. = V ( bˆ ) +V ( )
Comparison of Least Square Estimators with Rank Regression Estimators of Weibull Distribution - A Simulation Study 9 Table : Performance of Least Square Estimates of scale (b) and shape (c) parameters of Weibull Distribution obtained from ungrouped sample as compared with the Rank Regression Estimates obtained from complete sample. (ARE RR ( ): ARE of ( ) as compared with, ARE RR ( ): ARE of ( ) as compared with ) Least Square Rank Regression Rank Regression Least Square Estimate TR RR TR LS Estimate Estimate K N C 6 0 100 10 00 Estimate BIAS/b MSE/b VAR/b (1) () () -.011.006.0060 -.0091.008.007 -.007.0016.001 -.0060.0010.0010 -.0061.00.00 -.006.001.001 -.007.0008.0008 -.000.000.000 -.0066.001.001 -.007.0009.0009 -.007.000.000 -.000.000.000 -.009.0017.0016 -.0066.0008.0007 -.000.000.000 -.000.000.000 BIAS/b MSE/b VAR/b () () (6).001.007.007.000.00.00.0000.001.001 -.0001.0009.0009.000.009.009.0000.001.001 -.0001.0007.0007 -.000.000.000.0000.0018.0018 -.000.0008.0008 -.000.000.000 -.000.000.000.000.001.001.0000.0006.0006 -.0001.000.000 -.0001.000.000 BIAS MSE VAR (7) (8) (9).0079.0779.0779.0119.17.17.019.118.11 -.0199.87.868 -.009.086.086 -.00.0868.0867 -.008.1.1 -.007.10.10 -.001.066.066 -.0077.098.098 -.010.106.106 -.018.166.1660 -.01.006.00 -.018.06.061 -.0.08.0819 -.00.189.180 BIAS MSE VAR (10) (11) (1) -.006.08.076.01.00.0.06.019.006.10.090.0780.19.17.179.88.800.10.1.99.89 -.01.0.0 -.018.071.077 -.06.1.18.08.09.0899.060.11.10.019.018.017.078.017.09.067.071.0700.0796.117.109 (1) (1).086.080.1778.1786.10.1.877.90.01.06.0880.076.19.17.1.081.08.0.0606.01.1067.090.166.108.019.0191.067.001.08.070.18.1097 ARE RR ( ) ( ) (1) (16) 9. 99.8 9.9 99.6 9.6 99. 9. 99. 87.1 116.1 86.6 116.1 86. 116.1 86. 116.1 89. 118. 88.9 118. 88.7 118. 88. 118. 87. 117.0 86.7 117.0 86. 117.0 86. 117.0 0.001.001.001.0008.0006.0006.000.000.000.000.000.000 -.000.0011.0011 -.000.000.000 -.000.000.000 -.000.000.000 -.011.0170.0169 -.018.08.080 -.0.068.0676 -.00.1066.106.0168.01.010.0.0.016.07.07.061.01.089.0877.0180.01.08.0.0679.06.108.0879 8. 10. 8.7 10. 8.8 10. 8.8 10.
Table : Performance of Least Square Estimates of scale (b) and shape (c) parameters of Weibull Distribution obtained from ungrouped sample as compared with the Rank Regression Estimates obtained from complete sample. (ARE RR ( ): ARE of ( ) as compared with, ARE RR ( ): ARE of ( ) as compared with ) Least Square Rank Regression Rank Regression Least Square Estimate TR RR TR LS K N C Estimate Estimate 8 100 10 00 0 00 Estimate BIAS/b MSE/b VAR/b (1) () () -.0160.00.001 -.011.001.001 -.0087.0009.0008 -.0070.0006.000 -.010.00.001 -.010.0011.0010 -.0079.0006.000 -.006.000.000 -.0010.0018.0018 -.0010.0008.0008 -.0009.000.000 -.0007.000.000 -.01.001.001 -.008.0007.0006 -.006.000.000 -.00.000.000 -.008.001.001 -.001.000.000 -.001.000.000 -.00.000.000 BIAS/b MSE/b VAR/b () () (6).000.009.009.0000.001.001 -.0001.0007.0007 -.000.000.000.0000.0018.0018 -.000.0008.0008 -.000.000.000 -.000.000.000.000.001.001.0000.0006.0006 -.0001.000.000 -.0001.000.000 -.000.0011.0011 -.000.000.000 -.000.000.000 -.000.000.000.0000.0010.0010 -.0001.000.000 -.0001.000.000 -.0001.000.000 BIAS MSE VAR (7) (8) (9) -.011.0170.0169 -.018.08.080 -.009.086.086 -.00.0868.0867 -.008.1.1 -.007.10.10 -.001.066.066 -.0077.098.098 -.010.106.106 -.018.166.1660 -.01.006.00 -.018.06.061 -.0.08.0819 -.00.189.180 -.0.068.0676 -.00.1066.106 -.01.01.011 -.0187.00.017 -.09.069.06 -.011.0890.0880 BIAS MSE VAR (10) (11) (1).10.098.09.16.110.0887.00.1990.178..109.67 -.0098.0.0 -.011.01.09 -.00.0981.0977 -.06.1.16 -.019.0189.0187 -.0.06.01 -.099.078.079 -.07.118.1171.019.019.01.091.08.09.088.066.061.08.099.0970.0097.01.01.01.081.079.019.000.096.01.078.0776 (1) (1).01.0.0880.0901.19.186.1.7.08.06.0606.09.1067.098.166.10.019.00.067.09.08.07.18.117.0180.0169.08.0.0679.06.108.097.011.016.01.08.066.099.088.0778 ARE RR ( ) ( ) (1) (16) 91. 97.9 90. 97.8 89.7 97.7 89. 97.7 86. 108.9 8.7 108.8 8. 108.8 8.0 108.8 77. 109. 77. 109. 77. 109. 77. 109. 8.0 108.9 81. 108.9 81.0 108.9 80.7 108.9 8. 11. 8.1 11. 8.9 11. 8.8 11. 10 Chandika Rama Mohan, Anantasetty Vasudeva Rao and Gollapudi Venkata Sita Rama Anjaneyulu
Comparison of Least Square Estimators with Rank Regression Estimators of 11 Weibull Distribution - A Simulation Study Acknowledgements The authors are highly thankful to the honorable referees for their valuable suggestions in improving the presentation of this work. The authors are also thankful to the authorities of Acharya Nagarjuna University and ASRAM Medical College for providing facilities for this work. References 1. Balakrishnan, N., Ng, H.K.T. and Kannan, N. (00). Goodness-of-Fit Tests based on spacing for Progressively Type-II censored data from a general Location-scale Distribution. IEEE Transactions on Reliability,, 9-6.. Cacciari, M. and Montanari, G. C. (1987). A method to estimate the Weibull parameters for progressively censored tests. IEEE Transactions on Reliability, 6, 87-9.. Cohen, A. C. (197). Multi-censored sampling in the three parameter Weibull Distribution. Technometrics, 17, 7-1.. Herd, G. R. (1960). Estimation of reliability from incomplete data. Proceedings of the 6 th National Symposium on Reliability and Quality Control. IEEE, New York, 0-17.. Johnson, L. G. (196). The Statistical treatment of fatigue experiments. Elserier, New York. U.S.A. 6. Kantam, R.R.L., Vasudava Rao, A. and Srinivasa Rao, G. (00). Optimum group limits for estimation in scaled Log-logistic Distribution from a grouped data. Statistical Papers, 6, 9-77. 7. Mann, N. R. (1969). Exact three-order-statistic confidence bounds on reliable life for a Weibull model with progressive censoring. Journal of American Statistical Association, 6, 06-1. 8. Oltean, D. and Freeman, L. (009). The Evaluation of Median Rank Regression and Maximum Likelihood Estimation techniques for a two-parameter Weibull Distribution. Quality Engineering, 6-7. 9. Shuo-jye, Wu. (00). Estimation of the parameters of Weibull Distribution with Progressively censored data. Journal of Japan Statistical Society, (), 1-16. 10. Srinivas, T. V. S. (199). On asymptotically optimal grouping of a sample in two-parameter Weibull Distribution. Unpublished M.Phil. Dissertation, Acharya Nagarjuna University, India.
1 Chandika Rama Mohan, Anantasetty Vasudeva Rao and Gollapudi Venkata Sita Rama Anjaneyulu 1. Viveros, R. and Balakrishna, N. (199). Interval Estimation of parameters of life from progressively censored data. Technometrics, 6, 8-91. 1. Wong, J. Y. (199). Simultaneously estimating the three Weibull parameters from progressively censored samples. Microelectronics and Reliability,, 17-. APPENDIX-I Symbols: Least Square Estimators of scale(b) and shape(c) are and. The Rank Regression Estimators of scale(b) and shape(c) are b ~ and c ~ Asymptotically relative efficiency of Least Square Estimators scale ( bˆ ) and shape( ĉ ) as compared with Rank Regression Estimators scale( bˆ ) and shape( c ~ ) are ARE RR (bˆ ) and ARE RR ( ĉ ), Trace of variance covariance matrix of Least Square Estimators is TR LS and Trace of variance covariance matrix of Rank Regression Estimators is TR RR, Asymptotically optimal group limits are y i s where i=1,, k-1.