TL- MOMENTS AND L-MOMENTS ESTIMATION FOR THE TRANSMUTED WEIBULL DISTRIBUTION

Transcription

1 Journal of Reliability and Statistical Studies; ISSN (Print): , (Online): Vol 0, Issue (07): 7-36 TL- MOMENTS AND L-MOMENTS ESTIMATION FOR THE TRANSMUTED WEIBULL DISTRIBUTION Ashok Kumar * and Rajiv Saksena, Department of Statistics, University of Lucknow, Lucknow E Mail: statsashok@gmailcom, saksenarajiv@gmailcom * Corresponding author Received February 09, 07 Modified October, 07 Accepted November 05, 07 Abstract Accurate estimation of parameters of a probability distribution is of massive importance in statistics Biased and vague estimation of parameters can lead to misleading results The Transmuted Weibull Distribution (TWD) has the advantage of bring capable of modeling various types of data, so the accurate estimation of the parameters of this distribution is required The main purpose of this paper is to develop the Trimmed Linear moments (TLmoments) of the TWD and use the TL-moments to estimate unknown parameters of the TWD An special case, linear moments (L-moments) will be obtained and used to estimate the unknown parameters of the TWD Monte Carlo Simulation technique is used to compare the L-moments and TL-moments of TWD Key Words: Transmuted Weibull Distribution, Order Statistics, L-moments, TL-moments, Monte Carlo simulation Introduction The method of L-moments was introduced Hosking (990), as a linear combination of probability weighted moments L-moments can also be applied in parameter estimation and hypothesis testing of parameter values of the theoretical probability distributions L-moments can be defined for any random variable whose mean exists The main lead of L-moments over conventional moments ie, product moments is that they can be estimated by linear functions of sample values and are more resistant to the influence of sample variability Furthermore, L-moments are less insightful in case of outliers in the data (Vogel and Fennessey (993)) Sometimes L- moments bring even more competent parameter estimates for the parametric distribution as compared to those estimated by method of maximum likelihood estimation for small samples (Hosking (990), Tomer and Kumar (04)) The method of L-moment estimators have recently appeared L-moment estimators for Log-Normal, Gamma and Generalized Extreme value distributions have been derived Kundu and Raqab (005) gave L-moment estimators for generalized Rayleigh distribution Tomer and Kumar (04) compared the L-moment estimators with other estimators for the Transmuted Exponentiated Lomax distribution A substitute version of L-moments was introduced by Elamir and Seheult (003) This version is called Trimmed Linear moments and it is termed as TLmoments In the TL-moments, a predetermined percentage of data is trimmed by

2 8 Journal of Reliability and Statistical Studies, December 07, Vol 0() assigning zero weight before estimating the moments The main advantage of TLmoments outweighs those of usual L-moments and central moments Even if the L moments or central moments of a probability distribution do not exist, TL moments may be derived Elamir and Seheult (003) derived TL-moment for Cauchy distribution, as its mean does not exist Sample TL-moments are more defiant to outliers in any data TL-moments are also used to obtain the most fitted distribution and to estimate the unknown parameters of the probability distributions TL-moments are frequently applied in modeling of different types of data sets particularly, in hydrologic studies and especially in modeling of flood frequency Since TL-moments are recently developed, they have not yet received significant attention The parameters of exponential and Weibull distribution were estimated by Abdul-Monien (007, 009) using the methods of L-moment and TL-moment estimation Abdul-Monien and Selim (009) applied TL-moments and L-moments estimation for estimating the parameters of Generalized Pareto distribution Bilkova (04) showed that the L-moments and TLmoments are substitute methods for statistical data analysis Parameter estimation of power function distribution with TL-moments was introduced by Neveed- Shahzad et al (05) In this article we obtain the L-moments and TL-moments estimators for the TWD The rest of the study is organized as follows: In Section, we introduce the population and sample L-moments and TL-moments Section 3 is about the Transmuted Weibull distribution The first four L-moments and TL-moments are derived in Section 4 To compare the properties of the L-moments and TL-moments for TWD, we have established a Monte Carlo simulation study in section 5 Finally, in Section 6, we have concluded our study Generalized TL-Moments Let,,, be a continuous random sample of size n that has a distribution with distribution function ( ), density function( )and quantile function () Further, let :, :,, : denote the corresponding order statistic The r th generalized TL-moments with smallest and largest trimming was given by Elamir and Seheult (003) as follows (, ) = ( ) E( :) where,, = 0,,, ;! =,, 3, The expression of the expected value of the (! # + ) th order statistics of the random sample of size (! + + ), ie, E( : ) is given by ()! E( : ) = & ()!()! () [ ( )] ) ()! = & ( )( ()!()! ) [ ( )] ) () where, () is the quantile function of random variable X with distribution function ( ) The generalized TL-moment ratios such as coefficient of variation, coefficient of skewness and coefficient of kurtosis computed from the first four generalized TL- ()

3 TL- moments and L-moments estimation for 9 (-,-,) moments, these ratios are defined as * = + (-,-,), (-,-,), * = + 0 (-,-,) and * = + + +, respectively (-,-,) (-,-,), +, L-moments L-moments were introduced by Hosking (990) He derived these moments for well-known distributions He proved that L-moments provide superior fit, parameter estimation, hypothesis testing and empirical description of the data He also proved many hypothetical advantages of L-moments above conservative moments Due to the advantage of L-moments over conventional moments, many distributions are analyzed by L-moments L-moments can be defined for any distribution variable whose mean exists L-moments are linear functions of the expected values of certain linear combination of order statistics Furthermore, these moments are less insightful in the case of outliers in the data (Vogel and Fennessey (993)) Hosking (990) defined r th population L-moment ( ) as follows = ( ) E( :) ;! =,,3, (3) Substituting! = and! = in (3), we get first and second L-moments and these moments provide the measures of location and dispersion, respectively The ratio of the third L-moment ( ) to second L-moment ( ) ie * + 45 = + 0 is the measure of +, skewness and ratio of the fourth L-moment ( ) to the second L-moment ( ) ie * + 4 = + the measure of kurtosis Note that the r th population L-moment can also be +, obtained from () by putting = = 0 The sample L-moments can be obtained from the formula given by Elamir and Seheult (003) as 9: 6 = ()8 9: <: 8 ; 9:8: =;>:< 8 = 8?@ A A: ;! =,,3, (4) > 9 Similarly L-skewness and L-kurtosis for the sample are defined as the ratio of the third L-moment (6 ) to the second L-moment (6 ) and the ratio of the fourth L- moment (6 ) to the second L-moment (6 ) of the sample That is, the sample B#CDECBB = F 0 and sample #G!HBIB = F, respectively These ratios so obtained F, F, are less biased as compared to the conventional moments in estimation TL-Moments An extension of L-moments called TL-moments was given by Elamir and Seheult (003) In TL-moments a predetermined percentage of data is trimmed by assigning zero weight before estimating the moments TL-moments for any distribution may exist regardless of the existence of the central moments of that distribution The TL-moments for Cauchy distribution were obtained by Elamir and Seheult (003), while the mean of this distribution did not exist These moments are also used in obtaining most fitted distribution and estimation unknown parameters for the distribution According to Elamir and Seheult (003), the r th population TL-moment can be obtained from () when = =

4 30 Journal of Reliability and Statistical Studies, December 07, Vol 0() () = ( ) E( :) The r th sample TL-moment is as follows 9: ;! =,,3, (5) 6 () = () 8 9: 8 ; <: 9:8J-: =;>:< -J8 = 8?@ A A: ;! =,,3, (6) 9J,- > The TL-skewness and TL-kurtosis of a sample are given as * () 45 = F 0 respectively (-) F, (-) and * 4 () = F (-) (-), F, 3 The Transmuted Weibull Distribution W Weibull, a Swedish physicist, established Weibull distribution in 939 to analyze the breaking strength of materials Since then, it has been widely used in reliability theory for studying the life time data Aryal and Tsokos (0) used quadratic rank transmutation map approach, recommended by Shaw and Buckley (007), to present a new generalization of Weibull distribution This new generalized distribution is termed as Transmuted Weibull distribution The cumulative distribution function (cdf) of random variable X having transmuted probability distribution is given by ( ) = ( + K)L( ) KL( ) ; K (7) where, L( ) is the cdf of the base distribution A random variable X follows transmuted Weibull distribution with parameters N > 0, P > 0 QE) K if its cdf is given by (, N, P, K) = R C ;S T =U V R + K C ;S T =U V (8) The corresponding probability density function (pdf) of X is given by (, N, P, K) = W X ;Y X =W C ;S T =U R K + KC ;S T =U V (9) 4 Generalized TL-moments for the TWD In this segment, the population L-moments and TL-moments for the TWD are discussed The sample L-moments and TL-moments are also discussed First, we obtain the generalized TL-moments for the TWD In the subsections 4 and 4, we have derived L-moments and TL-moments respectively Using formula given in () and the cdf of X given in (8) and the pdf of X given in (9), the r th generalized TL-moment with smallest and largest trimming for the TWD is given by (,) = X Z; U = ( ) \ ( ) ] ] K) ]A ^ ()! ()!()! ] ; A + (]A) U J (]A) U = ] K A ( ] A J` (0) 4 L-moments of TWD The population L-moment is a special case of generalized TL-moment of order r for the TWD by putting = = 0 in (0) The r th population L-moment for the TWD is

5 TL- moments and L-moments estimation for 3 = = X Z; U = ( ) E( :) ( ) K A ( K) ]A ^! \ ()!! ] ( )] + (]A) U J (]A) U ] ; = ] ] A A J` () The first four population L-moments for the TWD are obtained by putting! =,, 3, 4 in () as = P b ; + = ^( K) + ` W U = Pb ; W + = ^( K) (, ) U (), U U = X Z; U = ^( K) + (5K K ) + U J (7K 38K + 7) + U J (7K 8K + 7) + f() U J (4 3K) +, U J g ` JV hu = X Z; U = ^( K) (6K 3K + 6) + () (5K 0K 5) +, hu U (5( K) 4K( K) 6K ) + () U J (5K ) gu (3K 7K + 3) 0 () g V, respectively First four sample L-moments can be obtained from (4) 6 = A: 6 = = fu ju U A (A) A () A: A ()() A: (A)(A)(A)l(A)(A)(A)l(A)(A)(A) l(a)(a)(a) A ()()() A: 6 = (A)(A)(A)(A)(A)(A) 6 = 4 TL-Moments of TWD TL-moments are the extension of L-moments TL-moments are more dynamic than L-moments as they truncate the outliers present in the data (Elamir and Seheult (003)) The population TL-moments of order r for the TWD as a special case of generalized TL-moments from (0) by taking = = and are given by () () = XZ; U = ( ) \ ( ) ] ] ^ ()! ()!()! ] ; A + (]A) U J (]A) U = ] K A ( K) ]A ] A J` ;,! =,,3, () Here, we take =, then the first four TL-moments for the TWD are obtained as

6 3 Journal of Reliability and Statistical Studies, December 07, Vol 0() () () = 6P b ; W + = ^(), () U J g, () 0 gu J JV hu () () = 6P b ; W + = ^(), () U J K) + K n + g() mk 7K + n + h, gu J () () = X Z; U = ^(), () U J K) + K n g h 9K + 4n h () jp lu J JV U () () = gx Z; U = ^(), () U J (K 5K + ) U J (K 5K + ) U J (K 3K + ) + m( U J K) U J h U J (5K 3K + 5) + U J m3k 7K + 3n + 6K( 0 () + fu J JV ju U J m4( K ) 0K( U J m4( K)g 40K( K) + 3K ( K) + K n m( K ) 36K( K) + 5K n gh,() U J (K 3K + ) h0 U J m4k U J f (3K 7K + 3) + m7( U J K) K( K) + U J K n g, () m3( K) g 4K( K) + 6K ( K)n + h gu J m( J hu K) h 5K( K) + K ( K) K n + mk( K)g U J 05K ( K) + 43K ( K)n + lu J j f U J m5k ( K) 60K ( K) + 7K n + 0 () (9K 5K + 9) + (5K K + 5) + hhp() U J U J First four sample TL-moments for the TWD are given by 6 () (A)(A) = 6 6 () = 6 6 () = A ()() A: (A)(A)(A)(A)(A)(A) A ()()() A: ()()()() A m(i )(I )(I 3)(E I) 3(I )(I )(E I)(E I ) + (I )(E I)(E I )(E I )n A: 6 () g = m(i )(I )(I 3)(I 4)(E I) ()()()()(g) A j + q JV U 6(I )(I )(I 3)(E I)(E I ) + 6(I )(I )(E I)(E I )(E I ) (I )(E I)(E I )(E I )(E I 3)n A: 5 Monte Carlo Simulation Study In this segment, a Monte Carlo simulation is performed for the comparison of estimators obtained by L-moment and TL-moment methods for TWD This comparison is based on measures of average estimators and mean square errors (MSEs) Here, we use R i386 3 Software for simulation study We have executed the experiments for different sample sizes (0, 30, 40, 60) as well as different values of the parameters The estimates are calculated from 5000 repeated samples In the estimation of η (shape parameter) and σ (scale parameter), we

7 TL- moments and L-moments estimation for 33 equate the sample moments to corresponding population moments for each case to obtain the average estimates and MSEs of η and σ of TWD The results of the average estimates and MSEs (in parenthesis) are listed in Table and 6 Results and Conclusion The results of the simulation study are very good in terms of average estimates and MSEs even in case of small sample sizes As the sample size n increases the average estimates of η and σ are get closer to the true values of the parameters η and σ and corresponding MSEs decrease L-moment estimates of η and σ are better than the TL-moment estimators of η and σ in respect of average estimates and MSEs for all combinations of sample sizes and different true values of the parameters We have observed that the MSE of η increases with the increase in true value of σ where as the MSEs of σ decreases with the increase in the true value of η L-Moment Estimates TL-Moment Estimates n η σ N Ps N Ps (0049) (00385) (006) (0059) (00430) (00839) (0063) (07) (0043) (05) (0064) (009) (00964) (0089) (0397) (0030) (00983) (00650) (0399) (0056) (000) (045) (0430) (0095) (007) (0057) (00366) (00357) (0069) (00586) (00364) (00804) (0077) (0047) (00374) (0440) (006) (0039) (0087) (005) (00584) (0054) (0083) (00343) (0099) (0048) (0049) (00573) (00607) (0095) (0093) (0095) (0090) (00785) (00836) (0056) (006) (0066) (006) (0080)

8 34 Journal of Reliability and Statistical Studies, December 07, Vol 0() (0046) 50 (0044) 5 (0045) 003 (005) 0077 (005) 0087 (005) 5076 (0075) 505 (0077) 53 (0080) 0889 (004) 309 (0048) 764 (00858) 0003 (007) 4996 (00300) 9968 (00506) (0095) 30 (00434) 765 (00757) 574 (00577) 57 (0059) 574 (0059) 03 (0059) 0093 (0063) 0080 (0060) 58 (00373) 556 (00374) 566 (00379) (009) 4960 (005) 9967 (00454) (0077) 4947 (0043) 996 (00694) (00078) 4975 (0073) 9988 (00303) Table : Estimates and MSEs for η and σ when t = u v n η σ L-Moment Estimates TL-Moment Estimates N Ps N Ps (004) (0050) (00594) (0066) (0043) (068) (0060) (0460) (00435) (0088) (006) (053) (00907) (003) (040) (00300) (0095) (0057) (040) (00665) (00933) (0097) (0445) (043) (005) (00346) (00344) (004) (0057) (00778) (00344) (00930) (0067) (038) (00363) (06) (00559) (0056) (00779) (0098) (00555) (00360) (00787) (00446) (00559) (00643) (0080) (00793)

9 TL- moments and L-moments estimation for (0089) 00 (0089) 030 (0090) 599 (0047) 574 (0047) 586 (004) 0067 (00) 0089 (007) 0068 (006) 57 (0064) 5 (0065) 53 (0065) (0048) 4985 (00588) 9949 (0038) (007) 4959 (0060) 0006 (0046) (0074) 498 (00399) 9893 (00695) (00079) 4973 (0073) 0000 (003) 050 (004) 030 (0044) 046 (0046) 583 (00567) 57 (00559) 57 (0056) 0074 (0056) 008 (0059) 0073 (006) 556 (00350) 575 (0035) 584 (00363) (00300) 5065 (00707) 0077 (068) (0044) 5008 (0038) 0089 (0059) (0004) 5058 (00467) 9980 (00809) 004 (00097) 4993 (00) 0033 (0038) Table : Estimates and MSEs for η and σ when t = w References Abdul-Moniem, I B (007) L-moments and TL-moments estimation for the exponential distribution, Far East J Theo Stat, 3() p 5-6 Abdul-Moniem, I B (009) TL-moments and L-moments estimation for the Weibull distribution, Advances and Applications in Statistics, 5(), p Abdul-Moniem, I B and Selim, Y M (009) TL-moments and L-moments estimation for the generalized Pareto distribution, Applied Mathematical Sciences, 3(), p Aryal, G R and Tsokos, C P (0) Transmuted Weibull distribution: a generalization of the Weibull probability distribution, European Journal of Pure and Applied Mathematics, 4(), p Bilkova, D (04) Alternative means of statistical data analysis: L-moments and TL-moments of probability distributions, Statistika, 94(), p Elamir, E A, Seheult, A H (003) Trimmed L-moments, Computational Statistics & Data Analysis, 43, p Hosking, J (990) L-moments: Analysis and estimation of distributions using linear combinations of order statistics, Journal of Royal Statistical Society, B 5(), p 05-4

10 36 Journal of Reliability and Statistical Studies, December 07, Vol 0() 8 Kundu, D and Raqab, M Z (005) Generalized Pareto distribution: different methods of estimations, Computational Statistics & Data Analysis, 49, p R Core Team (05) R: A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, Austria URL 0 Shaw, W and Buckley, I (007) The alchemy of probability distributions: beyond Gram-Charlier expansions and a skew-kurtotic-normal distribution from a rank transmuted map, Research report Shazad, N M, Asghar, S, F and Shazadi, M (05) Parameter estimation of power function distribution with TL-moments, Revista Colombiana de Estadistica, 38(), p Tomer, S and Kumar, A (04) Traditional moments and L-moments estimation for the transmuted Exponentiated Lomax distribution, Anushandhan Anveshika, 4, p Vogel, R M and Fennessey, N M (993) L-moments diagrams should replace product moments diagrams, Water Resources Research, 9, p