Some Theoretical Properties and Parameter Estimation for the Two-Sided Length Biased Inverse Gaussian Distribution

Transcription

1 Journal of Probability and Statistical Science 14(), 11-4, Aug 016 Some Theoretical Properties and Parameter Estimation for the Two-Sided Length Biased Inverse Gaussian Distribution Teerawat Simmachan Kamon Budsaba Andrei I Volodin Thammasat University University of Regina ABSTRACT The new lifetime distribution based on non-classical parametrization model called the two-sided length biased inverse Gaussian distribution is introduced The physical phenomena of this situation can be explained in the case when a crack develops from two sides Some statistical properties of the distribution such as reciprocal properties and the first four moments are investigated The conventional point estimation, method of moment, is developed for estimating the parameters of the distribution together with asymptotic property of the proposed estimators In order to evaluate the performance of the suggested estimators, Monte Carlo simulation studies are conducted Additionally, real data sets in a practical setting are used to illustrate the presented estimation method Concluding remarks and discussions are also provided Keywords Asymptotic property; Lifetime distribution; Method of moment estimate; Parametrization; Reciprocal property 1 Introduction Lifetime distributions are frequently studied in reliability aspects It is easy to consider a lifetime or failure time of physical objects such as coins, electric light bulbs, some pieces of machines, etc They provide useful information on certain practical problems Since some machines or systems are very important and extremely expensive, this information motivates practitioners to prevent financial or industrial damages occurring after the failure time terminates One of the interesting views of lifetime distributions in reliability analysis is in the situation when a failure of the object under consideration occurs from a fatigue crack development The common distributions used in practical applications of this area are Birnbaum-Saunders, inverse Gaussian, and length biased inverse Gaussian [7, 13] Received August 015, revised March 016, in final form April 016 Teerawat Simmachan ( moisture_dew@outlookcom) and Kamon Budsaba are affiliated to the Department of Mathematics and Statistics at Thammasat University, Rangsit Center, Pathum Thani 111, Thailand Andrei Volodin ( andrei@ureginaca) is affiliated to the Department of Mathematical and Statistics at University of Regina, Regina, Saskatchewan, Canada 016 Susan Rivers Cultural Institute, Hsinchu City, Taiwan, Republic of China JPSS: ISSN

2 1 JPSS Vol 14 No August 016 pp 11-4 These distributions had been studied in various cases Birnbaum and Saunders [] introduced the two-parameter Birnbaum-Saunders (BS) distribution as a lifetime distribution for fatigue failure caused by periodic loading Ahmed et al [1] proposed the new parametrization of BS distribution Importantly, the physical situation under this study are fitted by this reparametrization since the suggested parameters correspond to the thickness of the object under study and the nominal treatment loading on the object, respectively The original parameters of the distribution do not give these characteristics Several studies regarding on the inverse Gaussian (IG) distribution are often referred to Chhikara and Folks [4], Seshadri [15, 16], Johnson et al [10] and Tweedi [17, 18] Recently, Lisawadi [13] presented two new distributions based on the re-parametrization model proposed in [1] and they were called the two-sided Birbaum-Saunders and inverse Gaussian lifetime distributions These distributions are considered in the situation of a crack develops from two sides A review of applications of length biased distributions was given in Gupta and Kirmani [9] Akman and Gupta [] had studied the length biased inverse Gaussian (LBIG) distribution They provided comparative simulation studies of different estimators for the mean of data from IG and LBIG distributions Gupta and Akman [8] offered statistical properties involving the arithmetic and harmonic means of the LBIG distribution Essentially, as seen in the reviewed literature, all of them were considered in the term of usual parameters except the studies of [1] and [13] In this article, we introduce a new lifetime distribution based on non-classical parameters presented by Ahmed et al [1] The new distribution is called the two-sided length biased inverse Gaussian lifetime distribution denoted as TS-LBIG distribution Interestingly, our contribution is in the investiga- tion process The probability model of the TS-LBIG distribution is formed by applying the approach of Lisawadi [13] The new distribution is considered in the case when a crack develops from two sides For example, on a metallic rectangular object which is fixed on two sides, a pressure is applied to both upper and lower sides of the object that leads to a crack development from two sides Some statistical properties of the distribution such as reciprocal properties and the first four moments are investigated The traditional point estimation, method of moment, is developed together with the asymptotic analysis of the proposed estimators Monte Carlo simulations are utilized to study the efficiency of the suggested estimators, and illustrative examples for explaining the given estimation method are also provided The article is organized as follows The pdf of IG and LBIG distributions based on nonclassical parameters are provided in Section Probability model of the TS-LBIG is introduced in Section 3 Theoretical results are given in Section 4 Numerical results are shown in Section 5 Finally, conclusions and discussion are reported in Section 6 Materials and Methods 1 Inverse Gaussian Distribution The usual pdf of inverse Gaussian distribution of a continuous random variable X is 3 ( x ) f( x;, ) x exp ; x0, x (1)

3 Some Theoretical Properties and Parameter Estimation for the Two-Sided T Simmachan et al 13 where 0 and 0 The parameter stands for the mean and represents the scale parameter The proposed parameters provided by Ahmed et al [1] are 0 and 0 standing for the thickness of the object under consideration and the nominal treatment loading on the object, correspondingly The relations between the classical and proposed parameters are:,,, and () Hence, the pdf of IG distribution based on the new parametrization is define as 3 1 x f x;, exp ; x0, x x where 0 and 0 (3) Length Biased Inverse Gaussian Distribution We start with the definition of a length biased pdf presented by Khattree [11] Definition 1 Let X be a non-negative random variable having an absolutely continuous pdf, f ( x ) Assuming EX ( ), the length biased random variable Y has a pdf defined as xf ( x) g() ; x 0 (4) It is known that, and by (3) and (4), finally, pdf of the length biased inverse Gaussian (LBIG) distribution is define by where 0 and x x x f x;, exp ; x0, (5) 3 Probability Model The physical phenomena of this situation can be explained in the case when a crack develops from two sides of the object under consideration Let F(), t t 0, be the distribution function of the moment of the object breakdown for one-sided loading We consider Ft () FLBIG ; t, Let Y k/ be the random variable interpreted as a speed of the crack evolution At the bottom side of the metallic block, a crack is developing with the distribution function of the time to reach the length k Simultaneously, at the top side of the block, a crack is developing with the same distribution function as the bottom side Then, we have two random variables 1 and, and they are assumed to be independent and identically distributed The 1 1 speed of the crack development for this two-sided case is Y1Y k1 k, and the random variable k /( Y1Y) ( 1 ) corresponds to a moment of the object under consideration break down The distribution function and pdf of the two-sided length biased inverse Gaussian distribution are presented in the following theorems

4 14 JPSS Vol 14 No August 016 pp 11-4 Theorem 31 A random variable has a two-sided length biased inverse Gaussian distribution denoted as TS-LBIG (, ), if it has a distribution function in the form: and a density function in the following form: u 1 u 1 t 1dt 1ds F ( u) 1 f f, t t s s u 1 1 dt 1 1 t u t t ( u t) 0 f ( u) u f f Proof Let be a random variable defined above, T 1/ 1 and S 1/ Then Similarly, Thus, F () t P( T t) P( 1/ t) 1 F (1/ t) and T 1 F () s 1 F (1/ s) S 1 and f () s f(1/ s)/ s s f () t F () t f(1/ t)/ t T T 1 1 1dtds F ( u) PT S u f ( ) ( ) 1 t s u T t fs s dtds 1 f f 1 tsu t s t s 1 1 u 1dt u t 1ds 1 f f 0 0 tt s s The density function is obtained by differentiating the F ( u) Therefore, 1 u 1 1 dt 1 1 t u t t ( u t) 0 f ( u) u f f Finally, we achieve the required expressions which complete the proof However, the pdf of has no explicit form This may be difficult to find main functions such as a characteristic function and a moment generating function Figures 1- show variety of Figure 1 The two-sided length biased inverse Gaussian density functions for Figure The two-sided length biased inverse Gaussian density functions for 1

5 Some Theoretical Properties and Parameter Estimation for the Two-Sided T Simmachan et al 15 the probability density functions for TS-LBIG It is indicated that the TS-LBIG is the positively skewed distribution which is a useful choice in this framework 4 Theoretical Results 41 Reciprocal Properties Proposition 411 If random variable 0 has the density function f ( x), then the reciprocal random variable 1/ has the pdf f ( x) x f (1/ x) 1 Proof For the reciprocal random variable 1/, applying a definition of a cumulative distribution function, then its distribution function is defined by F1 ( x) P x P 1 F, x x and applying the chain rule, the density function is f x F x x f x 1( ) 1( ) (1/ ) Proposition 41 If random variable 0 has LBIG (, ) random variable 1/ is IG [,1/( )] distributed Proof By Proposition 411, x 1 f1 ( x) x f LBIG ;, exp ( x) ( x) x (1 / ) x 1 3 exp 1 1/( ) x x 1/( ) distribution, then the reciprocal 1 1 IG[ x;,1/( )] 4 The First Four Moments Finding the first four moments of the TS-LBIG distribution, we deal with the following way The reciprocal property is necessarily needed The characteristic function of the IG distribution is required Because of the difficulty of direct derivation, Maclaurin expansion is applied The first four terms are considered to obtain the first four cumulants Then, they are modified to gain the first four moments The required theorems are presented as follows Theorem 41 If a continuous random variable X is inverse Gaussian distributed with parameters and denoted as IG (, ), then its characteristic function is 1 ( x ;, IG ) exp{ [1 (1 it ) ]} Proof By definition, the characteristic function of the IG(, ) IG ( x;, ) exp( itx) fig ( x;, ) dx is defined as 3 x exp( itx) x exp 0 x dx

6 16 JPSS Vol 14 No August 016 pp it e x exp x dx 0 x By applying the formula 3475 from Gradsthteyn and Ryzhik [6], (p 369), we have then we get 1 i t p and q, 1 i t IG x;, e exp 1 e exp 1i t exp [1 (1 it) ] Theorem 4 If a random variable X is two-sided length biased inverse Gaussian distributed with parameters and denoted as TS-LBIG (, ), then the first four moments are 6 ( X) k1( X), ( X) k( X), 3 3( X) k3( X), 5 3 and ( X) k4( X) 3 ( X) 7 4 Proof If () t is a characteristic function, then its cumulants kj; j 1,,, m, are defined from the Maclaurin expansion of the logarithm of the characteristic function: m k j j m ln ( t) ( it) o( t ) j! Hence, the logarithm of the characteristic function of IG(, ) j1 ln ( t) (1 ) distribution is 1 i t (6) For the Maclaurin expansion, we use (1 x) 1 x x x x O( x ), (7) where x < 1 In this case, x i t and we assume that 0 t 1/( ) Applying equations (6) and (7), we obtain 3 4 ( it) 4( it) 8( it) 5 16( it) 5 t O t ln ( ) 1 ( ) ( it) ( it) ( it) ( it) Ot ( ) ( it) ( it) ( it) 3 ( it) Ot 1!! 3! 4! This expression gives us the cumulants of the IG distribution: k k 1,, k and , k

7 Some Theoretical Properties and Parameter Estimation for the Two-Sided T Simmachan et al 17 By proposition 41, if LBIG(, ), then the reciprocal 1/ by 1/( ), 1/ has the following cumulants: IG[,1/ ( )] Substituting 1 1 k1(1 / ), k(1 / ), k (1 / ), and k (1 / ) Because of the iid property, the moments for the random variable X 1 will be obtained by combining the cumulants for 1/ Finally, the first four moments for the two-sided length biased inverse Gaussian distribution are: and ( X) k1( X), ( X) k( X), 3 6 ( X) k ( X), ( X) k4( X) 3 ( X) Method of Moment Estimation Estimation of the parameters and by the method of moment for the TS-LBIG distribution can be derived in the following way In the previous section, the first four moments are investigated, and thus we obtain formulas for the expectation and variance; Hence, the method of moment estimation is n 1 X Xi and n ( X) ( X) and Var( X ) ( X ) 3 i1 1 S X X n ( i ) (8) & (9) 3 n 1 i1 Solving equations (8) and (9) for and, we obtain the desired estimates Let Then, 1 T 3 4 Therefore, the method of moment estimators are ˆ 1 and T ˆ ˆX The asymptotic variances and covariance are presented in the theorem below Theorem 431 As n, Var( ˆ ), Var( ˆ ), and Cov( ˆ, ˆ ) are given by 5 ˆ 76 1 Var( ) O, n n 4 4 ˆ Var( ) O, and n n 4 3 ˆ ˆ Cov(, ) O n n T S / X

8 18 JPSS Vol 14 No August 016 pp 11-4 Proof The asymptotic distribution of an estimate which smoothly depends on sample moments is commonly obtained by their decomposition into the Taylor series expansion For our case, ˆ( X, S ), ˆ( X, S ), ( X ) is the true value of X, and ( X ) is that of S Let a 1 and a be the values of partial derivatives of ˆ by X and S at the point ( ( X ), ( X )), correspondingly The b 1 and b are denoted as the values of analogous derivatives of ˆ Therefore, the Taylor expansion can be written as The X n Op(1 / n), a constant C 0 such that where ˆ a X ( X ) a S ( X ) O (1/ n ), 1 ˆ b X ( X ) b S ( X ) O (1/ n ) 1 X, 1 n n is a sequence of random variables, means that there exists lim sup PXn( C/ n) t 0 x n 1 These expressions are understood in terms of convergence in probability Thus, the vector ( n( ˆ ), n( ˆ ) ) has limit in distribution, the two-dimensional normal distribution with zero means, variances ˆ Var( ) a Var( X ) a Var( S ) a a Cov( X, S ), (10) ˆ 1 1 Var bvar 1 ( X) bvar ( S ) bbcov 1 ( X, S ), and (11) Cov( ˆ, ˆ) a bvar( X ) a b Var( S ) a b a b Cov( X, S ) (1) As presented results in section 4, we apply the formulas outlined in Cramér [5], (p ) and we obtain: Var( X ), Var( S ) O, n n n and (13) 6 1 Cov( X, S ) O 5 3 n n We firstly consider the derivatives of the estimate ˆ and we have p p ˆ d ˆ T 1 T X dt X T X Exchanging X and S on and, we get ˆ 4 a1 X Similarly, based on the appropriate substitution, we obtain a ˆ 4 S 4 We secondly provide the analogous derivations of the derivatives of the ˆ and we have

9 Some Theoretical Properties and Parameter Estimation for the Two-Sided T Simmachan et al 19 ˆ d ˆ ˆ ˆ ˆ ˆ 1 d d X ( 1) X ˆ ˆ ˆ ˆ ˆ X d X X X X X Note that Equivalently, ˆ (, ) a1 X, and hence ˆ 3 b1 (, ) X b 3 3 ˆ (, ) S Then, we substitute a1, a, b1, b and (13) into equations (10)-(1) to obtain Cov( ˆ, ˆ ) Hence, 4 Var( ˆ ), ˆ 301 Var( ) ( ) n n 6 ( ) O n n O O n n n n n n Thus, ˆ Var( ) n n O n n O O n n n n n n Therefore, ˆ ˆ Cov(, ) ( ) n n ( ) O 5 3 n n O O n n n n n n Finally, the proof is complete 5 Numerical Results 51 Simulation Study Var( ˆ ), and We studied the properties of the presented estimators by using the numerical method The results were reported to investigate the behavior of the estimators via calculating the estimated

10 0 JPSS Vol 14 No August 016 pp 11-4 bias and mean square error Monte Carlo simulations were performed for different sample sizes The R program version 31 was used to generate and analyze the data The number of iterations was fixed at 5,000 for each combination of, and sample sizes n We considered all combinations of the following values of,, and n as follows: =, 5, 10 and 50, = 1, 5, 10 and 50, n = 10, 50 and 100 Tables 1-3 show the estimated bias and MSE of ˆ and ˆ for n = 10, 50 and 100, respectively Table 1 The estimated bias and mean square error of ˆ and ˆ for n = 10 ˆ ˆ ˆ ˆ MSE( ˆ ) MSE( ˆ ) Table The estimated bias and mean square error of ˆ and ˆ for n = 50 ˆ ˆ ˆ ˆ MSE( ˆ ) MSE( ˆ )

11 Some Theoretical Properties and Parameter Estimation for the Two-Sided T Simmachan et al 1 Table 3 The estimated bias and mean square error of ˆ and ˆ for n = 100 ˆ ˆ ˆ ˆ MSE( ˆ ) MSE( ˆ ) As seen in Tables 1-3, importantly, the simulated bias for ˆ has a negative bias for all situations, so it is an overestimate In contrast, that for ˆ are systematically positive, then it is an underestimate When sample sizes are small, the amount of bias is quite large particularly when the true value of at least one parameter is sufficient big Furthermore, the bigger parameters, the larger are the bias For instance, at n = 100, and 5, the bias of ˆ is 00869, while for 50 and 50, the bias of ˆ is However, the magnitude of the bias may be assumed to be relatively small Tables 1-3 reveal that the more increasing values of the parameters, the more growing are the mean square error As the results from the numerical study, it is observed that ˆ and ˆ are consistent Thus, they are asymptotically unbiased estimators The simulated bias corresponds to the theoretical background as it is a decreasing function of sample sizes n That is, when sample sizes increase, the amount of the bias decreases and tends to zero as n Similarly, the MSE is a decreasing function of n The larger sample size, the smaller is the MSE, and it approaches to zero as n 5 Illustrative Examples The practical applications of the suggested estimators are illustrated in this section Two real data sets are considered as the followings Example 1 The following data set was provided by Lieblein and Zelen [1] on the fatigue life of the 3 deep groove ball bearings: The above data had been analyzed by Gupta and Akman [8] and the result indicated that the data set comes from the length biased inverse Gaussian distribution For two-sided case, we divide all

12 JPSS Vol 14 No August 016 pp 11-4 data in pairs, we have only 11 pairs of observations, and the last one is dropped We obtain 11 observations: ( y1, y),( y3, y4),,( y1, y) Let u, u,, u11, y y 1 y1 y y3 y4 1 be the observations drawn from the TS-LBIG distribution In this example, the point estimates are reported in Table 4 Moreover, using the relationship given in Section 1, the point estimates for parameters and are also computed Example This example was taken from Nichols and Padgett [14] consisting of 100 observations on breaking stress of carbon fibers (in GPa) These data had been analyzed by assuming the Weibull distribution In our case, the data was ascendingly ordered Dealing with the analogous manner of the example 1, finally, we obtain 50 observations drawn from the TS-LBIG distribution The point estimates for the proposed and original parameters are shown in Table 5 Most importantly, the estimators ˆ and ˆ represent the thickness of the object under study and the nominal treatment pressure on the object, correspondingly On the other hand, the estimators ˆ and ˆ lack this physical explanation Table 4 Point estimates for Example 1 Table 5 Point estimates for Example Proposed Parameters Usual Parameters Proposed Parameters Usual Parameters Conclusions and Discussion The new lifetime distribution based on re-parametrization model called the two-sided length biased inverse Gaussian distribution is proposed The reciprocal properties and the first four moments of the distribution are investigated The conventional point estimation, method of moment, is developed to estimate the parameters of the distribution and the asymptotic variances and covariance of the suggested estimators are also provided In this article, we discussed on some statistical properties of the distribution The results ensure us that the method of moment estimators works and provides consistent statistics Importantly, we estimate the parameters which reflect the physical nature of an object analyzed by a statistical view of the distribution Accordingly, one could continue investigations for other point estimation schemes such as maximum likelihood estimators, although we expect to encounter mathematical difficulties The density function of the TS-LBIG distribution involves an integral sign, and finding a maximum of their products even numerically is not easy work Furthermore, we may consider new other estimators, ie, a regression-quantile (least square) This method is dealt with the regression analysis of sample quantiles However, the method of

13 Some Theoretical Properties and Parameter Estimation for the Two-Sided T Simmachan et al 3 moment estimation can be examined as a preliminary topic of studying for the TS-LBIG distribution since it has a more satisfying property, eg simple computation Interval estimation and hypothesis testing issues remain to be interesting for further investigation As presented results of the asymptotic analysis, tests and confident interval estimation procedures will be explored regarding on a power of the test and coverage probabilities Nevertheless, they are above the scope of this article and the investigation will be separately communicated Acknowledgements The authors gratefully acknowledge the financial support provided by Thammasat University under the TU Research Scholar, Contact No 46/558 Special thanks are also extended to the referees for providing valuable recommendations to complete the manuscript References [1] Ahmed S E, Budsaba K, Lisawadi S, and Volodin A (008) Parametric estimation for the Birnbaum-Saunders lifetime distribution based on a new parametrization, Thailand Statistician, 6, [] Akman O and Gupta R C (199) A comparison of various estimators of the mean of an inverse Gaussian distribution, Journal of Statistical Computation and Simulation, 40(1-), [3] Birnbaum Z W and Saunders S C (1969) A new family of life distribution, Journal of Applied Probability, 6, [4] Chhikara, R S and Folks L (1989) The Inverse Gaussian Distribution: Theory, Methodology and Applications, Marcel Dekker, New York [5] Cramér H (1946) Mathematical Methods of Statistics, 6 th ed, Princeton University Press, Princeton [6] Gradsthteyn, I S and Ryzhik, I M (007) Table of Integrals, Series, and Products, 7 th ed, Elsevier Academic Press [7] Gupta, R C and Akman, O (1995) On the reliability studies of a weighted inverse Gaussian model, Journal of Statistical Planning and Inference, 48, [8] Gupta, R C and Akman, O (1998) Statistical inference based on the length biased data for the inverse Gaussian distribution, Journal of Theoretical and Applied Statistics, 31(4), [9] Gupta, R C and Kirmani, S (1990) The role of weighted distributions in stochastic modeling, Comunications in Statistics Theory and Methods, 19(9), [10] Johnson, N L, Kotz S, and Balakrishnan N (1994) Continuous Univariate Distributions: Volum 1, Wiley, New York [11] Khattree, R (1989) Characterization of inverse-gaussian and gamma distributions through their length-biased distributions, IEEE Transactions on Reliability, 38(5),

14 4 JPSS Vol 14 No August 016 pp 11-4 [1] Lieblein, J and Zelen, M (1956) Statistical investigation of the fatigue life of deep groove ball bearings, Journal of Research of the National Bureau of Standards, 57, [13] Lisawadi, S (009) Parameter Estimation for the Two-Sided BS and IG Distributions: and Laws of Large Numbers for Arrays under a Condition of Uniform Integrability, VDM Verlag, 009 [14] Nichols, M D and Padgett, W J (006) A bootstrap control chart for Weibull percentiles, Quality and Reliability Engineering International,, [15] Seshadri, V (1993) The Inverse Gaussian Distribution: A Case Study in Exponential Families, Claredon Press, New York [16] Seshadri, V (1999) The Inverse Gaussian Distribution: Statistical Theory and Applications Springer, New York [17] Tweedi, M C K (1957) Statistical properties of the inverse Gaussian distributions I, Annals of Mathematical Statistics, 8, [18] Tweedi, M C K (1957) Statistical properties of the inverse Gaussian distributions II, Annals of Mathematical Statistics, 8,