CONDITIONAL CONFIDENCE INTERVAL FOR THE SCALE PARAMETER OF A WEIBULL DISTRIBUTION. Smail Mahdi

Transcription

1

2 Serdia Math. J. 30 (2004), CONDITIONAL CONFIDENCE INTERVAL FOR THE SCALE PARAMETER OF A WEIBULL DISTRIBUTION Smail Mahdi Communiated by N. M. Yanev Abstrat. A two-sided onditional onfidene interval for the sale parameter θ of a Weibull distribution is onstruted. The onstrution follows the rejetion of a preliminary test for the null hypothesis: θ = θ 0 where θ 0 is a given value. The onfidene bounds are derived aording to the method set forth by Meeks and D Agostino (1983) and subsequently used by Arabatzis et al. (1989) in Gaussian models and more reently by Chiou and Han (1994, 1995) in exponential models. The derived onditional onfidene interval also suits non large samples sine it is based on the modified pivot statisti advoated in Bain and Engelhardt (1981, 1991). The average length and the overage probability of this onditional interval are ompared with whose of the orresponding optimal unonditional interval through simulations. The study has shown that both intervals are similar when the population sale parameter is far enough from θ 0. However, when θ is in the viinity of θ 0, the onditional interval outperforms the unonditional one in terms of length and also maintains a reasonably high overage probability. Our results agree with the findings of Chiou and Han and Arabatzis et al. whih ontrast with whose of Meeks and D Agostino stating that the unonditional interval is always shorter than the onditional one. Furthermore, we derived the likelihood ratio onfidene interval for θ and ompared numerially its performane with the two other interval estimators Mathematis Subjet Classifiation: 62F25, 62F03. Key words: Weibull distribution, rejetion of a preliminary hypothesis, onditional and unonditional interval estimator, likelihood ratio interval, overage probability, average length, simulation.

3 56 Smail Mahdi 1. Introdution. The estimation following rejetion of a preliminary hypothesis used, for instane, in Chiou and Han[5, 6], Meeks and D Agostino [14], and, Arabatzis et al. [1] is ertainly the losest inferene proedure to the one initiated in Banroft [4]. The distintion between these two proedures has been pointed out, among others, in Mahdi [9]. For a detailed aount on the use of preliminary test proedures, see, for instane, Jugde and Bok [8], Mahdi [10, 12], Rai and Srivastava [15], Giles et al. [7] and referene therein. In Chiou and Han [5, 6], the effet of the use of onditional interval estimation for the shape and sale parameters, following rejetion of a preliminary test, in a two-parameter exponential population has been investigated. The inferene is based on a type II ensored single sample. The authors have shown that the onditional onfidene interval is more aurate than the usual unonditional interval for some values of the parameter spae. On the other hand, the onditional interval estimation based on two-sample data from exponential populations has been reently investigated in Mahdi [9], and, the ase of normal populations has been treated in Mahdi and Gupta [11]. In this paper, we onsider the problem of interval estimation for the sale parameter θ of a Weibull distribution when it is suspeted that θ = θ 0 for some given θ 0. This interval is ompared in terms of length and overage probability to the optimal orresponding unonditional interval with same targeted onfidene level. Furthermore, we ompare these onfidene intervals to the likelihood ratio based onfidene interval reommended in Meeker and Esobar [13] for the shape parameter of the Weibull distribution. It is worth noting that the Weibull distribution was introdued in 1939 by a Swedish sientist on empirial ground in the statistial theory of the strength of materials. This law has proven, sine then, to be a suessful analytial model for many phenomena in reliability engineering, infant mortality and extreme value problems. We organize this paper as follows. In Setion 2, we state the onsidered problem and in Setion 3, we derive the bounds for the optimal unonditional interval for θ. The bounds for the onditional onfidene interval are derived in Setion 4. In Setion 5, we ompute the atual overage probability of the unonditional interval and in Setion 6 we present the likelihood ratio based onfidene interval for θ. Simulations results are disussed in Setion 7. Table 1 and Figure 1, illustrating the main simulation results, are displayed in Appendix. We finally onlude in Setion 8.

4 Conditional onfidene interval for the sale parameter Statement of the problem. Suppose that X 1,,X n onstitute a random sample from a Weibull distribution with shape parameter β and sale parameter θ. It is suspeted that θ θ 0 where θ 0 is a prefixed value. The null hypothesis H 0 : θ = θ 0 versus the alternative H 1 : θ > θ 0 is then tested and in the ase of rejetion a two sided onfidene interval for θ is thus onstruted. Otherwise, θ 0 is substituted for θ. To test H 0 and onstrut the onfidene interval bounds for θ we use the pivotal quantity derived in Bain and Engelhardt [2, 3], that is, (1) n 1 ˆβ ln ˆθ θ t n 1 where = 1.053, ˆβ is the maximum likelihood estimator of β and ˆθ is the maximum likelihood estimator of θ. The ritial region assoiated with the preliminary test of H 0 versus H 1 performed at the signifiane level α is given by R = { [ ]} tn 1 (α) ˆθ : ˆθ > θ0 exp n 1ˆβ where t n 1 (α) denote the quantile of order (1 α)100% of the Student variable with n 1 degrees of freedom. Remark 1. The value of the statisti ˆβ is independent of θ and its estimator ˆθ. by (2) P r o o f. The two-parameter Weibull probability density funtion is given [ ( x ) ] β g(x;θ,β) = βθ β x β 1 exp θ for x > 0, β > 0 and θ > 0. The orresponding log-likelihood funtion based on the observed random sample x 1,,x n is (3) LnL(θ,β) = n(ln(β) β ln(θ)) + (β 1) n ln(x i ) ( xi ) β. θ After some algebrai simplifiations, we find that the maximum likelihood esti-

5 58 Smail Mahdi mator of β is given by the solution β = ˆβ of the non linear equation ( n )( n ) 1 1 β = x β i ln(x i) x β i 1 n (4) ln(x i ). n The maximum likelihood estimator for θ is then obtained as (5) ˆθ = [ n 1 n ] 1ˆβ xˆβ i. We derive below the bounds of the optimal length unonditional onfidene interval for θ. For the benefit of using optimal onfidene intervals, see, for instane, Wardell [16]. 3. Optimal unonditional onfidene interval. The upper bound θ U and lower bound θ L of a 100(1 p)% unonditional onfidene interval, based on the pivot statisti (1) for θ are given by [ ] θ L = ˆθ tn 1 (1 p 1 ) (6) exp ˆβ n 1 and (7) [ ] θ U = ˆθ tn 1 (p 2 ) exp ˆβ n 1 for all p i, i = 1,2 satisfying 0 < p = p 1 + p 2 1. However, we an state the following result. Theorem 1. For any positive p 1 and p 2 suh that 0 < p = p 1 + p 2 < 1, the (1 p)100% unonditional onfidene interval for θ has optimal length when p 1 = p 2 = p. In suh a ase, the lower and upper onfidene bounds are, 2 respetively, given by [ θ L = ˆθ exp t ] n 1(p/2) (8) ˆβ ˆθ n 1 and (9) [ ] θ U = ˆθ tn 1 (p/2) exp ˆβ ˆθ. n 1

6 Conditional onfidene interval for the sale parameter (10) and (11) P r o o f. Consider the intervals [ ] [ ] I = ˆθ tn 1 (1 p/2) exp ˆβ < θ < ˆθ tn 1 (p/2) exp n 1 ˆβ n 1 [ ] [ ] I = ˆθ tn 1 (1 ((k 1)p)/k) exp ˆβ < θ < ˆθ tn 1 (p/k) exp n 1 ˆβ n 1 where k > 2. Both I and I are (1 p)100% onfidene intervals for θ. Let LI and LI denote the length of I and I, respetively. To prove that LI LI, it suffies to prove that (12) t n 1 (p/2) t n 1 (1 p/2) t n 1 (p/k) t n 1 (1 ((k 1)p)/k) whih is equivalent to prove the inequality, (13) t n 1 (1 ((k 1)p)/k)) t n 1 (1 p/2) t n 1 (p/k) t n 1 (p/2). Inequality (13) is true sine the intervals (t n 1 (1 p/2),t n 1 (1 ((k 1)p)/k)) and (k 2)p (t n 1 (p/2),t n 1 (p/k)) interept the same area under the even probability density funtion of T n 1 and that the urve of this probability density funtion 2k is higher above the interval (t n 1 (1 p/2),t n 1 (1 ((k 1)p)/k)). This proves then inequality (13) and therefore Theorem 1 in the ase k > 2. The proof in the ase 1 < k < 2 an be done in a similar way. Corollary 1. The two-sided optimal length unonditional (1 p)100% onfidene interval for θ has a smaller length than the usual one-sided unonditional interval of the form ( 0, ˆθ exp [ tn 1 (p) ˆβ n 1 ]). Proof. The intervals (0,t n 1 (1 p/2)) and (t n 1 (p),t n 1 (p/2)) have the same probability value but the length of the former is larger aording to the position of the points 0, t n 1 (1 p/2),t n 1 (p) and t n 1 (p/2) under the urve of the probability density funtion of the Student variable with n 1 degrees of freedom.

7 60 Smail Mahdi 4. Conditional onfidene interval. The bounds θl C and θc U of the (1 p)100% onditional onfidene interval for θ are omputed using the onditional sampling distribution of the statisti T = θ given rejetion of ˆθ n 1 ˆβ ln n 1ˆβ ln(ψ) H 0, that is, T > t n 1 (α) + where ψ = θ 0. The onditional umulative funtion of T, given rejetion of H 0, is F C (t) = P[T t T > t 1 ] where θ n 1ˆβ ln(ψ) t 1 = t n 1 (α) +. Thus, 0 if t < t 1 (14) F C (t) = F(t) F(t 1 ) if t t 1 1 F(t 1 ) where F is the umulative funtion of a Student variable with n 1 degrees of freedom. The orresponding probability density funtion is 0 if t < t 1 (15) f C (t) = f(t) if t t 1 1 F(t 1 ) where f denote the probability density funtion of a Student variable with n 1 degrees of freedom. The bounds of the onditional onfidene set are solutions of the system of inequations (16) ( ˆθ ) n 1ˆβ ln F θ C = ( ˆθ ) ( n 1ˆβ ln ) F θ n 1ˆβ ln(ψ) F t n 1 (α)+ ( ) n 1ˆβ ln(ψ) 1 F t n 1 (α) + p 1 (17) ( ˆθ ) n 1ˆβ ln F θ C = ( n 1ˆβ ln ˆθ F θ ( 1 F ) ( ) n 1ˆβ ln(ψ) F t n 1 (α)+ ) n 1ˆβ ln(ψ) t n 1 (α) +

8 Conditional onfidene interval for the sale parameter p 2 suh that p 1 + p 2 = p. Now from the monotony property of the funtion ξ(θ) = ˆθ n 1ˆβ ln F C ( θ ), the above system redues to simplified following system of equations (18) ( n 1ˆβ ln ˆθ ) θu 1 F ( ) = p 1 n 1ˆβ ln(ψ) 1 F t n 1 (α) + and (19) ( n 1ˆβ ln ˆθ ) θl 1 F ( ) = 1 p 2 n 1ˆβ ln(ψ) 1 F t n 1 (α) + whih gives the upper and lower onditional onfidene bounds. In the ase α = 1, that is, we always rejet H 0, the above system redues to the following system (20) ( n 1ˆβ ln F ( ˆθ θ U ) ) = p 2 and (21) ( n 1ˆβ ln( ˆθ θ )) L F = 1 p 2

9 62 Smail Mahdi whih also guarantees that the onditional onfidene set to be an interval sine (22) df ( n 1ˆβ ln dθ ( ) ˆθ ) θ = ˆβ n 1 f θ ( n 1ˆβ ln ( ) ˆθ ) θ for any value θ > 0. Note that the onditional onfidene bounds are the same as the unonditional onfidene bounds in this ase. Setting again p 1 = p 2 = p/2 in the above system of equations yields the bounds of the minimum length unonditional onfidene interval Coverage probability of the unonditional interval. The overage probability of the onditional interval is 1 p sine it is derived under this nominal level. However, the atual overage probability of the unonditional onfidene interval has to be omputed under the onditional probability funtion of the pivot statisti (1), given rejetion of H 0. This yields the following result. Theorem 2. The overage probability CP of the unonditional interval (θ L,θ U ) is given by CP = 0 if t n 1 (p/2) < t 1 ; CP = 1 p/2 F(t 1) if 1 F(t 1 ) t n 1 (p/2) < t 1 < t n 1 (p/2) and by CP = 1 p 1 F(t 1 ) if t 1 < t n 1 (p/2) < t n 1 (p/2) where t 1 is the previously defined quantity. (23) P r o o f. The overage probability is given by CP = f C (t)dt A where f C is defined in formula (15) and A = {t : { t n 1 (p/2) < t < t n 1 (p/2)} and {t > t 1 }}. Now, if t n 1 (p/2) < t 1, then A = Ø and therefore CP = 0. On the other hand, if t n 1 (p/2) < t 1 < t n 1 (p/2), then A = (t 1,t n 1 (p/2)) and the formula (23) gives CP = F(t n 1(p/2)) F(t 1 ) = 1 p/2 F(t 1). Finally, 1 F(t 1 ) 1 F(t 1 ) when t 1 < t n 1 (p/2) < t n 1 (p/2), then A = ( t n 1 (p/2),t n 1 (p/2)) and thus CP = F(t n 1(p/2)) F( t n 1 (p/2)) = 1 p 1 F(t 1 ) 1 F(t 1 ) 1 p.

10 Conditional onfidene interval for the sale parameter Remark 2. The overage probability CP may exeed the onfidene level 1 p when 1 F(t 1 ) < 1 and this ours over a signifiant region of the parameter spae. We derive below the likelihood ratio onfidene interval. 6. Likelihood ratio based onfidene interval. The likelihood funtion based on the two parameter Weibull distribution and the random sample x 1,...,x n is given by ] (24) Lw(θ,β) = β n θ nβ n x β 1 i exp [ n ( x i θ )β For a fixed value θ, the maximum likelihood estimator of β is given by the solution β of the profile likelihood gradient equation [ n (25) ln(lw(θ, β)) β = β whih has the following property. (26) ln( x [ i θ ) ( x ] ] i θ )β 1 n = 0. Theorem 3. The profile likelihood gradient equation [ n ( xi ) [ ( xi ) ] ] β β ln 1 n = 0. θ θ admits a unique solution. (27) where y i = x i θ (28) Proof. Consider the funtion n g(β) = ln(y β i )[yβ i 1] n for i = 1,,n. The derivative of the funtion g is given by g (β) = n ln(y i )[y β i [ln(yβ i ) + 1] 1] whih is greater or equal to zero as sum of non negative terms. Indeed, in the ase, 0 < y i < 1, ln(y i ) < 0 and y β i ln(yβ i ) + yβ i 1 0 sine yβ i ln(yβ i ) < 0 and.

11 64 Smail Mahdi y β i 1 0. Thus ln(y i)[y β i [ln(yβ i ) + 1] 1] 0 for 0 < y i < 1. Now in the ase y i > 1, we have that ln(y i ) > 0, y β i 1 and ln(y β i ) > 0. Thus ln(y i )[y β i [ln(yβ i ) + 1] 1] 0. Finally in the ase y i = 1, it is easy to see that ln(y i )[y β i [ln(yβ i ) + 1] 1] = 0. We onlude then that the ontinuous funtion g is non dereasing for β 0 provided that not all y i worth 1; whih ours almost surely. On the other hand, we have that lim g(β) = sine not all y i = 1 almost surely. Thus β lim β g(β) = as sum of almost sure infinite positive quantities. Futhermore, for β = 0, g(β) = n < 0. Consequently, the equation g(β) = 0 admits a unique positive root β by virtue of the intermediate property of ontinuous funtions. The profile likelihood ratio for θ is then given by (29) PLR(θ) = Lw(θ, β) Lw(ˆθ, ˆβ), where (ˆθ, ˆβ) denote the unrestrited maximum likelihood estimate of (θ,β). Thus (1 p)100% likelihood ratio onfidene interval for θ is then given by the solution set of the inequation [ ] χ 2 (1 p,1) (30) PLR(θ) > exp 2 where χ 2 (1 p,1) is the (1 p)100% perentile of the hi-squared distribution whith one degree of freedom. 7. Simulation results. To illustrate the performane of the onditional interval over the orresponding unonditional one and the likelihood ratio interval, we present simulation results obtained in the ases of n = 50, α =.05, 0.10, 0.20, 0.40, 0.80, 1.0, Ψ = 0.90(0.1)1, β = 5 and p =.10. Note that the sale parameter θ is often set to unity in real life appliations. We therefore took θ 0 = 1 without loss of generality. Probability plots, as illustrated in Figure1, have onfirmed the fit auray of the pivot (1) statisti by the Student variable

12 Conditional onfidene interval for the sale parameter with n 1 degrees of freedom for moderate and large sample sizes and for suffiiently large β in the ase of small samples. We have therefore hosen β = 5. The hosen values for α and ψ = 0.01(0.01)1 aount for small to large possible values. Now, under eah possible value of the parameter spae, we randomly seleted 1000 times random samples of size n and for eah sample we arried out the testing of H 0 and the omputation of the onditional, unonditional and likelihood ratio onfidene bounds in the ase of rejetion of H 0. The Newton-Raphson proedure using the starting point for the sale parameter advoated in Zanakis [18] has been suessfully used to solve equations (4) and (5). Furthermore, the bisetion method has been used for solving equations (18) and (19) in order to find the bounds of the onditional interval. The Newton-Raphson method has also been used to ompute the bounds of the likelihood ratio interval. The results have shown that for α < 1, the respetive ratios of the overage probability and length of the onditional interval and the likelihood ratio interval to the unonditional interval are just slightly smaller than unity, with equality, when α = 1. Therefore, these ratios are not signifiantly affeted by the parameter α. Furthermore, ψ values below.89 yielded similar lengths and overage probabilities for the onditional and unonditional onfidene intervals, see, Table1 in Appendix. The likelihood ratio onfidene interval has a slighly smaller average length but also a smaller overage probability with respet to the unonditional interval whih always maintain a overage probability lose to the nominal level. However, when.90 ψ 1.0, the length of the onditional interval i s signifiantly smaller than the length of the unonditional one. Furthermore, the overage probability of the onditional onfidene interval is reasonably lose to the nominal level when Ψ <.95 as illustrated in Table 1 displayed in Appendix. Thus the unonditional onfidene interval does not always outperform the orresponding onditional interval. Note that the ritial region R an also be expressed using the estimator ˆΨ = θ 0 ˆθ as R = { [ ˆΨ : ˆΨ < exp t ] } n 1(α) 1 n 1ˆβ where the bound Ψ = exp [ t n 1(α) n 1ˆβ ] inreases rapidly towards 1 as n beomes large. We have notied that the region over whih the onditional interval performs very well orresponds to the viinity of the point Ψ = Ψ in the ritial

13 66 Smail Mahdi region R. This agrees with the results of Chiou and Han [5, 6]. There is then indeed a gain in using the onditional interval based on the tehnique advoated in Meeks and D Agostino [14]. Contrarily to the finding of Meeks and D Agostino [14], the length of the obtained onditional onfidene interval never exeeds the length of the unonditional onfidene interval. On the other hand, the onditional interval also performs better than the likelihood ratio interval over the region.90 Ψ <.95. However, as ψ inreases from Ψ =.95, the overage probability of the onditional interval dereases signifiantly and the unonditional interval and the likelihood ratio interval beome very similar in terms of length and overage probability. We reommand then to just use the unonditional interval in suh a ase sine it is less omputational. Consequently, we reommand to always use the unonditional onfidene interval based on Bain and Engelhardt statisti expet when Ψ or its estimate ˆΨ is lose to the ritial point Ψ. 8. Conlusion. We have investigated here the onditional estimation by onfidene interval of the sale parameter θ in Weibull model. The ase of shape parameter will be treated in a separate paper. The onfidene interval is onstruted only after rejetion of a one-sided preliminary test of signifiane for the null hypothesis H 0 : θ = θ 0. Conditional onfidene bounds are obtained following the proedure set forth by Meeks and D Agostino [14]. This interval is ompared in terms of overage probability and average length to the optimal orresponding unonditional interval and to the likelihood ratio onfidene interval. The simulation study has also shown that the likelihood ratio interval, reommended in Meeker and Esobar for the estimation of the shape parameter, is not very appropriate for the estimation of the sale parameter. The overage probability of the unonditional onfidene interval, evaluated under the onditional distribution of the pivot statisti (1) is also given. It has been notied that as the values of ψ move away from the viinity of Ψ = Ψ both intervals beome similar in terms of average length and overage probability. However, in the neighborhood of ψ = Ψ, the study has shown that the length of the onditional onfidene interval is signifiantly smaller than the length of the unonditional one. Moreover, it has a reasonably high overage probability. It is then worth using the onditional interval in suh situations. The study has also shown that none of these two intervals outperforms ompletely the other over the whole parameter spae. Therefore, the always use of unonditional onfidene intervals

14 Conditional onfidene interval for the sale parameter independently of the preliminary test outome may lead to inaurate estimates. The numerial study has been arried out with Gauss, SPSS and Mathematia [17]. Aknowledgments. The finanial support of UWI is gratefully aknowledged. Appendix A Table Table 1. Empirial length and overage probability ratios of 90% onfidene intervals based on 1000 simulations. LCCI/LU CI and LLRCCI/LU CI represent, respetively, the average length ratios of the onditional onfidene interval and the likelihood ratio onfidene interval to the unonditional onfidene interval. CP CI/CP U CI and CP LLRCI/CP U CI represent, respetively, the average overage probability ratios of the onditional onfidene interval and the likelihood ratio onfidene interval to the unonditional onfidene interval. Ψ LCCI/LU CI CP CI/CP U CI LLRCCI/LU CI CP LLRCI/CP U CI

15 68 Smail Mahdi Appendix B Figure Fig. 1. Student t Q-Q Plot Pivot REFERENCES [1] A. A. Arabatzis, T. G. Gregoire, M. R. Reynolds. Conditional interval estimation of the mean following rejetion of a two sided test. Comm. Statist. Theory Methods 18 (1995), [2] L. J. Bain, M. Engelhardt. Simple approximate distributional results for onfidene and tolerane limits for the Weibull distribution based on maximum likelihood estimators. Tehnometris 23, 1 (1981),

16 Conditional onfidene interval for the sale parameter [3] L. J. Bain, M. Engelhardt. Statistial Analysis of Reliability and Life- Testing Models. Deker, New York, [4] T. A. Banroft. On biases in estimation due to the use of preliminary tests of signifiane. Ann. Math. Statist. 15 (1944), [5] P. Chiou, C. P. Han. Conditional interval estimation of the exponential sale parameter following rejetion of a preliminary test. J. Statist. Res. 28 (1994), [6] P. Chiou, C. P. Han. Conditional interval estimation of the exponential loation parameter following rejetion of a pre-test. Comm. Statist. Theory Methods 24 (1995), [7] D. E. A. Giles, O. Lieberman, J. A. Giles. The optimal size of a preliminary test of linear restritions in a misspeified regression model. J. Amer. Statist. Asso. 87 (1992), [8] G. G. Judge, M. E. Bok. The Statistial Impliation of Pre-Test and Stein-Rule Estimators in Eonometris. New York, North-Holland, [9] S. Mahdi. Estimation in Exponential Models. Mat. Vesnik 52, (2000), [10] S. Mahdi. Monte Carlo studies on the auray of an interval estimator after a preliminary test of signifiane proedure. Bulletin of the International Statistial Institute, 52nd session, Book 2, Helsinki, (1999), [11] S. Mahdi, V. P. Gupta. Conditionally speified onfidene interval for the variane of a normal population. Bull. So. Math. Belg. B 45, 3 (1993), [12] S. Mahdi. Estimation onditionnelle par intervalle de onfiane après test préliminaire de signifiation. Magister Thesis, Constantine University, [13] W. Q. Meeker, L. A. Esobar. Statistial Methods for Reliability Data. Wiley, New York, [14] S. L. Meeks, R.B. D Agostino. A note on the use of onfidene limits following rejetion of a null hypothesis. Amer. Statist., 37 (1983),

17 70 Smail Mahdi [15] A. Rai, A. K. Srivastava. Estimation of regression oeffiient from survey data based on test of signifiane. Comm. Statist. Theory Methods 27, 3 (1998), [16] D. G. Wardell. Small sample interval estimation of Bernoulli and Poisson parameters. Amer. Statist. 51, 4 (1997), [17] S. Wolfram. Mathematia: a system for doing mathematis by omputer (2nd ed.). Addison-Wesley Publishing Company, USA, [18] S. H. Zanakis. Extended pattern searh with transformation for the threeparameter Weibull MLE problem. Manag. Si. 25 (1979), Department of Computer Siene, Mathematis and Physis, University of the West Indies Cave Hill Campus, Barbados smahdi@uwihill.edu.bb Reeived August 14, 2003