Exponentiated Exponential Family: An Alternative to Gamma and Weibull Distributions

Transcription

1 BiometricaJourna43 (2001) 1, Exponentiated Exponentia Famiy: An Aternative to Gamma and Weibu Distributions Rameshwar D. Gupta* Department of Mathematics, Statistics and Computer Sciences The University of New Brunswick Saint John Canada Debasis Kundu Department of Mathematics Indian Institute of Technoogy Kanpur India Summary In this artice we study some properties of a new famiy of distributions, namey Exponentiated Exponentiadistribution, discussed in Gupta, Gupta, and Gupta (1998). The Exponentiated Exponentia famiy has two parameters (scae and shape) simiar to a Weibu or a gamma famiy. It is observed that many properties of this new famiy are quite simiar to those of a Weibu or a gamma famiy, therefore this distribution can be used as a possibe aternative to a Weibu or a gamma distribution. We present two reaife data sets, where it is observed that in one data set exponentiated exponentia distribution has a better fit compared to Weibu or gamma distribution and in the other data set Weibu has a better fit than exponentiated exponentia or gamma distribution. Some numerica experiments are performed to see how the maximum ikeihood estimators and their asymptotic resuts work for finite sampe sizes. Key words: Gamma distribution; Weibu distribution; Likeihood ratio ordering; Hazard rate ordering; Stochastic ordering; Fisher Information matrix; Maximum Likeihood Estimator. 1. Introduction Two-parameter gamma and two-parameter Weibu are the most popuar distributions for anayzing any ifetime data. Gamma has a ong history and it has severadesirabe properties, see Johnson, Kotz, and Baakrishnan (1994) for * Part of the work has been supported by the NSERC Grant, Canada, Cerant No. OCP

2 118 R. D. Gupta, D. Kundu: An Aternative to Gamma and Weibu Distribution the different properties of the two-parameter gamma distribution. It has ots of appications in different fieds other than ifetime distributions, some of the references can be made to Aexander (1962), Jackson (1963), Kinken (1961) and Masuyama and Kuroiwa (1952). The two parameters of a gamma distribution represent the scae and the shape parameters and because of the scae and shape parameters, it has quite a bit of fexibiity to anayze any positive rea data. It has increasing as we as decreasing faiure rate depending on the shape parameter, which gives an extra edge over exponentiadistribution, which has ony constant faiure rate. Since sum of independent and identicay distributed (i.i.d.) gamma random variabes has a gamma distribution, it has a nice physica interpretation aso. If a system has one component and n-spare parts and if the component and each spare parts have i.i.d. gamma ifetime distributions, then the ifetime distribution of the system aso foows a gamma distribution. Another interesting property of the famiy of gamma distributions is that it has ikeihood ratio ordering, with respect to the shape parameter, when the scae parameter remains constant. It naturay impies the ordering in hazard rate as we as in distribution. But one major disadvantage of the gamma distribution is that the distribution function or survivafunction cannot be expressed in a cosed form if the shape parameter is not an integer. Since it is in terms of an incompete gamma function, one needs to obtain the distribution function, survivafunction or the faiure rate by numerica integration. This makes gamma distribution itte bit unpopuar compared to the Weibu distribution, which has a nice distribution function, surviva function and hazard function. Weibu distribution was originay proposed by Weibu (1939), a Swedish physicist, and he used it to represent the distribution of the breaking strength of materias. Weibu distribution aso has the scae and shape parameters. In recent years the Weibu distribution becoming very popuar to anayze ifetime data mainy because in presence of censoring it is much easier to hande, at east numericay, compared to a gamma distribution. It aso has increasing and decreasing faiure rates depending on the shape parameter. Physicay it represents a series system, because the minimum of i.i.d. Weibu distributions aso foows a Weibu distribution. Severa appications of the Weibu distribution can be found in Pait (1962) and Johnson (1968) athough some of the negative points of the Weibu distribution can be found in Gorski (1968). One of the disadvantages can be pointed out that the asymptotic convergence to normaity for the distribution of the maximum ikeihood estimators is very sow (Bain, 1976). Therefore most of the asymptotic inferences (for exampe asymptotic unbiasedness or asymptotic confidence interva) may not be very accurate uness the sampe size is very arge. Some ramifications of this probem can be found in Bain (1976). It aso does not enjoy any ordering properties ike gamma distribution. In this paper we consider a two-parameter exponentiated exponentiadistribution and study some of its properties. The two parameters of an exponentiated

3 BiometricaJourna43 (2001) Fig. 1 exponentia distribution represent the shape and the scae parameter ike a gamma distribution or a Weibu distribution. It aso has the increasing or decreasing faiure rate depending of the shape parameter. The density function varies significanty depending of the shape parameter (see Figure 1). It is observed that it has ots of properties which are quite simiar to those of a gamma distribution but it has an expicit expression of the distribution function or the surviva function ike a Weibu distribution. It has aso ikeihood ratio ordering with respect to the shape parameter, when the scae parameter is kept constant. We aso observe that for fixed scae and shape parameters there is a stochastic ordering between the three distributions. The main aim of this paper is to introduce a new famiy of distributions and make comments both positive and negative of this famiy with respect to a Weibu famiy and a gamma famiy and give the practitioner one more option, with a hope that it may have a better fit compared to a Weibu famiy or a gamma famiy in certain situations. The rest of the paper is organized as foows. In Section 2, we introduce the exponentiated exponentiadistribution and compare its properties with the Weibu and the gamma distributions. Some of the stochastic ordering resuts are presented in Section 3. The maximum ikeihood estimators and their asymptotic properties have been discussed in Section 4. We anayze two data sets in Section 5 and some numerica experimenta resuts are presented in Section 6. Finay we draw concusions in Section 7.

4 120 R. D. Gupta, D. Kundu: An Aternative to Gamma and Weibu Distribution 2. Exponentiated ExponentiaDistribution The exponentiated exponentia (EE) distribution is defined in the foowing way. The distribution function, F E ðx; a; Þ, ofeeis F E ðx; a; Þ ¼ð1 e x Þ a ; a; ; x > 0 ; therefore it has the density function f E ðx; a; Þ ¼að1 e x Þ a 1 e x : The corresponding survivafunction is S E ðx; a; Þ ¼1 ð1 e x Þ a ; and the hazard function is h E ðx; a; Þ ¼ að1 e x Þ a 1 e x 1 ð1 e x Þ a : Here a is the shape parameter and is the scae parameter. When a ¼ 1, it represents the exponentia famiy. Therefore, a three famiies, namey gamma, Weibu and EE, are generaization of the exponentia famiy but in different ways. The EE distribution has a nice physicainterpretation aso. Suppose, there are n-components in a parae system and the ifetime distribution of each component is inde- Fig. 2

5 BiometricaJourna43 (2001) pendent and identicay distributed. If the ifetime distribution of each component is EE, then the ifetime distribution of the system is aso EE. As opposed to Weibu distribution, which represents a series system, EE represents a parae system. The typicaee density and hazard functions with ¼ 1, are shown in Figure 1 and Figure 2 respectivey. It is an unimoda density function and for fixed scae parameter as the shape parameter increases it is becoming more and more symmetric. For any, the hazard function is a non-decreasing function if a > 1, and it is a non-increasing function if a < 1. For a ¼ 1, it is constant. In this paper we use the foowing notations of the gamma ðf G Þ distribution and the Weibu ðf W Þ distribution: f G ðxþ ¼ a GðaÞ xa 1 e x ; a; ; x > 0 ; f W ðxþ ¼aðxÞ a 1 e ð xþa ; a; ; x > 0 : Therefore the parameters a and represent the shape and scae parameters respectivey in a the three different cases. A comparison of the three different hazard functions are given in Tabe A beow. Tabe A Hazard Function Parameters Gamma Weibu EE a ¼ 1 Constant Constant Constant a > 1 Increasing Increasing Increasing from 0 to from 0 to 1 from 0 to a < 1 Decreasing from 1 to Decreasing from 1 to 0 Decreasing from 1 to Therefore the hazard function of the EE distribution behaves ike the hazard function of the gamma distribution, which is quite different from the hazard function of the Weibu distribution. For the Weibu distribution if a > 1, the hazard function increases from zero to 1 and if a < 1, the hazard function decreases from 1 to zero. Many authors point out (see Bain, 1976) that since the hazard function of a gamma distribution (for a > 1) increases from zero to a finite constant, the gamma may be more appropriate as a popuation mode when the items in the popuation are in a reguar maintenance program. The hazard rate may increase initiay, but after some times the system reaches a stabe condition because of maintenance. The same comments hod for the EE distribution aso. Therefore, if it is known that the data are from a reguar maintenance environment, it may make more sense to fit the gamma distribution or the EE distribution than the Weibu distribution.

6 122 R. D. Gupta, D. Kundu: An Aternative to Gamma and Weibu Distribution Now et us consider the different moments of the EE distribution. Suppose X denote the EE random variabe with parameter a and, then Eðx k Þ¼a Ð1 0 x k ð1 e x Þ a 1 e x dx : Now since 0 < e x < 1, for > 0 and x > 0, therefore by using the series representation (finite or infinite) of ð1 e x Þ a 1 ð1 e x Þ a 1 ¼ P1 where cða 1; iþ ¼ EðX k Þ¼ i¼0 ð 1Þ i cða 1; iþ e ix ; ða 1Þ...ða iþ, we obtain i! agðk þ 1Þ k P 1 i¼0 ð 1Þ i cða 1; iþ 1 ði þ 1Þ kþ1 : ð2:1þ Since (2.1) is a convergent series for any k 0, therefore a the moments exist and for integer vaues of a, (2.1) can be represented as a finite series representation. Therefore putting k ¼ 1, we obtain the mean as EðXÞ ¼ a P 1 ð 1Þ i 1 cða 1; iþ i¼0 ði þ 1Þ 2 ; and putting k ¼ 2, we obtain the second moment as EðX 2 Þ¼ 2a P 1 2 ð 1Þ i 1 cða 1; iþ i¼0 ði þ 1Þ 3 : It is aso possibe to express the moment generating function in terms of the gamma function, which in turn can be used to obtain different moments. The moment generating function, MðtÞ, ofx for 0 < t < can be written as MðtÞ ¼Eðe tx Þ¼a Ð1 0 ð1 e x Þ a 1 e ðt Þ x dx : ð2:2þ Making the substitution y ¼ e x, (2.2) reduces to MðtÞ ¼a ð 1 0 Gða þ 1Þ G 1 t ð1 yþ a 1 y t dy ¼ G a t : ð2:3þ þ 1 Differentiating n ðmðtþþ and evauating at t ¼ 0, we get the mean and the variance of X as EðXÞ ¼ 1 ðyða þ 1Þ yð1þþ and var ðxþ ¼ 1 2 ðy0 ð1þ y 0 ða þ 1ÞÞ ; ð2:4þ

7 BiometricaJourna43 (2001) where yð:þ is the digamma function and y 0 ð:þ is its derivative. The higher centra moments can be obtained in terms of the poygamma functions. 3. Some Ordering Properties Ordering of distributions, particuary among the ifetime distributions, pays an important roe in statistica iterature. Johnson, Kotz, and Baakrishnan (1995, Chap 33) have a major section on the ordering of various positive vaued distributions. Pecaric, Proschan, and Tong (1992) aso provide a detaied treatment of stochastic ordering, highighting their growing importance and iustrating their usefuness in numerous practica appications. It might be usefu to obtain the bounds in survivafunctions, hazard functions or on the moments depending on the circumstances. In this section we discuss some of the ordering properties within each famiy of distribution and between the three famiies aso. In this section we take the scae parameter to be one throughout. It is we known that gamma famiy has increasing ikeihood ratio ordering in the shape parameter for fixed scae parameter so it has the ordering in hazard rate as we as in distribution functions. Since the gamma famiy has the ikeihood ratio ordering, it has the monotone ikeihood ratio property. This impies there exists a uniformy most powerfu test (UMP) for any one-sided hypothesis or uniformy most powerfu unbiased test (UMPU) for any two-sided hypothesis on the shape parameter if the scae parameter is known. Unfortunatey the Weibu famiy does not have the ordering even in distribution, so naturay it does not have the ordering in hazard rate or in ikeihood ratio. It can be easiy checked that for EE famiy it has the ordering in ikeihood ratio, so it has the ordering in hazard rate as we as in distribution function simiary as the gamma famiy. Therefore, for EE famiy aso if the scae parameter is known, there exists a UMP test for any one-sided hypothesis or UMPU test for any two-sided hypothesis on the shape parameter. Now consider some ordering properties between the famiies, when the shape parameter is kept at a constant vaue. Since f G ðxþ=f E ðxþ is an increasing function for a 1 and a decreasing function for a 1, therefore we can say that gamma is arger (smaer) than EE in terms of ikeihood ratio ordering if a 1ð 1Þ and they are equawhen a ¼ 1. It is interesting to observe that a ¼ 1 pays an important roe. When a ¼ 1 a the three distributions become equa to the exponentia distribution. It can be easiy seen that there is no ikeihood ratio ordering between Weibu and gamma or between Weibu and EE. But the foowing can be easiy observed for a vaues of x; h W ðxþ h E ðxþ 0 if a > 1 ; h W ðxþ h E ðxþ 0 if a < 1 :

8 124 R. D. Gupta, D. Kundu: An Aternative to Gamma and Weibu Distribution Since there is a hazard rate ordering between the gamma and EE, we immediatey obtain the foowing h W ðxþ h E ðxþ h G ðxþ if a > 1 ; h W ðxþ h E ðxþ h G ðxþ if a < 1 : Therefore we have an ordering in distribution aso between the three as foows; F W ðxþ F E ðxþ F G ðxþ for a > 1 ; F W ðxþ F E ðxþ FGðxÞ for a < 1 : 4. Maximum Likeihood Estimators and the Fisher Information Matrix In this section we discuss the maximum ikeihood estimators (MLE s) of a twoparameter EE distribution and their asymptotic properties. Let x 1 ;...; x n be a random sampe from EE, then the og ikeihood function can be written as: Lða; Þ ¼n n a þ n n þða 1Þ Pn n ð1 e x i Þ Pn x i : ð4:1þ Therefore, to obtain the MLE s of a and, either we can maximize (4.1) directy with respect to a and or we can sove the non-inear norma equations which are ¼ n a ¼ n Pn þða 1Þ n ð1 e x i Þ¼0 ; x i e x i 1 e x Pn x i ¼ 0 : i From (4.2), we obtain the MLE of a as a function of, say ^aðþ, as ^aðþ ¼ P n n : n ð1 e x i Þ ð4:2þ ð4:3þ ð4:4þ Therefore, if the scae parameter is known, the MLE of the shape parameter, ^a, can be obtained directy from (4.4). If both the parameters are unknown, first the estimate of the scae parameter can be obtrained by maximizing directy gðþ ¼Lð^aðÞ; Þ ¼C n n Pn n ð1 e x i Þ þ n n ðþ Pn n ð1 e x i Þ Pn x i : ð4:5þ

9 BiometricaJourna43 (2001) with respect to. Here C is a constant independent of. Once ^ is obtained, ^a can be obtrained from (4.4) as ^að^þ. Therefore it reduces the two dimensiona probem to a one dimensiona probem which is reativey easier to sove. In this situation we use the asymptotic normaity resuts to obtain the asymptotic confidence interva. We can state the resuts as foows: pffiffi n ð^q qþ!n 2 ð0; I 1 ðqþþ ð4:6þ where IðqÞ is the Fisher Information matrix, i.e. 2 IðqÞ ¼ L E @ 2 and ^q ¼ð^a; ^Þ, q ¼ða; Þ. Since for a > 0, the EE famiy satisfies a the reguarity conditions (see Bain, 1976), therefore (4.6) hods. Now, we provide the eements of the negative Fisher Information matrix, which might be usefu in practice. For a > 2 2 ¼ n a 2 2 L E ¼ n a ðyðaþ yð1þþ yða þ ða 1Þ 2 and for 0 < a 2, E 2 ¼ n 2 1 þ aða 1Þ ða 2Þ ðy0 ð1þ y 0 ða 1Þþðyða 1Þ yð1þþ 2 na 2 ½ðy0 ð1þ yðaþþðyðaþ yð1þþ 2 ÞŠ ¼ n a 2 ; 2 L ¼ n naða 1Þ 2 2 ð1 0 ð1 0 x e 2x ð1 e x Þ a 2 dx < 1 ; x e 2x ð1 e x Þ a 2 dx < 1 : Since q is unknown in (4.6), I 1 ðqþ is estimated by I 1 ð^qþ and this can be used to obtain the asymptotic confidence intervas of a and. In presence of Type I or Type II censoring the resuts can be suitaby modified. It may be mentioned that for Type I censored data the Fisher Information matrix aso can be obtained aong the same ine as the compete sampe case but unfortunatey in case of Type II censoring it is not possibe to obtain the Fisher Information matrix in a cosed form.

10 126 R. D. Gupta, D. Kundu: An Aternative to Gamma and Weibu Distribution 5. Data Anaysis In this section we use two uncensored data sets and fit the three modes namey; gamma, Weibu and Exponentiated Exponentia. Data Set 1: The first data set is as foows; (Lawess, 1986 page 228). The data given here arose in tests on endurance of deep groove ba bearings. The data are the number of miion revoutions before faiure for each of the 23 ba bearings in the ife test and they are 17.88, 28.92, 33.00, 41.52, 42.12, 45.60, 48.80, 51.84, 51.96, 54.12, 55.56, 67.80, 68.64, 68.64, 68.88, 84.12, 93.12, 98.64, , , , and We have fitted Gamma, Weibu and EE to this data set. We present the estimates, the Log-ikeihood (LL), the observed and the expected vaues and the c 2 statistics. The resuts are as foows. For Gamma distribution ^ ¼ 0:0556 ; ^a ¼ 4:0196 ; LL ¼ 113:0274 ; c 2 ¼ 1:040 : For Weibu distribution ^ ¼ 0:0122 ; ^a ¼ 2:1050 ; LL ¼ 113:6887 ; c 2 ¼ 1:791 : For EE distribution ^ ¼ 0:0314 ; ^a ¼ 5:2589 ; LL ¼ 112:9763 ; c 2 ¼ 0:783 : The observed and the expected frequencies are as given beow; Tabe 1 Intervas Observed EE Weibu Gamma Data set 2: (Linhart and Zucchini (1986, page 69). The foowing data are faiure times of the air conditioning system of an airpane: 23, 261, 87, 7, 120, 14, 62, 47, 225, 71, 246, 21, 42, 20, 5, 12, 120, 11, 3, 14, 71, 11, 14, 11, 16, 90, 1, 16, 52, 95. In this case aso we have fitted a the three distributions. The resuts are as foows: For Gamma distribution ^ ¼ 0:0136 ; ^a ¼ 0:8134 ; LL ¼ 152:2312 ; c 2 ¼ 3:302 :

11 BiometricaJourna43 (2001) For Weibu distribution ^ ¼ 0:0183 ; ^a ¼ 0:8554 ; LL ¼ 152:007 ; c 2 ¼ 3:056 : For EE distribution, ^ ¼ 0:0145 ; ^a ¼ 0:8130 ; LL ¼ 152:264 ; c 2 ¼ 3:383 : The observed and the expected frequencies are as given beow; Tabe 2 Intervas Observed EE Weibu Gamma It is observed that EE fits the best in the first data set whereas Weibu fits the best in the second data in terms of ikeihood and in terms of Chi-square. Therefore, it is not guaranteed the EE wi behave aways better than Weibu or gamma but at east it can be said in certain circumstances EE might work better than Weibu or gamma. 6. NumericaExperiments and Discussions In this section we perform some numericaexperiments to see how the MLE s and their asymptotic resuts work for finite sampe. A the numerica works are performed on PC-486 using the random deviate generator by Press et a. (1994). We consider the foowing different mode parameters: Mode1: a ¼ 2:0, ¼ 0:1, Mode2: a ¼ 1:0, ¼ 0:1, Mode3: a ¼ 0:5, ¼ 0:1, Mode4: a ¼ 2:0, ¼ 0:2, Mode5: a ¼ 1:0, ¼ 0:2, Mode6: a ¼ 0:5, ¼ 0:2, We consider the foowing sampe size (SS), n ¼ 10, 15 (sma), 40, 50 (moderate), and 100 (arge). For each mode parameters and for each sampe size, we compute the MLE s of a and, we aso compute the asymptotic confidence intervain each repications. We repeat this process 1000 times and compute the average estimators (AE), the square root of the mean squared errors (SMSE) and the coverage probabiities (CP). The resuts are reported in Tabes 3 4. Some of the points are very cear from the numerica experiments. It is observed that for a the parametric vaues the MSE s and the biases decrease as the sampe size increaes. It verifies the consistency properties of the MLE s as mentioned in (4.6). For fixed as a increases the MSE s and the biases of ^a increase

12 128 R. D. Gupta, D. Kundu: An Aternative to Gamma and Weibu Distribution Tabe 3 SS Par a ¼ 2:0, ¼ 0:1 a ¼ 1:0, ¼ 0:1 a ¼ 0:5, ¼ 0:1 AE SMSE CP AP SMSE CP AE SMSE CP 10 a a a a ß a Tabe 4 SS Par a ¼ 2:0, ¼ 0:2 a ¼ 1:0, ¼ 0:2 a ¼ 0:5, ¼ 0:2 AE SMSE CP AP SMSE CP AE SMSE CP 10 a a a a a where as the corresponding MSE s and the biases of ^ decrease for a the sampe sizes. Therefore, estimation of a becomes better as a decreases where as the estimation of becomes more accurate as a increases. On the other hand for fixed a as increases the MSE s and the biases of both ^a and ^ increase. Note that for arge sampe sizes ^ remains constant for a a. It is not very surprising because is the scae parameter and it aso foows from (4.6). Interestingy for moderate or arge sampe sizes it is observed that for fixed a the MLE s of a and the corresponding MSE s remain constant for different. It is cear that the MLE s of a and are positivey biased athough biases go to zero as sampe size increases. It is aso interesting to observe that the asymptotic confidence interva maintains the nomina coverage probabiities even for sma sampe sizes. Therefore, the MLE s

13 BiometricaJourna43 (2001) and their the asymaptotic resuts can be used for estimation and for constructing confidence intervas even for sma sampe sizes. 7. Concusions In this artice we consider EE famiy of distributions. It is observed that the twoparameter EE famiy are quite simiar in nature to the other two-parameter famiy ike Weibu famiy or gamma famiy. It is observed that most of the properties of a EE distribution are quite simiar in nature to those of a gamma distribution but computationay it is quite simiar to that of a Weibu distribution. Therefore, it can be used as an aternative to a Weibu distribution or a gamma distribution and it is expected that in some situations it might work better (in terms of fitting) than a Weibu distribution or a gamma distribution athough it can not be guaranteed. We present two reaife data sets, where in one data set it is observed that EE has a better fit compare to Weibu or gamma but in the other the Weibu has a better fit than EE or gamma. Moreover it is we known that gamma has certain advantages compare to Weibu in terms of the faster convergence of the MLE s. It is expected that EE aso shoud enjoy those properties. Extensive simuations are required to compare the rate of convergences of the MLE s of the different distributions. More work is needed in that direction. Primary numericaexperiments confirm that for EE famiy asymptotic resuts can be used even for sma sampe sizes for different a s and s. References Aexander, G. N., 1962: The use of gamma distribution in estimating reguated output from storages. Transactions in Civi Engineering, Institute of Engineers, Austraia 4, Bain, L. J., 1976: Statistica anaysis of reiabiity and ife testing mode. Marceand Dekker Inc., New York. Freudentha, A. M. and Gumbe, E. J., 1954: Minimum ife in fatigue. Journa of the American Statistica Association 49, Gorski, A. C., 1968: Beware of the Weibu euphoria. Transactions of IEEE Reiabiity 17, Gupta, R. C., Gupta, P. L., and Gupta, R. D., 1998: Modeing faiure time data by Lehman aternatives. Communications in Statistics, Theory and Methods 27, Jackson, O. A. Y., 1963: Fitting a gamma or og norma distribution to fiber diameter measurements of wootops. Appied Statistics 1, Johnson, L. G., 1968: The probabiistic basic of cumuative damage. Transactions of the 22nd. Technica Conference of the American Society of Quaity Contro, Johnson, N. L., Kotz, S., and Baakrishnan, N., 1994: Continuous Univariate Distributions-1, 2nd edition. John Wiey and Sons, New York. Johnson, N. L., Kotz, S., and Baakrishnan, N., 1995: Continuous Univariate Distributions-2, 2nd edition. John Wiey and Sons, New York. Kinken, J. van, 1961: A method for inquiring whether the gamma distribution represents the frequency distribution of industriaaccident costs. Actuaree Studien 3,

14 130 R. D. Gupta, D. Kundu: An Aternative to Gamma and Weibu Distribution Lawess, J. F., 1982: Statistica Modes and Methods for Lifetime Data. John Wiey and Sons, New York. Linhart, H. and Zucchini, W., 1986: Mode Seection. Wiey, New York. Masuyama, M. and Kuroiwa, Y., 1952: Tabes for the ikeihood soutions of gamma distribution and its medicaappications. Reports of Statistica Appications Research (JUSE) 1, Pait, A., 1962: The Weibu distribution with tabes. Industria Quaity Contro 19, Pecaric, J. E., Proschan, F., and Tong, Y. L., 1992: Convex Functions, Partia Ordering and Statistica Appications. Academic Press, San Diego. Press, W. H., Teukosky, S. A., Veering, W. T., and Fannery, B. P., 1994: Numerica Recipes, 2nd. edition. University Press, Cambridge. Weibu, W., 1939: A statisticatheory of the strength of materia. Ingeniors Vetenskaps Akademiens, Stockhom 151. Debasis Kundu Received, May 1999 Department of Mathematics Revised, Apri2000 Indian Institute of Technoogy Kanpur Accepted, Apri 2000 Kanpur, Pin India E-mai: kundu@iitk.ac.in

15