Testing the Equality of Two Pareto Distributions

Transcription

1 Proceeding of the World Congre on Engineering 07 Vol II WCE 07, July 5-7, 07, London, U.K. Teting the Equality of Two Pareto Ditribution Huam A. Bayoud, Member, IAENG Abtract Thi paper propoe an overlapping-baed tet for the equality of two Pareto ditribution with different cale and hape parameter. The propoed tet tatitic i defined a the maximum likelihood etimate of Weitzman overlapping coefficient, which etimate the agreement of two denitie. Simulated critical point are provided for the propoed tet for variou ample ize and ignificance level. Statitical power of the propoed tet are computed via imulation tudie. Type- I error robutne of the propoed tet i tudied alo via imulation tudie when the underlying ditribution are not Pareto. Index Term MLE Monte Carlo Simulation, Overlapping coefficient, Simulated Powe T I. INTRODUCTION eting the imilarity of two population i needed in variou practical field including pharmaceutical cience, laboratory management, engineering cience, food technology, management cience, quality control and many other field. Such tet are needed, for example, to ae the acceptability of a newly developed proce\deign\approach to a gold tandard one. In thi paper, we propoe a tet for the equality of two Pareto ditribution of the firt kind. Pareto ditribution i important in practice becaue of it adequacy in modeling variou ditribution, for example, in the commercial ector, including city population ditribution, tock price fluctuation, and oil field location. In addition, thi ditribution i highly applicable in the Air Force ector for modeling, for example, the failure time of equipment component, ee Davi and Michael (979), maintenance ervice time, ee Harri (968), nuclear fallout particle' ditribution, ee Freiling (966), and error cluter in communication circuit, ee Berger and Mandelbrot (963). Manucript received Jan 6, 07; revied Feb 7, 07. Thi work wa financially upported by Fahad Bin Sultan Univerity, Tabuk, Saudi Arabia. Huam Awni Bayoud, College of Science and Humanitie, Fahad Bin Sultan Univerity, Tabuk, Saudi Arabia, P.O. Box 5700 Tabuk Phone: ; fax: ; huam.awni@yahoo.com. A decribed by Johnon and Kotz (970), the probability denity function (pdf) and the cumulative ditribution function (cdf) of Pareto ditribution with hape and cale parameter and, repectively, are: f ( x, ) ; x x 0, 0, () and F ( x, ) ; x 0, 0, () x Throughout thi article, we denote Pareto ditribution by Pareto,. Conider for teting the imilarity of two Pareto population: H0 : F( x, ) F( x, ) v H : F( x, ) F( x, ), (3) which i equivalent to tet: H 0 : and v. H : or. (4) In general, teting the imilarity of two ditribution can be tatitically linked to one-ided teting problem. For intance, teting the hypothei: H o : F G at a ignificance level can be done by teting the hypothei: H o : SF, G ( ) c, where S F, G i a uitable meaure of imilarity (diimilarity) of F and G, and c i the cut-off point of the meaure S at a ignificance level. In thi paper we propoe an overlapping (OVL) coefficientbaed tet for the imilarity of two Pareto ditribution. The propoed tet i defined a the Maximum Likelihood Etimate (MLE) of the Weitzman OVL coefficient; thi coefficient wa firt propoed by Weitzman (970) to tudy the agreement of two income ditribution. Thi OVL coefficient ha received the attention of different author in the literature for it importance in meauring the agreement of two tatitical ditribution. Helu and Samawi (0) derived three Weitzman' coefficient (970), Matuita Coefficient (955) and Moriita Coefficient (959), for two Pareto ditribution of kind II, and propoed point and confidence interval etimate for thee coefficient baed on Simple Random and Ranked Set Sample, eparately. Inman and Bradley (989) ued the OVL coefficient to meaure the agreement between two normal ditribution auming equal variance. Mulekar and Mihra (994) tudied variou OVL coefficient for two normal population with the ame mean. Reier and Faraggi (999) derived the confidence interval of the OVL coefficient of two normal ditribution auming equal variance in term of non central t and non central F ditribution eparately. Chaubey et. al. (008) tudied the OVL coefficient for two invere Gauian population. Al- Saleh and Samawi (007) tudied OVL coefficient for two exponential population. Recently, OVL-baed tet have been propoed for teting the equality of two exponential ditribution and two normal ditribution eparately by Bayoud (05) and Bayoud and Kittaneh (06), ISBN: ISSN: (Print); ISSN: (Online) WCE 07

2 Proceeding of the World Congre on Engineering 07 Vol II WCE 07, July 5-7, 07, London, U.K. repectively. Thi paper propoe the ame technique for teting the equality of two Pareto ditribution. The ret of thi paper i organized a follow: In ection, the propoed tet tatitic i derived in analytical form. Simulated critical point for the propoed tet tatitic are provided in Section 3 for variou ample ize and level of ignificance. An approximation to the null ditribution of the propoed tet tatitic i propoed in Section 4. In ection 5, tatitical power of the propoed tet i tudied under variou cenario. Variou real dataet are analyzed in Section 6. The paper i concluded in Section 7. II. PROPOSED TEST STATISTIC The propoed tet tatitic i defined a the etimated OVL coefficient. The OVL coefficient of f ( x, ) and f x, ) i defined a: ( min( f ( x, ), f ( x, )) dx (5) x The coefficient range from zero to one. A value cloe to zero indicate no common area between the two denity function. A value cloe to one indicate that the two denitie are identical. It can be eaily proved that two Pareto denitie f x, and f x, with different parameter either interect at only one point in the upport,, where Max,, or they do not interect at any point in,. Note that, if and, then the two denitie are clearly identical, and then. If and, then the two denitie do not interect. If then, regardle the value of and, the two denitie interect only at x 0 given by: x 0, (6) provided that thi value belong to,. Algebraically, the value of x 0 in Eq(8) might be outide the interval, at ome value of,, and, For example, when.,, 5 and, the value of x , which doe not belong to.,, thi mean that the two denitie do not interect in their upport; and hence, there i no interection point. In uch a cae, we conclude that one of the denitie lie above the other at all value in,. Therefore, two different Pareto denitie either interect at x only one point 0 in the interval,, or they do not interect at all value x,. If the pdf f x, and f x, interect at x0,, then in Eq (7) can be repreented a: -, if f( ) f ( ) x0 x0, (7) -, if f( ) f ( ) x0 x0 where fi ( x) f x i, i defined in Eq (), Fi ( x) Fx i, i defined in Eq (), for i,, and x 0 i the interection point defined in Eq(8). On the other hand, if f x and x, f, do not interect in,, then one of the denitie i greater than the other at all value in,, then in Eq (7) can be implified to: f ( x, ) dx, if f( ) f ( ) f ( x, ) dx, if f( ) f ( ), if, if f ( ) f f ( ) f ( ) ( ). (8) Clearly, Eq(0) i the limit of Eq(9) when x 0 approache infinity. However, the propoed tet tatitic for teting the null hypothei in Eq (3) i defined a the MLE of the overlapping coefficient in Eq (7) which can be implified either to Eq (9) or Eq (0) depending on the interection point of f x, and f x,. n, Let, x,..., x ~ Pareto x and, x,..., xn ~ Pareto, x be two random ample, not necearily independent, from Pareto ditribution. The MLE of the cale and hape parameter baed on the random ample above are: ˆ x() minx, x,..., xn, ˆ x() minx, x,..., xn n n, ˆ and ˆ n. n ln x i ln x ln x i ln x i () By the invariant property of the MLE, the propoed tet tatitic, denoted by ˆ, can be obtained by replacing the parameter,,,,, f, f and x 0 in Eq (9) or Eq (0) by their MLE. The null hypothei H o in Eq (4) i rejected at ignificance level if ˆ ˆ, n, n, where Pˆ ˆ, n, H0 n ; i.e. ˆ th, n, n i the percentile of the ditribution of ˆ at n and n under the aumption that H o i true. Since, deriving the null ditribution of ˆ in analytic form i complicated. Therefore, percentile of ˆ ( ˆ, n, n ) are imulated in Section 3 at different ample ize and i () ISBN: ISSN: (Print); ISSN: (Online) WCE 07

3 Proceeding of the World Congre on Engineering 07 Vol II WCE 07, July 5-7, 07, London, U.K. ignificance level. Moreover, the null ditribution of the propoed tet tatitic i approximated by Beta ditribution in Section 4. III. SIMULATED CRITICAL POINTS OF ˆ Since the null ditribution of the propoed tet tatitic could not be obtained in cloed form. Therefore, imulated percentile for thi ditribution are generated at different ignificant level, and for different ample ize auming, for implicity, n n. Table how the imulated t,.5th, 5th, and 0th percentile of the null ditribution of the propoed tet tatitic, thee percentile give the critical point of the propoed tet tatitic. Thee percentile have been computed baed on 0,000 Monte Carlo Simulation generated at variou ignificance level under the aumption of equal ample ize n n n by uing the following algorithm: Algorithm I: i) Arbitrary, chooe and, baed on the aumption that H in Eq (3) i true. ii) Generate, x,..., ~ Pareto x,..., xn ~ Pareto, o x n, x,. x and iii) Find the MLE ˆ ˆ,, ˆ ˆ, ˆ Max ˆ ˆ, from the ample obtained in tep. Hence, ˆ and ˆ are not necearily equal, and ˆ and ˆ a well. iv) Compute the MLE of x 0 defined in Eq (8) by replacing the parameter by their MLE thoe have been computed in tep (3). v) If x 0 belong to, then the propoed tet tatitic and ˆ i computed uing Eq (9). Otherwie, the two denitie do not interect in,, and then the propoed tet tatitic ˆ i computed uing Eq (0). vi) Repeat tep (-5) m 0, 000 iteration, to get ˆ, ˆ,..., ˆ m. vii) t,.5th, 5th, and 0th percentile are computed for the et obtained in tep (6) which are the imulated critical point of the propoed tet tatitic at ignificance level of 0. 0, 0. 05, and 0., repectively. IV. APPROXIMATED NULL DISTRIBUTION of ˆ In hi ection, we propoe an approximation for the null ditribution of the propoed tet. Under the aumption that the null hypothei i true, it wa oberved from the imulation tudie in the previou ection that the ditribution of the tet tatitic ˆ i kewed to the left with value between zero and, far away from zero and cloe to. Eventually, Table how that 00 th percentile are much cloer to than zero for fixed n. Moreover, the percentile become much cloer to when n increae. Conequently, one can approximate the null ditribution of ˆ by beta ditribution with parameter p and q auming p q 0, to inure kewed to left ditribution, with pdf: ˆ p q ˆ p q f ˆ provided that p q 0 and p q 0 ˆ. TABLE I SIMULATED 00 th PERCENTILE ˆ,n, n OF THE NULL DISTRIBUTION of ˆ n Clearly, the parameter p and q are function in n and n. A imulation tudy i performed by uing Algorithm II to tet whether beta ditribution adequately fit the null ditribution of the tet tatitic ˆ. For implicity, we aume n n n. Algorithm II: i) For a fixed n, Step (i-v) in Algorithm I are performed with m 5000 iteration to get ˆ ˆ,..., ˆ, m, a et of 5000 ˆ value. ii) The parameter p and q of beta ditribution are etimated by the method of moment etimation from the et ˆ ˆ,..., ˆ, m obtained in Step (i). iii) The frequency hitogram for the imulated et ˆ ˆ,..., ˆ, m i plotted along with the fitted beta pdf uing the etimated p and q in Step (ii). Table how the frequency hitogram along with the fitted beta pdf obtained from Algorithm II for variou value of n. It i evident from Table that beta ditribution adequately fit the null ditribution of ˆ. Obviouly, there i a poitive linear relationhip between n and p, and a ISBN: ISSN: (Print); ISSN: (Online) WCE 07

4 Proceeding of the World Congre on Engineering 07 Vol II WCE 07, July 5-7, 07, London, U.K. negative linear relationhip between n and q. Hence, one can approximate the parameter p and q by fitting uitable function for p and q in term of n. For illutrative purpoe, we tudy the relationhip between n, p and q by analyzing a ample of the form n p, q;, n 0(5)00 that i generated by uing Algorithm II for n 0(5) 00. For thi ample, the etimated linear correlation coefficient between n and p and between n and q are 0.98 and - 0.9, repectively. Thi how that linear model could adequately fit p and q in term of n a follow: p ( n) n and q(n) n Therefore, the null ditribution of the tet tatitic ˆ can be approximated by a beta ditribution with parameter p ( n) n and q(n) n, where n i the ample ize, i.e. Beta n, n. For intance, at n n n 30, the parameter p and q are etimated by and. 664, repectively. Hence, the null ditribution of ˆ i approximately ,.664 TABLE II FITTED BETA TO THE NULL DISTRIBUTION OF ˆ n p q Beta. Beta, Fitted Beta Curve 0 Beta 8.4,.3 0 Beta 5.53,.4 30 Beta 0.48,. 50 Beta 7.74,. 0 Beta 39.6,.96 0 Note that the 5th and 0th percentile of Beta ,.664 are and which are very cloe to the imulated 5th and 0th percentile ˆ and ˆ 0. 8, repectively. 0.05,30,30 0.0,30,30 It i worth mentioning that nonlinear model might alo adequately fit p and q in term of n. However, more tudie are needed in thi direction to tudy other model when n n. V. SIMULATED POWERS Statitical power of a tet i defined a the probability of rejecting the null hypothei H 0 when the null hypothei i fale. Unfortunately, power function of the propoed tet cannot be obtained in explicit form, then imulated tatitical power i computed under different alternative, keeping Type-I error of each tet at the ame level. However, Table 3 how the imulated power of the propoed tet baed on 5000 Monte Carlo imulation for two ample ize n 0 and 30 and variou alternative auming the ignificance level 0. 05, and the Pareto parameter and. TABLE III SIMULATED POWERS OF ˆ, ASSUMING, AND 0.05 LEVEL OF SIGNIFICANCE n n n n n n It i obviou from Table 3 that the tatitical power of the propoed are atifactory. Clearly, the tatitical power of thi tet increae a the ample ize increae. Furthermore, it appear from Table 3 that thi tet efficiently control the ignificance level, i.e. tatitical power achieve the correct ignificance level in imulation under the aumption that the null hypothei i true. However, ince the propoed tet i derived under the aumption that the ditribution are Pareto, it i ignificant to tudy the Type-I error robutne of the propoed tet for teting the imilarity of non-pareto ditribution. Table 4 ummarize the ignificance (nominal) level of the propoed tet baed on 0,000 imulated pair of ample ISBN: ISSN: (Print); ISSN: (Online) WCE 07

5 Proceeding of the World Congre on Engineering 07 Vol II WCE 07, July 5-7, 07, London, U.K. when the null hypothei H0 i true for the elected population. Pareto, Normal, Gamma, Chi-quare, and Exponential ditribution are conidered. Variou aumption of the ample ize and the ignificance level are conidered. It i worth mentioning that Pareto ditribution i conidered again for the ake of teting the nominal level of the tet at variou Type-I error and ample ize. The robutne of a tet could be aeed by the cloene of the imulated nominal level to the correct Type-I erro It i evident from Table 4 that the propoed tet i highly enitive for violation of the Pareto aumption; which mean that the tet i not robut againt departure from the Pareto aumption. TABLE IV NOMINAL LEVELS FOR ˆ WHEN H0 IS TRUE FOR THE GIVEN POPULATION Significance Level Population n n 0 Pareto (9,) N (5,) Gamma (,3) Exp (9) n n 0 Pareto (9,) N (5,) Gamma (,3) Exp (9) n n 30 Pareto (9,) N (5,) Gamma (,3) Exp (9) VI. REAL APPLICATION In thi ection, a real data obtained from Watthanacheewakul and Suwattee (00) i analyzed, thi data ummarize the major rice crop (in Kilogram) in the crop year 00/00 (April t, 00 to March 3t, 00) from two Tambol, Nongyang and Nongjom, of Amphoe Sanai in the Chiang Mai province, Thailand. Sample (hown in Table 5) of ize 8 and 30 were drawn from Nongyang and Nongjom Tambol, repectively. Firt, it wa checked whether Pareto ditribution can be ued or not to analyze thee data et. For Nongyang' data, the MLE of and are and 3000, repectively. The Kolmogorov-Smirnov (KS) ditance between the empirical ditribution function and the fitted ditribution function ha been ued to check the goodne of fit. The KS tatitic value i 0.07, and the KS critical value i 0.50 at at n 8 and Accordingly, one cannot reject the hypothei that the data are coming from T-L ditribution. For Nongjom' data, the MLE of and are and 3600, repectively. The KS tatitic value i 0.33, and the KS critical value i 0.76 at at n 30 and Accordingly, one cannot reject the hypothei that the data are coming from T-L ditribution. TABLE V THE MAJOR RICE CROP IN KILOGRAMS FROM THE TWO TAMBOLS FOR THE CROP YEAR 00/00 (APRIL ST, 00 TO MARCH 3ST, 00) Nongyang Nongjom Now, we need to tet the null hypothei that the rice crop of Nongyang and Nongjom are the ame by uing the propoed tet. The propoed tet tatitic i equal to which i le than the critical value ˆ , thi mean that the hypothei that the 0.05,8,30 two crop are the ame i rejected at ignificance level of 0.05, which agree with what wa concluded by Watthanacheewakul and Suwattee (00). VII. CONCLUSIONS In thi paper, a non claical tet ha been propoed to tet the imilarity of two Pareto population. The propoed tet tatitic i defined a the MLE of Weitzman OVL coefficient, which etimate the agreement of two denitie. Thi tet tatitic ha been derived in cloed form. Since deriving the exact null ditribution of the propoed tet tatitic i not imple, imulated percentile have been provided at variou equal ample ize and ignificance level. However, the null ditribution of the propoed tet tatitic wa approximated by beta ditribution, but till further tudie are needed in thi direction. Simulated power of the propoed tet have been computed. It ha been concluded that the propoed tet almot ha atifactory power and it efficiently control the ignificance level. Furthermore, we tudied Type-I error robutne of the propoed tet when the underlying ditribution are non- ISBN: ISSN: (Print); ISSN: (Online) WCE 07

6 Proceeding of the World Congre on Engineering 07 Vol II WCE 07, July 5-7, 07, London, U.K. Pareto. It ha been noticed that thi tet the propoed tet i highly enitive for violation of the Pareto aumption; which mean that the tet i not robut againt departure from the Pareto aumption. REFERENCES [] H. T. Davi, and M. L. Feldtein, "The Generalized Pareto Law a a Model for Progreively Cenored Survival Data," Biometrika, vol. 66, pp , 979. [] C. M. Harri, "The Pareto Ditribution a a Queue Service Dicipline," Operation Reearch, vol. ii, pp , January- February 968. [3] E. C. Freiling, A Comparion of the Fallout Ma-Size Ditribution Calculated by Lognormal and Power-Law Model, San Francico: U.S. Naval Radiological Defene Laboratory, 966 (AD ). [4] J. M. Berger and B. Mandelbrot, "A New Model for Error Clutering in Telephone Circuit," IBM Journal of Reearch and Development, vol. 7, pp. 4-36, July 963. [5] N. L. Johnon and S. Kotz, Continuou Uniwriute Ditri- hution-i, Houghton Mifflin Co [6] M.S. Weitzman, Meaure of overlap of income ditribution of white and Negro familie in the United State, Technical paper No., Department of Commerce, Bureau of Cenu, Wahington. U.S. (970). [7] A. Helu and H. Samawi, On Inference of Overlapping Coefficient in Two Lomax Population Uing Different Sampling Method, Journal of Statitical Theory and Practice, vol. 5(4), pp , 0. [8] K. Matuita, "Deciion rule baed on the ditance for problem of fir, two ample, and Etimation," Ann. Math. Statit. vol. 6, pp , 955. [9] M. Moriita, "Meauring interpecific aociation and imilarity between communitie. Memoir of the faculty of Kyuhu Univerity," Serie E. Biology, vol. 3, pp , 959. [0] H.F. Inman and E.L. Bradley, The overlapping coefficient a a meaure of agreement between probability ditribution and point etimation of the overlap of two normal denitie, Commun Stat Theory Method, vol. 8(0), pp , 008. [] M. Mulekar and S. N. Mihra, Overlap coefficient of two normal denitie: equal mean cae, J. Japan. Statit. Soc., vol. 4, pp , 994. [] B. Reier and D. Faraggi, Confidence Interval for the overlapping coefficient: the normal equal variance cae, The Statitician, vol. 48 (3), pp , 999. [3] Y. P. Chaubey, D. Sen, and S. N. Mihra, Inference on Overlap for Two Invere Gauian Population: Equal Mean Cae, Communication in Statitic-Theory and Method, vol. 37,, , 008. [4] M. F. AL-Saleh and H. M. Samawi, "Inference on overlapping coefficient in two exponential population," J. of Modern Statit. Sci., vol. 6 (), pp , 007. [5] H. A. Bayoud and O. A. Kittaneh, "Teting the equality of Two exponential Ditribution," Communication in Statitic-Simulation and Computation, vol. 45(7), pp , 06. [6] H. A. Bayoud, "Teting the Similarity of Two Normal Population with Application to the Bioequivalence Problem," Journal of Applied Statitic, vol. 43(7), pp , 06. [7] L. Watthanacheewakul and S. Suwattee, "Comparion of Several Pareto Population Mean," Proceeding of International MultiConference of Engineer and Computer Scientit, vol III, 00. ISBN: ISSN: (Print); ISSN: (Online) WCE 07