COMPARATIVE SIMULATION STUDY OF LIKELIHOOD RATIO TESTS FOR HOMOGENEITY OF THE EXPONENTIAL DISTRIBUTION

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1 ACTA UIVERSITATIS AGRICULTURAE ET SILVICULTURAE MEDELIAAE BRUESIS Volume LX 33 umber 7, 2012 COMPARATIVE SIMULATIO STUDY OF LIKELIHOOD RATIO TESTS FOR HOMOGEEITY OF THE EXPOETIAL DISTRIBUTIO L. Střelec, M. Stehlík Received: August 27, 2012 Abstract STŘELEC, L., STEHLÍK, M.: Comparative simulation stud of likelihood ratio tests for homogeneit of the exponential distribution. Acta univ. agric. et silvic. Mendel. Brun., 2012, LX, o. 7, pp The aim of this paper is to present and discuss the power of the exact likelihood ratio homogeneit testing procedure of the number of components k in the exponential mixture. First we present the likelihood ratio test for homogeneit (), the likelihood ratio test for homogeneit against twocomponent exponential mixture (2), and finall the likelihood ratio test for homogeneit against three-component exponential mixture (3). Comparative power stud of mentioned homogeneit tests against three-component subpopulation alternative is provided. Therein we concentrate on various setups of the scales and weights, which allow us to make conclusions for generic settings. The natural propert is observed, namel increase of the power of exact likelihood ratio, 2 and 3 tests with scale parameters considered in the alternative. We can state that the differences in power of, 2 and 3 tests are small therefore using of the computationall simpler 2 test is recommended for broad usage rather than computationall more expensive 3 test in the cases when unobserved heterogeneit is modelled. Anhow caution should be taken before automatic usage of 3 in more informative settings, since the application of automatic methods hoping that the data will enforce its true structure is deceptive. Application of obtained results in reliabilit, finance or social sciences is straightforward. exponential distribution, homogeneit testing, likelihood ratio, mixture models, Monte Carlo simulations, power stud Currentl, man homogeneit tests exist see Stehlík and Wagner (2012) and references therein. In this paper we focus primaril on likelihood ratio tests. The exact likelihood ratio test for scale and homogeneit in the complete sample from gamma famil was derived in Stehlík (2003). The exact distribution of the likelihood ratio test for homogeneit was derived in Stehlík (2006) for the exponential and Weibull distribution and for the generalized gamma distribution was derived in Stehlík (2008). Exact likelihood testing for homogeneit of the number of components in the Raleigh mixture for k = 2 and k = 3 components was introduced in Stehlík and Ososkov (2003), and for k = 2 in exponential mixture was studied b Stehlík and Wagner (2011), and finall for k = 3 in the Raleigh famil was studied in Střelec and Stehlík (2012). The aim of this paper is to present and discuss the power of the likelihood ratio homogeneit testing procedure for the number of components k in the exponential mixture. In other words, the aim of this paper is to present and discuss the power of exact procedure for testing exponential homogeneit against alternatives of exponential heterogeneit. Therefore, the rest of this paper is organized as follows. In this section, the exact likelihood ratio homogeneit tests are introduced. In Section 2, alternatives to homogeneit and simulations setup are specified. In Section 3, a comparative power stud of exact likelihood ratio tests for homogeneit against the three-component subpopulation alternative is provided. Last sections are Conclusions and Summar. 307

2 308 L. Střelec, M. Stehlík Exact likelihood ratio tests for homogeneit Firstl, we present exact likelihood ratio tests for homogeneit used for comparative power stud. test Let 1, be independentl distributed with exponential densities with unknown scale parameter θ. Then following Stehlík (2006, Theorem 3), the test statistic lnλ (), where λ () is the formula for likelihood ratio, has the following form ln ln ln ln. (1) i i i1 i1 The test statistic has some important properties e.g. scale invariance, i.e. the distribution of the test statistic under null hpothesis is independent of the unknown scale parameter (Stehlík, 2006) and it is optimal in the Bahadur sense (see Rublík, 1989a, 1989b). 2 test 2 is test constructed for testing of homogeneit of the number of components k in mixture for k = 2 components, firstl introduced b Stehlík and Ososkov (2003). Therefore, we consider the testing problem of the form : k= 1 vs. H 1 : k= 2, (2) which can be, following Stehlík and Ososkov (2003), in the mixture model approximated b the hpothesis of the subpopulation model : θ 1 = θ vs. H 1 : non empt disjoint subsets M 1, M 2, M 1 M 2 = {1,, }, (3) where M 1 M 2 =, M 1, M 2, j M 1 : j = 1, j M 2 : θ j = θ 2, where θ 1 and θ 2 are different scale parameters, i.e. θ 1 θ 2. Following Stehlík and Ososkov (2003) and Stehlík and Wagner (2011) we introduce the exact likelihood ratio test of the hpothesis (3) which approximates the hpothesis (2). Let 1, be independentl distributed with exponential densities and suppose that { i1,, ik }, 0 < K < are the observations from exponential distribution with scale parameter θ 1 and the other observations are distributed according to the exponential distribution with scale parameter θ 2 and where i k denotes indices from set {1,,} for 1 i K. Then following Stehlík and Ososkov (2003), the formula for likelihood ratio has the following form min K K i 1 i K i K1 i 0 K, pp K K K K 1 K where P(K) for 0 < K < denotes all partitions of {1,,K} in two non-empt subsets., (4) Then 2 test statistic lnλ (), where λ () is given b formula (4), has the following form K ln Kln K Kln KKln i n n1 ln min. 0 K, ppk K Kln i ln n n n1 n1 (5) The 2 test statistic is scale invariant under the null hpothesis (see Stehlík and Ososkov, 2003). Following Stehlík and Wagner (2011, Lemma 3.1), 2 test statistic can be also determined as ln ln ln H n, (6) min n1 where H min can be determined as sums of order statistics (i), i.e. K Hmin min Kln K Kln KKln Kln 0K i i. i1 ik1 (7) 3 test 3 is test constructed for testing of homogeneit of the number of components k in mixture for k = 3 components, similarl as 2 test introduced b Stehlík and Ososkov (2003). Therefore, we consider the testing problem of the form : k= 1 vs. H 1 : k = 3 (8) which can be, following Stehlík and Ososkov (2003), in the mixture model approximated b the hpothesis of the subpopulation model : θ 1 = θ vs. H 1 : non empt disjoint subsets M 1, M 2, M 3 (9) of the set 1,, such that j M 1 : θ j = θ 1, j M 2 : θ j = θ 2, j M 3 : θ j = θ 3, where θ 1, θ 2, and θ 3 are different scale parameters. Following Stehlík and Ososkov (2003) we introduce the exact likelihood ratio test of the hpothesis (9) which approximates the hpothesis (8). Let 1, be independentl distributed with exponential densities and suppose that { i1,, ik }, 0 < K < 1 are the observations from exponential distribution with scale parameter θ 1, { j1,, jl }, 0 < L < K are the observations from exponential distribution with scale parameter θ 2, and finall the other observations are distributed according to the exponential distribution with scale parameter θ 3 and where i k denotes indices from set {1,,} for 1 i K and j l denotes indices from set {1,,} for 1 j L. Then following Stehlík and Ososkov (2003), the formula for likelihood ratio has the following form

3 Comparative simulation stud of likelihood ratio tests for homogeneit of the exponential distribution 309 K L KL K L K L min K L KL 0K1, 0 LK, i 1 i K j 1 j L l 1 lkl ppk, L 1, (10) where P(K,L) for 0 < K < 1 and 0 < L < K denotes all disjoint pairs of K-subsets { i1,, ik } and L-subsets { j1,, jl } of the set {1,,}, where i k denotes indices from set {1,,} for 1 i K and j l denotes indices from set {1,,} for 1 j L. Then 3 test statistic lnλ (), where λ () is given b formula (10), has the following form ln Kln K Lln L K Lln K L ln min K L K L KL Kln i Lln ln n j K L n ln. n1 n1 n1 1, 0 K 0, ppk, L ln n n1 MATERIALS AD METHODS Alternatives to homogeneit (11) As Stehlík and Wagner (2011) state, alternatives to homogeneit are often specified as mixture models and the most popular alternative to homogeneit is the mixture model with exponential components. The joint densit of a sample 1, from a general k-component mixture of exponential components is k f 1,..., pj j exp ji, (12) i1 j1 where 0 < p j < 1, j p j = 1. In this paper we will present and discuss the power of the exact likelihood ratio homogeneit testing procedure of the number of components k in the exponential mixture onl for k = 3 components, introduced b Stehlík and Ososkov (2003). Therefore we suppose following hpothesis : 1,, Exponential (θ) vs. (13) H 1 : 1,, follow a mixture of three exponential components, i.e. we suppose mixture of three exponential components with following probabilit densit function (pdf) K1exp 1 L2exp 2 1 K L3exp 3 (14) f p p p p, where, and 1 are weights of components such that 0 <,, 1 < 1. Setup of simulation stud A simulation stud was performed to compare the power of the exact likelihood ratio tests, 2 and 3 for the following parameters setup: {10, 20, 50}, θ 1 = 1, θ 2 {1, 3, 5, 7} and θ 3 {1, 5, 7, 10} and different component weights {0.2, 0.4, 0.5, 0.6, 0.8} and {0.1, 0.2, 0.3, 0.4, 0.6}. For mentioned parameters setup M = samples were generated and the proportion of rejections of, 2 and 3 tests was determined. ote that the, 2 and 3 tests have non standard asmptotic distributions but we can simulate their exact distributions. Therefore, critical values of the, 2 and 3 tests can be simpl obtained b Monte Carlo simulations, i.e. we generated M = samples of size {10, 20, 50} from the standard exponential distribution, then we computed the test statistic for each sample, and finall critical values c 1 were determined. RESULTS AD DISCUSSIO Critical values of the, 2 and 3 test statistics for {0.01, 0.05, 0.10} based on M = samples of size {10, 20, 50} from the standard exponential distribution are presented in Tab. I. Consequentl, sizes of the, 2 and 3 test statistics for = 0.05 (i.e. power of the, 2 and 3 test statistics against mixture of three exponential components with pdf from (14) for θ 1 = θ 2 = θ 3 = 1) are presented in Tab. II. As we can see from Tab. II, the, 2 and 3 tests hold the chosen size = 0.05 even for small samples. The biggest difference between theoretical and empirical sizes is for 3 test for = 50 and and for this parameters setup the difference between theoretical and empirical sizes is Finall, power of exact likelihood ratio tests, 2 and 3 against mixture of three exponential components with probabilit densit I: Critical values of, 2 and 3 test statistics = 0.01 = 0.05 = 10 = 20 = 50 = 10 = 20 = 50 = 10 = 20 = Source: own simulations

4 310 L. Střelec, M. Stehlík II: Size of the, 2 and 3 tests for = 0.05 θ 1 = θ 2 = θ 3 = 1, = 10 = 20 = 50, = 10 = 20 = = function (14) for parameters setup mentioned above is reported in Tab. III V. Firstl, Tab. III presents power of, 2 and 3 tests against mixture of three exponential components for θ 1 = 3, θ 3 = 5 and θ 1 = 7. In Tab. IV power of, 2 and 3 tests against mixture of three exponential components for θ 1 = 10 and θ 1 = 7 is presented. Finall, last Tab. V presents power of, 2 and 3 tests against mixture of three exponential components for θ 1 = 10 and θ 1 = 7, θ 3 = 10. As it can be seen from Tab. III V, the power of the, 2 and 3 tests increases with scale parameters θ = (θ 2, θ 3 ). For example the power of the most powerful 3 test against mixture of three exponential components for θ 1 = 3, = 50 and component weights and is for θ 3 = 5, for θ 3 = 7, and finall for θ 3 = 10. For fixed θ the highest power is obtained for component weights, and θ 2 < θ 3. On the other hand the lowest power is obtained for high component weight = 0.80, which means that first component of mixture of three exponential components is predominant and this lower contamination is for, 2 and 3 tests hard to detect. The differences in power of, 2 and 3 tests for fixed θ, and component weights and are small the highest difference in power of mentioned tests is against mixture of three exponential components for θ 1 = 7, = 50 and component weights and where power of test is and power of the most powerful 2 test is 0.662, i.e. difference between the most powerful 2 test and the less powerful test is Small differences between the most and less powerful tests mean that powers of analzed, 2 and 3 tests are comparable. It can be also demonstrated b following percent proportion of cases in which analzed tests show the highest power as it is evident from Tab. III V, the test shows the highest power in 3.6 percent of analzed cases, the 2 test in 58.8 percent of analzed cases, and finall the 3 shows the highest power in 37.6 percent of analzed cases. Based on simulation results reported above, we can also state that the 3 test outperforms the and 2 tests in most cases for < 0.40, while the 2 test outperforms the and 3 tests for COCLUSIOS Using of exponential distribution is broad e.g. the exponential distribution is one of the most widel used lifetime distribution in reliabilit engineering (see Stehlík and Wagner, 2011). Homogeneit testing provided in this paper can be also of importance for studing of mixed risks in portfolio (see e.g. Potocký, 2008). As can be seen from results presented above, the power of analzed, 2 and 3 tests is

5 Comparative simulation stud of likelihood ratio tests for homogeneit of the exponential distribution 311 III: Power of the, 2 and 3 tests against mixture of three exponential components for = 0.05, θ 1 = 5 and θ 1 = 1, θ 2 = 7, = 0.80 test θ 1 = 5 θ 1 = 7 = 10 = 20 = 50 = 10 = 20 = comparable, i.e. differences in power of analzed tests are ver small max. difference in power is between 2 and tests. Similarl, the 2 test shows the highest power in almost 59 percent of analzed cases and the 3 test in almost 38 percent of analzed cases. These comparable results mean that using of the computationall simpler 2 test is recommended for broad usage rather than computationall more expensive 3 test, especiall for large sample sizes.

6 312 L. Střelec, M. Stehlík IV: Power of the, 2 and 3 tests against mixture of three exponential components for = 0.05, θ 1 = 10 and θ 1 = 1, 2 = 5, 3 = 7, = 0.80 test 1 = 1, 2 = 3, 3 = 10 θ 1 = 7 = 10 = 20 = 50 = 10 = 20 = SUMMARY In this stud, power of three likelihood ratio tests for homogeneit testing procedure of the number of components k in the exponential mixture for k = 3, firstl introduced b Stehlík and Ososkov (2003). Therefore, in this stud, we presented and discussed the power of the exact likelihood ratio tests, 2 and 3 against three-component subpopulation alternative i.e. mixture of three exponential components for various parameters setup. For purpose of power comparison M = samples were generated and the proportion of rejections of, 2 and 3 tests was determined.

7 Comparative simulation stud of likelihood ratio tests for homogeneit of the exponential distribution 313 V: Power of the, 2 and 3 tests against mixture of three exponential components for = 0.05, θ 1 = 10 and θ 1 = 1, θ 2 = 7, θ 3 = 10, = 0.80 test θ 1 = 10 θ 1 = 7, θ 3 = 10 = 10 = 20 = 50 = 10 = 20 = As can be seen from results of Monte Carlo simulations the power of the exact likelihood ratio, 2 and 3 tests increases with scale parameters θ = (θ 2, θ 3 ). We found that for fixed the highest power is obtained for component weights, and θ 2 < θ 3. On the other hand the lowest power is obtained for high component weight = But for fixed θ, and component weights and used in this paper we can state that the differences in power of, 2 and 3 tests are small therefore using of the computationall simpler 2 test is recommended for broad usage rather than computationall more expensive 3 test.

8 314 L. Střelec, M. Stehlík Acknowledgement Authors are thankful for constructive comments of the referees. REFERECES POTOCKÝ, R., 2008: On a dividend strateg of insurance companies. E+M Ekonomie a management: 11, 4: ISS RUBLÍK, F., 1989a: On optimalit of the LR tests in the sense of exact slopes. I. General case. Kbernetika, 25, 1: ISS RUBLÍK, F., 1989b: On optimalit of the LR tests in the sense of exact slopes. II. Application to individual distributions. Kbernetika, 25, 2: ISS STEHLÍK, M., 2003: Distribution of exact tests in the exponential famil. Metrika, 57, 2: ISS STEHLÍK, M., 2006: Exact likelihood ratio scale and homogeneit testing of some loss processes. Statistics and Probabilit Letters, 76, 1: ISS STEHLÍK, M., 2008: Homogeneit and scale testing of generalized gamma distribution. Reliabilit Engineering & Sstem Safet, 93, 12: ISS STEHLÍK, M., OSOSKOV, G., 2003: Efficient testing of homogeneit, scale parameters and number of components in the Raleigh mixture. JIR Rapid Communications, E STEHLÍK, M., WAGER, H., 2011: Exact Likelihood Ratio Testing for Homogeneit of the Exponential Distribution. Communications in Statistics Simulation and Computation, 40, 5: ISS STŘELEC, L., STEHLÍK, M., 2012: On Simulation of Exact Tests in Raleigh and ormal Families. Accepted for AIP (American Institute of Phsics) Conference Proceedings ICAAM Address Ing. Luboš Střelec, Ph.D., Ústav statistik a operačního výzkumu, Mendelova univerzita v Brně, Zemědělská 1, Brno, Česká republika, Assoz. Univ.-Prof. Mag. Dr. Milan Stehlík, Institut für angewandte Statistik, Johannes Kepler Universit in Linz, Altenbergerstraße 69, Linz, A-4040, Austria, lubos.strelec@ mendelu.cz, Milan.Stehlik@jku.at