TESTING THE VALIDITY OF THE EXPONENTIAL MODEL BASED ON TYPE II CENSORED DATA USING TRANSFORMED SAMPLE DATA

Size: px

Start display at page:

Download "TESTING THE VALIDITY OF THE EXPONENTIAL MODEL BASED ON TYPE II CENSORED DATA USING TRANSFORMED SAMPLE DATA"

Brian Gallagher
5 years ago
Views:

1 STATISTICA, anno LXXVI, n. 3, 2016 TESTING THE VALIDITY OF THE EXPONENTIAL MODEL BASED ON TYPE II CENSORED DATA USING TRANSFORMED SAMPLE DATA Hadi Alizadeh Noughabi 1 Depatment of Statistics, Univesity of Bijand, Bijand, Ian 1. Intoduction Alizadeh Noughabi and Aghami (2011a) wee the fist to popose a test fo the exponential distibution using tansfomed data based on the complete samples. They tansfomed the oiginal sample into a unifom sample ove [0,1] and then used testing unifomity fo this tansfomed sample. Next, Dhumal and Shike (2012) suggested anothe test fo unifomity of tansfomed obsevations and impoved the exponentiality test using tansfomed data. Moeove, Alizadeh Noughabi and Aghami (2011b) poposed othe tansfomations on the oiginal sample and then poposed othe tests fo the exponential distibution. It is impotant to know that they showed these tests dont depend on the scale paamete of the exponential distibution and hence all these tests ae exact. Also, to computing the test statistics based on tansfomed data, we dont need to estimate the scale paamete unlike the othe tests such as KolmogoovSminov, Camevon Mises, and Andeson-Daling. Recently, again Alizadeh Noughabi and Aghami (2013) intoduced a test fo the nomal distibution using tansfomed data based on the complete samples. Then, they compaed this test with competing tests and showed that thei test has a good pefomance. In this aticle, we ae going to constuct a test fo the exponential distibution using tansfomed data with Type-II censoed data. Goodness of fit tests has been widely used in pactice and many eseaches have poposed diffeent tests. Fo example, see DAgostino and Stephens (1986), Hube-Caol et al. (2002), Thode (2002), Pak and Pak (2003), and Alizadeh Noughabi and Aghami (2012) fo the complete samples and Balakishnan et al. (2007), Lim and Pak (2007), Lin et al. (2008), Habibi Rad et al. (2011) and Pakyai and Balakishnan (2013) fo the censoed samples. It is noticeable that thee ae lots of tests fo the complete samples but thee ae few tests fo censoed samples. In the next section, we apply the popula classic goodness of fit tests. Theefoe a summay of these tests ae given as follows. Let X 1,, X n be a andom sample fom a distibution F with density function 1 Coesponding Autho alizadehhadi@bijand.ac.i

2 222 H. Alizadeh Noughabi f. Also, let g(x; θ) be a paametic family of distibutions. We wish to test the hypothesis H 0 : f(x) = g(x; θ), against the geneal altenative H 1 : f(x) g(x; θ). Came von Mises (1931) poposed a test statistic fo the above hypotheses based on the empiical distibution function given by CH = 1 n 12n + ( ) 2 2i 1 2n p (i), whee p (i) = G(x (i) ; ˆθ) and ˆθ is an estimato of θ. Kolmogoov-Sminov (1933) based on maximum distance between the theoetical distibution and the empiical distibution suggested the following test statistic. KS = max { D +, D }, whee D + = max { } i n p (i) { ; D = max p (i) i 1 }, i = 1,..., n. n Andeson-Daling (1954) used the weight ψ(p) = [p(1 p)] 1 in the following integal n (F n (x) G(x)) 2 ψ(g(x))dg(x) and then obtained a test statistic as A 2 = n 1 n (2i 1) { log(p (i) ) + log(1 p (n i+1) ) }. n In Section 2, some test statistics using tansfomed data fo testing exponentiality based on Type-II censoed data ae intoduced. In Section 3, by Monte Calo simulations, the citical values of the poposed test ae obtained and the powe values of the poposed test ae evaluated and compaed with competing tests. We finally conclude that the tests using tansfomed data have a good pefomance in detecting non-exponentiality. In Section 4, the tests ae employed to analyze eal data sets in the liteatue. A bief conclusion in pesented in Section The poposed tests Given the fist ode statistics of a andom sample of size n, X (1:n),..., X (:n), fom a population with unknown distibution F and pobability function f ove a non-negative suppot with mean θ. We ae inteested in a goodness of fit test fo H 0 : f(x) = 1 ( θ exp x ), θ

3 Testing the Validity of the Exponential Model 223 vesus H 1 : f(x) 1 ( θ exp x ), θ whee θ is unknown. In ode to obtain a test statistic, the following theoem is applied. Theoem 1. Let X 1 and X 2 be two obsevations unde Type-II censoed sample fom a continuous distibution F. Then U = X1 X 1+X 2 is distibuted as g(u) if and only if F is exponential, whee { ( ) 4 1 g(u) = n u + n 0 u 0.5, 4 ( n 1) (u 1) + n 0.5 < u 1, and is the sample size of Type-II censoed sample. Remak 2. When = n, the above theoem educe to the theoem pesented by Alizadeh Noughabi and Aghami (2011a) based on the complete samples. Let X (1:n),..., X (:n) be a Type-II censoed sample. Fist, we tansfom the sample data to U ij = X (1:n),..., X (:n) U ij = X (i:n) X (i:n) + X (j:n), i j, i, j = 1, 2,...,. By the above theoem, unde the null hypothesis, each U i has a distibution with density g(u), and it seems to be appopiate to use the classic tests to test that the distibution of U, is ae g(u) and thus the exponentiality of the distibution of X, is. Theefoe, summay of the poposed tests can be pesented as whee { } i D + = max p (i:n ) X (i:n) X (i:n) + X (j:n), i j CH = ( ) 2 2i 1 2 p (i:n ), KS = max { D +, D }, ; D = max CH KS A 2 { p (i:n ) i 1 }, i = 1,...,, A 2 = 1 (2i 1) { log(p (i:n )) + log(1 p ( i+1:n )) }, and n = n(n 1), = ( 1), p (i:n ) = G(u (i:n )), i = 1,...,. Also, G(u) is the distibution function coesponding to the density function g above, given by { ( 2 1 G(u) = n) u 2 + n u 0 u 0.5, ( n 1) (u 1) 2 + n (u 1) 0.5 < u 1.

4 224 H. Alizadeh Noughabi Lage values of the test statistics indicate that ou censoed sample follows a non-exponential distibution. Remak 3. When = n, ou pocedue educe to the method intoduced by Alizadeh Noughabi and Aghami (2011a) fo testing exponentiality based on the complete samples. 3. Simulation study Fo diffeent sample sizes and diffeent values of, Monte Calo simulations with 30,000 eplicates used to obtain the citical values of ou pocedue. Since the test statistics dont depend on the scale paamete, we obtain the citical values fom Type-II censoed samples of the exponential distibution with mean 1. These values ae pesented in Table 1. We chose the competito tests the same tests discussed in Pak (2005). The test statistics of these tests ae as follows. Bain and Shapio (1983) intoduced two test statistics fo testing exponentality based on Type-II censoed data as z = ( ) ( Z = z ( + 1)( 2)( 3) whee Y 1 = nx (1:n), and ) ( i 2) Yi+1 1 Y i+1 ( i 2, ) 2Yi+1 ( 2) 1 1 Y i+1 Y i = (n i + 1) ( X (i:n) X (i 1:n) ), i = 2,...,. They showed that thei tests outpefom othe tests. Samanta and Schwaz (1988) suggested the following test statistic. W = +1 i=2 +1 j=2 ( Y i) 2 min(i,j) 1 min(i,j)+2 Y. i 1Y j 1 Y i+1 Then they showed that this test has a good pefomance in compaed to z and Z. Pak (2005), based on Kullback-Leible infomation, poposed the following test statistic fo testing exponentiality with Type-II censoed data. ( ( ( )) ) T (n, m, ) = H(n, m, ) + 1 log X (i:n) + (n )X (:n) + 1, n whee H(n, m, ) = 1 n { n } ( log 2m (X (i+m) X (i m) ) 1 ) ( log 1 ), n n and the window size m is a positive intege /2, and X (i) = X (1) if i 1, X (i) = X () if i 1. He showed that his test in compaed to the above tests is poweful against the altenatives with monotone inceasing hazad functions.,

5 Testing the Validity of the Exponential Model 225 TABLE 1 Citical points of the test statistics. CH KS A 2 n α = 0.1 α = 0.05 α = 0.1 α = 0.05 α = 0.1 α =

6 226 H. Alizadeh Noughabi To facilitate compaisons of the powe values of the pesent tests with the powe values of the published tests, we chose the same altenatives pesented in Pak (2005). These altenatives accoding to the type of hazad function ae as: 1. Monotone deceasing hazad: Gamma (shape paamete 0.5), Weibull (shape paamete 0.5, 0.8). 2. Monotone inceasing hazad: Unifom, Gamma (shape paamete 1.5, 2), Weibull (shape paamete 2), Beta (shape paametes: 1 and 2, 2 and 1). 3. Non-monotone hazad: Log-nomal (shape paamete: 0.6, 1.0, 1.2), Beta (shape paametes: 0.5 and 1). The powe values of the pesent tests and also the othe tests fo diffeent values of n and at significance level α = 0.10, wee computed by Monte Calo simulations. The powe esults of the tests ae summaized in Tables 2, 3 and 4. It is evident fom tables that thee is not a unifom most poweful test against all altenatives. Theefoe, we discuss on the powe values of the tests in tem of type of altenative. A unifom supeioity of the poposed tests to all competitos against the altenatives with monotone deceasing hazad functions is clealy evident. Fo this type of altenatives, the poposed test A 2 has the most powe and the Pak, s test T has the smallest powe. Fo these altenatives, all poposed tests (CH, KS, A 2 ) have a high powe and the diffeence powe between these tests with the competitos ae substantial. Fo altenatives with inceasing hazad functions the Pak, s test T has the most powe. Howeve, the diffeence powe between the poposed tests and T is not substantial and when inceases the diffeence is deceases. Also, the poposed tests have bette powes than z and Z tests. Finally, fo the altenative with a non-monotone hazad function, no single test unifomly dominates othe tests and it is seen that sometimes the poposed tests ae poweful and sometimes the othe tests. Fo example, the test T has highe powe than othes against log-nomal(0,1) and the poposed tests ae poweful than the competitos against Beta(0.5,1). 4. Illustative examples Two eal examples ae analyzed below to show the behavio of the poposed tests when only a Type-II censoed sample is available. Example 4. We apply the data pesented by Balakishnan and Chen (1999) which ae fom a life expectancy expeiment. They consideed twenty thee ball beaings in the expeiment and ecoded the data coesponding to the millions of evolutions befoe failue fo each of the beaings. The expeiment was teminated once the twentieth ball failed. The data ae 17.88, 28.92, 33.00, 41.52, 42.12, 45.60, 48.48, 51.84, 51.96, 54.12, 55.56, 67.80, 68.64, 68.65, 68.88, 84.12, 93.12, 96.64, , The poposed pocedues ae employed fo investigating that this dataset follows an exponential distibution. In this example, n = 23, = 20 and the values of

7 Testing the Validity of the Exponential Model 227 TABLE 2 Powe compaison at 10% significance level: monotone deceasing hazad altenatives. T ests altenative n z Z W T CH KS A 2 Gamma(0.5) Weibull(0.5) Weibull(0.8) Gamma(0.5) Weibull(0.5) Weibull(0.8)

8 228 H. Alizadeh Noughabi TABLE 3 Powe compaison at 10% significance level: monotone inceasing hazad altenatives. T ests altenative n z Z W T CH KS A 2 Unifom Weibull(2) Gamma(1.5) Gamma(2) Beta(1,2) Beta(2,1) Unifom Weibull(2) Gamma(1.5) Gamma(2) Beta(1,2) Beta(2,1)

9 Testing the Validity of the Exponential Model 229 TABLE 4 Powe compaison at 10% significance level: non-monotone hazad altenatives. T ests altenative n z Z W T CH KS A 2 Log-nomal(0.6) Log-nomal(1.0) Log-nomal(1.2) Beta(0.5,1) Log-nomal(0.6) Log-nomal(1.0) Log-nomal(1.2) Beta(0.5,1)

10 230 H. Alizadeh Noughabi the test statistics ae CH = 6.743, KS = , A 2 = Example 5. The following data wee oiginally pesented by Xia et al.(2009) and then wee analyzed by Saaoglu et al. (2012) unde pogessively Type-II censoing. These data ae beaking stengths of jute fibe fo diffeent gauge lengths. Hee, we assume that only the 24/30 smallest beaking stengths fo 20 mm gauge length wee obseved , 45.58, 48.01, 71.46, 83.55, 99.72, , , , , , , , , , , , , , , , , , In this example, we want to know this dataset can be modeled by an exponential distibution. In this case, n = 30, = 24 and the values of the poposed test statistics ae CH = , KS = , A 2 = At the 5% significance level, the citical values fo CH, KS and A 2 ae 2.118, and , espectively. Since the values of the test statistics ae smalle than the citical points, the exponential hypothesis is not ejected at the significance level of So, thee is evidence that shows that the data can be modeled by using an exponential distibution. This esult agees with the conclusion eached by othe tests mentioned above. 5. Concluding emaks In this pape, we have developed a new pocedue fo testing exponentiality based on Type-II censoed data using tansfomed data. Then, based on the famous classic tests, thee tests have been poposed. We have shown by Monte Calo simulation study that the poposed tests outpefom all othe consideed tests fo altenatives with deceasing monotone hazad functions, which ae used in many eal-life applications. Fo othe altenatives also the poposed tests have a consideable powe against some distibutions. Moeove, the data examples showed that the poposed tests ae easonable when used in eal obsevations. Hee, exponentiality tests have been constucted based on Type-II censoed data. Fo complete samples these tests educe to the pocedue intoduced by Alizadeh Noughabi and Aghami (2011a). As we know, in eliability and life-testing expeiments thee ae diffeent censoing schemes and hence testing exponentiality fo obsevations of othe censoing schemes like the pogessively Type-II and hybid censoed data is an impotant issue. Some authos developed tests fo the pogessively Type-II censoing such as Balakishnan et al. (2004), Balakishnan et al. (2007), Habibi Rad et al. (2011), and Pakyai and Balakishnan (2012, 2013). Moeove, we know that thee is not any goodness of fit test unde hybid censoed data. Theefoe, in futue wok, it would be inteesting to deive a goodness of fit test fo the obsevations obtained by the hybid censoing scheme. We ae cuently woking on the poposed tests hee to extend fo obsevations obtained unde pogessively censoing o hybid censoing and hope to epot these finding in futue papes.

11 Testing the Validity of the Exponential Model 231 Acknowledgements The autho thanks the Edito and anonymous efeees fo making some valuable suggestions which led to a consideable impovement in the pesentation of this manuscipt. Refeences H. Alizadeh Noughabi N. R. Aghami (2011a). Testing exponentiality using tansfomed data. Jounal of Statistical Computation and Simulation, 81, H. Alizadeh Noughabi N. R. Aghami (2011b). Testing exponentiality based on chaacteizations of the exponential distibution. Jounal of Statistical Computation and Simulation, 81, H. Alizadeh Noughabi N. R. Aghami (2012). Geneal teatment of goodness-of-fit tests based on Kullback-Leible infomation. Jounal of Statistical Computation and Simulation, 83, H. Alizadeh Noughabi N. R. Aghami (2013). Testing nomality using tansfomed data. Communication in Statistics: Theoy and Methods, 42, H. Alizadeh Noughabi A goodness of fit test fo Laplace distibution against symmetic altenatives, using tansfomed data. (Submitted). T. W. Andeson, D. A. Daling (1954). A test of goodness of fit. Jounal of Ameican Statistical Association, 49, N. Balakishnan, W. Chen (1999). Handbook of tables fo ode statistics fom Lognomal distibutions with applications. Spinge. N. Balakishnan, A. Habibi Rad, N. R. Aghami (2007). Testing exponentiality based on KullbackLeible infomation with pogessively type-ii censoed data. IEEE Tansactions on Reliability, 56, N. Balakishnan, H.K.T. Ng, N. Kannan (2004). Goodness-of-fit tests based on spacings fo pogessively Type-II censoed data fom a geneal locationscale distibution. IEEE Tansactions on Reliability, 53, C. W. Bain, S. S. Shapio (1983). A egession test fo exponentiality: censoed, complete samples. Technometics, 25, R.B. D Agostino, M. A. Stephens (1986). Goodness-of-Fit Techniques. New Yok, Macel Dekke. B. R. Dhumal, D. T. Shike (2012). A modified test fo testing exponentiality using tansfomed data. Jounal of Statistical Computation and Simulation, 84, A. Habibi Rad, F. Yousefzadeh, N. Balakishnan (2011). Goodness-of-fit test based on KullbackLeible infomation fo pogessively Type-II censoed data. IEEE Tansactions on Reliability, 60,

12 232 H. Alizadeh Noughabi C. Hube-Caol, N. Balakishnan, M. S. Nikulin, M. Mesbah (2002). Goodness-of-fit tests and model validity. Boston, Basel, Belin: Bikhuse. A. N. Kolmogoov (1933). Sulla Deteminazione Empiica di une legge di Distibuzione. Gionale dellintituto Italiano degli Attuai, 4, J. Lim, S. Pak (2007). Censoed Kullback-Leible infomation and goodness-offit test with type II censoed data. Jounal of Applied Statistics, 34, C-T. Lin, Y-L. Huang, N. Balakishnan (2008). A new method fo goodness-of-fit testing based on Type-II ight censoed samples. IEEE Tansactions on Reliability, 57, R. Pakyai, N. Balakishnan (2012). A geneal pupose appoximate goodness-of-fit test fo pogessively Type-II censoed data. IEEE Tansactions on Reliability, 61, R. Pakyai, N. Balakishnan (2013). Goodness-of-fit tests fo pogessively Type-II censoed data fom location-scale distibutions. Jounal of Statistical Computation and Simulation, 83, S. Pak (2005). Testing exponentiality based on the Kullback-Leible infomation. IEEE Tansactions on Reliability, 54, S. Pak, D. Pak (2003). Coecting moments fo goodness of fit tests based on two entopy estimates. Jounal of Statistical Computation and Simulation, 73, M. Samanta, C. J. Schwaz (1998). The Shapio-Wilk test fo exponentiality based on censoed data. Jounal of Ameican Statistical Association, 83, C. von Mises (1931). Wahscheinlichkeitsechnung und ihe Anwendung in de Statistik und theoetischen Physik. Deuticke, Leipzig and Vienna. Summay In this aticle, we popose a new appoach to testing exponentiality fo Type-II censoed samples. Ou method convets the oiginal obsevations into tansfomed data which follow fom a special distibution. It is shown that the poposed tests do not depend on the scale paamete of the exponential distibution and hence the tests ae exact. Powe pefomance of the poposed tests is compaed with the existing tests against vaious altenatives. The esults ae impessive and the poposed tests have highe powe than the competing tests. Finally, two eal data fom eliability liteatue as illustate examples ae pesented. Keywods: Testing exponentiality, Type-II censoing, Powe study, Monte Calo simulation.

Central Coverage Bayes Prediction Intervals for the Generalized Pareto Distribution

Central Coverage Bayes Prediction Intervals for the Generalized Pareto Distribution Statistics Reseach Lettes Vol. Iss., Novembe Cental Coveage Bayes Pediction Intevals fo the Genealized Paeto Distibution Gyan Pakash Depatment of Community Medicine S. N. Medical College, Aga, U. P., India