On the Power Performance of Test Statistics for the Generalized Rayleigh Interval Grouped Data
Author(s)
Hatim Solayman Migdadi
Affiliation(s)
Department of Mathematics, Faculty of Science and Information Technology, Jadara University, Irbid, Jordan.
Department of Mathematics, Faculty of Science and Information Technology, Jadara University, Irbid, Jordan.
ABSTRACT
In this paper, the weighted Kolmogrov-Smirnov, Cramer von-Miss and the Anderson Darling test statistics are considered as goodness of fit tests for the generalized Rayleigh interval grouped data. An extensive simulation process is conducted to evaluate their controlling of type 1 error and their power functions. Generally, the weighted Kolmogrov-Smirnov test statistics show a relatively better performance than both, the Cramer von-Miss and the Anderson Darling test statistics. For large sample values, the Anderson Darling test statistics cannot control type 1 error but for relatively small sample values it indicates a better performance than the Cramer von-Miss test statistics. Best selection of the test statistics and highlights for future studies are also explored.
In this paper, the weighted Kolmogrov-Smirnov, Cramer von-Miss and the Anderson Darling test statistics are considered as goodness of fit tests for the generalized Rayleigh interval grouped data. An extensive simulation process is conducted to evaluate their controlling of type 1 error and their power functions. Generally, the weighted Kolmogrov-Smirnov test statistics show a relatively better performance than both, the Cramer von-Miss and the Anderson Darling test statistics. For large sample values, the Anderson Darling test statistics cannot control type 1 error but for relatively small sample values it indicates a better performance than the Cramer von-Miss test statistics. Best selection of the test statistics and highlights for future studies are also explored.
KEYWORDS
Generalized Rayleigh Distribution, Interval Grouped Data, Goodness of Fit Tests, Empirical Type 1 Error, Power Function
Generalized Rayleigh Distribution, Interval Grouped Data, Goodness of Fit Tests, Empirical Type 1 Error, Power Function
Cite this paper
Migdadi, H. (2015) On the Power Performance of Test Statistics for the Generalized Rayleigh Interval Grouped Data. Open Journal of Statistics, 5, 474-482. doi: 10.4236/ojs.2015.55049.
Migdadi, H. (2015) On the Power Performance of Test Statistics for the Generalized Rayleigh Interval Grouped Data. Open Journal of Statistics, 5, 474-482. doi: 10.4236/ojs.2015.55049.
References
[1] Pipper, C.B. and Ritz, C. (2006) Cheking the Grouped Data Version of Cox Model for Interval Grouped Survival Data. Scandinavian Journal of Statistics, 10, 1467-1469. [2] Aludaat, K.M., Alodat, M.T. and Alodat, T.T. (2008) Parameter Estimation of Burr Type X Distribution for Grouped Data. Applied Mathematical Sciences, 2, 415-423. [3] Migdadi, H.S. and Al-Batah, M.S. (2014) Bayesian Inference Based on the Interval Grouped Data from the Weibull Model with Application. British Journal of Mathematics & Computer Science, 4, 1170-1183.
http://dx.doi.org/10.9734/BJMCS/2014/7930 [4] Pearson, K. (1900) On a Criterion That a Given System of Deviations from the Probable in the Case of a Correlated System of Variables Is Such That It Can Be Reasonably Supposed to Have Arisen from Random Sampling. Philosophical Magazine, 50, 157-175.
http://dx.doi.org/10.1080/14786440009463897 [5] Best, D.J. and Rayner, J.C.W. (2007) Chi Squared Components for Tests of Fit and Improved Models for the Grouped Exponential Distribution. Computational Statistics and Data Analysis, 51, 3946-3954.
http://dx.doi.org/10.1016/j.csda.2006.03.014 [6] Best, D.J. and Rayner, J.C.W. (2006) Improved Testing for the Binomial Distribution Using Chi Squared Components with Data Dependent Cells. Journal of Statistical Computation and Simulation, 76, 75-81.
http://dx.doi.org/10.1080/00949650412331320891 [7] Choulakian, V., Lockhart, R.A. and Stephens, M.A. (1994) Cramer-von Mises Test of Discrete Distribution. The Canadian Journal of Statistics, 22, 125-137.
http://dx.doi.org/10.2307/3315828 [8] Spinelli, J.J. and Stephens, M.A. (1997) Cramer-von Mises Tests of Fit for the Poisson Distribution. The Canadian Journal of Statistics, 25, 257-268.
http://dx.doi.org/10.2307/3315735 [9] Spinelli, J.J. (2001) Testing of Fit for the Grouped Exponential Distribution. The Canadian Journal of Statistics, 29, 451-458.
http://dx.doi.org/10.2307/3316040 [10] Baklizi, A. (2006) Weighted Kolmogrov-Simirnov Type Tests for Grouped Rayleigh Data. Applied Mathematical Modeling, 30, 437-445.
http://dx.doi.org/10.1016/j.apm.2005.05.012 [11] Schmid, P. (1988) On the Kolmogrov and Smirnov Limit Theorems for Discontinuous Distribution Functions. The Annals of Mathematical Statistics, 29, 1011-1027.
http://dx.doi.org/10.1214/aoms/1177706438 [12] Pettitt. A.N. and Stephens. M.A. (1977) The Kolmogrov-Smirnov Goodness-of-Fit Statistic with Discrete and Grouped Data. Technometrics, 19, 205-210.
http://dx.doi.org/10.1080/00401706.1977.10489529 [13] Conover, W.J. (1992) A Kolmogrov Goodness-of-Fit Test for Discontinuous Distributions. Journal of the American Statistical Association, 6, 591-596.
http://dx.doi.org/10.1080/01621459.1972.10481254 [14] Reidwyl, H. (1967) Goodness of Fit. Journal of the American Statistical Association, 6, 390-398. [15] Maag, U.R., Streit. P. and Drouilly, P.A. (1973) Goodness-of-Fit Test for Grouped Data. Journal of the American Statistical Association, 68, 462-465.
http://dx.doi.org/10.1080/01621459.1973.10482456 [16] Damianou, C. and Kemp, A.W. (1990) New Goodness of Fit Statistics for Discrete and Continuous Data. American Journal of Mathematics, and Management Science, 10, 275-307. [17] Gulati, S. and Neus, J. (2003) Goodness of Fit Statistics for the Exponential Distribution When the Data Are Grouped. Communications in Statistics, Theory and Methods, 32, 681-700.
http://dx.doi.org/10.1081/STA-120018558 [18] Richard, A., Lockhart, J., Spinelli, J.J. and Stephens, M.A. (2007) Cramér-Von Mises Statistics for Discrete Distributions with Known Parameters. The Canadian Journal of Statistics, 35, 125-133. [19] Ampai, T. and Kanisa, C. (2011) A Power of Comparison of Goodness of Fit Tests for Exponential Distribution with Grouped Data. Thailand Statistician, 9, 37-49. [20] Raqab, M.Z. and Kundu, D. (2006) Burr Type X Distribution: Revisited. Journal of Probability and Statistical Sciences, 4, 179-193. [21] Kundua, D. and Raqabb, M.Z. (2005) Generalized Rayleigh Distribution. Different Methods of Estimations. Computational Statistics & Data Analysis, 49, 187-200. [22] Bradley, J.V. (1978) Robustness. British Journal of Mathematics and Statistical Psychology, 31, 144-151.
http://dx.doi.org/10.1111/j.2044-8317.1978.tb00581.x
[1] Pipper, C.B. and Ritz, C. (2006) Cheking the Grouped Data Version of Cox Model for Interval Grouped Survival Data. Scandinavian Journal of Statistics, 10, 1467-1469. [2] Aludaat, K.M., Alodat, M.T. and Alodat, T.T. (2008) Parameter Estimation of Burr Type X Distribution for Grouped Data. Applied Mathematical Sciences, 2, 415-423. [3] Migdadi, H.S. and Al-Batah, M.S. (2014) Bayesian Inference Based on the Interval Grouped Data from the Weibull Model with Application. British Journal of Mathematics & Computer Science, 4, 1170-1183.
http://dx.doi.org/10.9734/BJMCS/2014/7930 [4] Pearson, K. (1900) On a Criterion That a Given System of Deviations from the Probable in the Case of a Correlated System of Variables Is Such That It Can Be Reasonably Supposed to Have Arisen from Random Sampling. Philosophical Magazine, 50, 157-175.
http://dx.doi.org/10.1080/14786440009463897 [5] Best, D.J. and Rayner, J.C.W. (2007) Chi Squared Components for Tests of Fit and Improved Models for the Grouped Exponential Distribution. Computational Statistics and Data Analysis, 51, 3946-3954.
http://dx.doi.org/10.1016/j.csda.2006.03.014 [6] Best, D.J. and Rayner, J.C.W. (2006) Improved Testing for the Binomial Distribution Using Chi Squared Components with Data Dependent Cells. Journal of Statistical Computation and Simulation, 76, 75-81.
http://dx.doi.org/10.1080/00949650412331320891 [7] Choulakian, V., Lockhart, R.A. and Stephens, M.A. (1994) Cramer-von Mises Test of Discrete Distribution. The Canadian Journal of Statistics, 22, 125-137.
http://dx.doi.org/10.2307/3315828 [8] Spinelli, J.J. and Stephens, M.A. (1997) Cramer-von Mises Tests of Fit for the Poisson Distribution. The Canadian Journal of Statistics, 25, 257-268.
http://dx.doi.org/10.2307/3315735 [9] Spinelli, J.J. (2001) Testing of Fit for the Grouped Exponential Distribution. The Canadian Journal of Statistics, 29, 451-458.
http://dx.doi.org/10.2307/3316040 [10] Baklizi, A. (2006) Weighted Kolmogrov-Simirnov Type Tests for Grouped Rayleigh Data. Applied Mathematical Modeling, 30, 437-445.
http://dx.doi.org/10.1016/j.apm.2005.05.012 [11] Schmid, P. (1988) On the Kolmogrov and Smirnov Limit Theorems for Discontinuous Distribution Functions. The Annals of Mathematical Statistics, 29, 1011-1027.
http://dx.doi.org/10.1214/aoms/1177706438 [12] Pettitt. A.N. and Stephens. M.A. (1977) The Kolmogrov-Smirnov Goodness-of-Fit Statistic with Discrete and Grouped Data. Technometrics, 19, 205-210.
http://dx.doi.org/10.1080/00401706.1977.10489529 [13] Conover, W.J. (1992) A Kolmogrov Goodness-of-Fit Test for Discontinuous Distributions. Journal of the American Statistical Association, 6, 591-596.
http://dx.doi.org/10.1080/01621459.1972.10481254 [14] Reidwyl, H. (1967) Goodness of Fit. Journal of the American Statistical Association, 6, 390-398. [15] Maag, U.R., Streit. P. and Drouilly, P.A. (1973) Goodness-of-Fit Test for Grouped Data. Journal of the American Statistical Association, 68, 462-465.
http://dx.doi.org/10.1080/01621459.1973.10482456 [16] Damianou, C. and Kemp, A.W. (1990) New Goodness of Fit Statistics for Discrete and Continuous Data. American Journal of Mathematics, and Management Science, 10, 275-307. [17] Gulati, S. and Neus, J. (2003) Goodness of Fit Statistics for the Exponential Distribution When the Data Are Grouped. Communications in Statistics, Theory and Methods, 32, 681-700.
http://dx.doi.org/10.1081/STA-120018558 [18] Richard, A., Lockhart, J., Spinelli, J.J. and Stephens, M.A. (2007) Cramér-Von Mises Statistics for Discrete Distributions with Known Parameters. The Canadian Journal of Statistics, 35, 125-133. [19] Ampai, T. and Kanisa, C. (2011) A Power of Comparison of Goodness of Fit Tests for Exponential Distribution with Grouped Data. Thailand Statistician, 9, 37-49. [20] Raqab, M.Z. and Kundu, D. (2006) Burr Type X Distribution: Revisited. Journal of Probability and Statistical Sciences, 4, 179-193. [21] Kundua, D. and Raqabb, M.Z. (2005) Generalized Rayleigh Distribution. Different Methods of Estimations. Computational Statistics & Data Analysis, 49, 187-200. [22] Bradley, J.V. (1978) Robustness. British Journal of Mathematics and Statistical Psychology, 31, 144-151.
http://dx.doi.org/10.1111/j.2044-8317.1978.tb00581.x