A k-sample homogeneity test: the Harmonic Weighted Mass index
Yıl 2012,
Cilt: 4 Sayı: 1, 17 - 39, 01.04.2012
Jeroen Hinloopen
Rien J.lm. Wagenvoort
Charles Van Marrewijk
Öz
We propose a quantification of the p-p plot that assigns equal weight to all distances between the respective distributions: the surface between the p-p plot and the diagonal. This surface is labelled the Harmonic Weighted Mass (HWM) index. We introduce the diagonal-deviation (d-d) plot that allows the index to be computed exactly under all circumstances. This two-dimensional d-d plot accommodates a straightforward extension to the k-sample HWM index, with k > 2. A Monte Carlo simulation based on an example involving long-term sovereign credit ratings illustrates the power of the HWM test.
Kaynakça
- Anderson, T.W. and D.A. Darling (1952). Asymptotic theory of certain goodness of fit criteria based on stochastic processes. Annals of Mathematical Statistics, 23, 193-212.
- Cramér, H. (1928). On the composition of elementary errors II: statistical applications. Skandinavisk Aktuarietidskrift, 11, 141-180.
- Dufour, J.M. (1995). Monte Carlo tests with nuisance parameters: a general approach to finite-sample inference and non-standard asymptotics in econometrics. Technical report C.R.D.E., Université de Montreal.
- Dufour, J.M. and J.F. Kiviet (1998). Exact inference methods for first-order autoregressive distributed lags models. Econometrica, 66, 79-104.
- Fisz, M. (1960). On a result by M. Rosenblatt concerning the von Mises-Smirnov test. Annals of Mathematical Statistics, 31, 427-429.
- Girling, A.J. (2000). Rank statistics expressible as integrals under PP- plots and receiver operating characteristics curves. Journal of the Royal Statistical Society B, 62, 367-382.
- Hinloopen, J. (1997). Research and development, product differentiation, and robust estimation. Ph.D.-thesis, European University Institute.
- Hinloopen, J. and R.J.L.M. Wagenvoort (2010). Identifying All Distinct Sample P-P Plots, with an Application to the exact Finite Sample Distribution of the L–FCvM Test Statistic. Tinbergen Institute Discussion Paper, TI 2010-083/1.
- Holmgren, E.B. (1995). The p-p plot as a method of comparing treatment effects. Journal of the American Statistical Society, 90, 360-365.
- IMF (2011). International Financial Statistics. http://www.imf.org (accessed March 31, 2012).
- Johnson, B.McK. and T. Killeen (1983). An explicit formula for the C.D.F. of the L1 norm of the Brownian bridge. The Annals of Probability, 11, 807-808.
- Kiefer, J. (1959). K-sample analogues of the Kolmogorov-Smirnov and Cramer-v.Mises tests. Annals of Mathematical Statistics, 30, 420-447.
- Kolmogorov, A.N. (1933). Sulla determinizazione empirica delle leggi di probabilita. Giornale dell ’Istituto Italiano Attuari, 4, 1-11.
- Kuiper, N.H. (1960). Tests concerning random points on a circle. Koninklijke Nederlandse Akademie van Wetenschappen, The Netherlands.
- Rosenblatt, M. (1952). Limit theorems associated with variants of the von Mises statistic. Annals of Mathematical Statistics, 23, 617-623.
- Scholz, F.W. and M.A. Stephens (1987). K-sample Anderson-Darling tests. Journal of the American Statistical Association, 82, 918-924.
- Schmidt, F. and M. Trede (1995). A distribution free test for the two sample problem for general alternatives. Computational Statistics & Data Analysis, 20, 409-419.
- Smirnov, N.V. (1939). On the deviation of the empirical distribution function. Mathematicheskii Sbornik, 6, 3-26.
- Von Mises, R. (1931). Wahrscheinlichkeitsrechnung. Vienna: Deuticke.
- Wilk, M.B. and R. Gnanadesikan (1968). Probability plotting methods for the analysis of data. Biometrika, 55, 1-17.
Yıl 2012,
Cilt: 4 Sayı: 1, 17 - 39, 01.04.2012
Jeroen Hinloopen
Rien J.lm. Wagenvoort
Charles Van Marrewijk
Kaynakça
- Anderson, T.W. and D.A. Darling (1952). Asymptotic theory of certain goodness of fit criteria based on stochastic processes. Annals of Mathematical Statistics, 23, 193-212.
- Cramér, H. (1928). On the composition of elementary errors II: statistical applications. Skandinavisk Aktuarietidskrift, 11, 141-180.
- Dufour, J.M. (1995). Monte Carlo tests with nuisance parameters: a general approach to finite-sample inference and non-standard asymptotics in econometrics. Technical report C.R.D.E., Université de Montreal.
- Dufour, J.M. and J.F. Kiviet (1998). Exact inference methods for first-order autoregressive distributed lags models. Econometrica, 66, 79-104.
- Fisz, M. (1960). On a result by M. Rosenblatt concerning the von Mises-Smirnov test. Annals of Mathematical Statistics, 31, 427-429.
- Girling, A.J. (2000). Rank statistics expressible as integrals under PP- plots and receiver operating characteristics curves. Journal of the Royal Statistical Society B, 62, 367-382.
- Hinloopen, J. (1997). Research and development, product differentiation, and robust estimation. Ph.D.-thesis, European University Institute.
- Hinloopen, J. and R.J.L.M. Wagenvoort (2010). Identifying All Distinct Sample P-P Plots, with an Application to the exact Finite Sample Distribution of the L–FCvM Test Statistic. Tinbergen Institute Discussion Paper, TI 2010-083/1.
- Holmgren, E.B. (1995). The p-p plot as a method of comparing treatment effects. Journal of the American Statistical Society, 90, 360-365.
- IMF (2011). International Financial Statistics. http://www.imf.org (accessed March 31, 2012).
- Johnson, B.McK. and T. Killeen (1983). An explicit formula for the C.D.F. of the L1 norm of the Brownian bridge. The Annals of Probability, 11, 807-808.
- Kiefer, J. (1959). K-sample analogues of the Kolmogorov-Smirnov and Cramer-v.Mises tests. Annals of Mathematical Statistics, 30, 420-447.
- Kolmogorov, A.N. (1933). Sulla determinizazione empirica delle leggi di probabilita. Giornale dell ’Istituto Italiano Attuari, 4, 1-11.
- Kuiper, N.H. (1960). Tests concerning random points on a circle. Koninklijke Nederlandse Akademie van Wetenschappen, The Netherlands.
- Rosenblatt, M. (1952). Limit theorems associated with variants of the von Mises statistic. Annals of Mathematical Statistics, 23, 617-623.
- Scholz, F.W. and M.A. Stephens (1987). K-sample Anderson-Darling tests. Journal of the American Statistical Association, 82, 918-924.
- Schmidt, F. and M. Trede (1995). A distribution free test for the two sample problem for general alternatives. Computational Statistics & Data Analysis, 20, 409-419.
- Smirnov, N.V. (1939). On the deviation of the empirical distribution function. Mathematicheskii Sbornik, 6, 3-26.
- Von Mises, R. (1931). Wahrscheinlichkeitsrechnung. Vienna: Deuticke.
- Wilk, M.B. and R. Gnanadesikan (1968). Probability plotting methods for the analysis of data. Biometrika, 55, 1-17.