Abstract
Khmaladze martingale transformation provides an asymptotically-distribution-free method for a goodness-of-fit test. With its usage not being restricted to testing for normality, it can also be selected to test for a location-scale family of distributions such as logistic and Cauchy distributions. Despite its merits, the Khmaladze martingale transformation, however, could not have enjoyed deserved celebrity since it is computationally expensive; it entails the complex and time-consuming computations, including optimization, integration of a fractional function, matrix inversion, etc. To overcome these computational challenges, this paper proposes a fast algorithm which provides a solution to the Khmaladze martingale transformation method. To that end, the proposed algorithm is equipped with a novel strategy, named integration-in-advance, which rigorously exploits the structure of the Khmaladze martingale transformation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Birnbaum ZW (1942) An inequality for Mill’s ratio. Ann Math Stat 13:245–246
D’Agostino RB (1971) An omnibus test of normality for moderate and large size samples. Biometrika 58:341–348
Dufour JM, Farhat A, Gardiol L et al (1998) Simulation-based finite sample normality tests in linear regressions. Econom J 1:154–173
Durbin J (1973) Distribution theory for tests based on the sample distribution function. SIAM, Philadelphia
Dong LB, Giles DEA (2004) An empirical likelihood ratio test for normality. Commun Stat Simul Comput 36(1):197–215
Fan J, Koul HL (2006) Frontiers in statistics. Imperial College Press, London
Gordon RD (1941) Values of Mill’s ratio of area to bounding ordinate of the normal probability integral for large values of the argument. Ann Math Stat 12:364–366
Jarque CM, Bera AK (1980) Efficient tests for normality, homoscedasticity and serial independence of regression residuals. Econ Lett 6(3):255–259
Khmaladze EV (1981) A martingale approach in the thory of goodness-of-fit tests. Theory Probab Its Appl 26:240–257
Khmaladze EV, Koul HL (2004) Martingale transforms of goodness-of-fit tests in regression models. Ann Stat 32:995–1034
Kim J (2018) A fast algorithm for the coordinate-wise minimum distance estimation. J Stat Comput Simul 3:482–497
Koul HL, Sakhanenko L (2005) Goodness-of-fit testing in regression: a finite sample comparison of bootstrap methodology and Khmaladze transformation. Stat Probab Lett 74(3):290–302
Koul HL, Zhu X (2015) Gooness-of-fit testing of error distribution in nonparametric ARCH(1) models. J Multivar Anal 137:141–160
Lilliefors H (1967) On the Kolmogorove–Smirnov test for the exponential distribution with mean unknown. J Am Stat Assoc 64:387–389
Owen AB (1988) Empirical lieklihood ratio confidence intervals for a single functional. Biometrika 75(2):237–249
Qin J, Lawless J (1994) Empirical likelihood and general estimating equations. Ann Stat 22:300–325
Tsigroshvili Z (1998) Some notes on goodness-of-fit tests and innovation martingales. Proc A Razmadze Math Inst 117:89–102
Acknowledgements
The author wants to express his deepest gratitude to the editor and two referees for their constructive comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Kim, J. Implementation of a goodness-of-fit test through Khmaladze martingale transformation. Comput Stat 35, 1993–2017 (2020). https://doi.org/10.1007/s00180-020-00971-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00180-020-00971-7