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Distribution free testing of goodness of fit in a one dimensional parameter space. (English) Zbl 1396.62094
Summary: We propose two versions of asymptotically distribution free empirical processes. When a composite null hypothesis contains a family of distributions indexed by a one dimensional parameter space, and when that single parameter is estimated by maximum likelihood, the resulting distribution free goodness of fit tests are simpler than tests applying the Khmaladze transformation. For the Pareto distribution, the process we advocate is especially simple. The theory is illustrated by fitting the Pareto distribution to threshold exceedances of stock returns, and the Weibull distribution to fibre strength data.

MSC:
62G30 Order statistics; empirical distribution functions
62G10 Nonparametric hypothesis testing
62P05 Applications of statistics to actuarial sciences and financial mathematics
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[1] Cramer, H., Mathematical methods of statistics, (1946), Princeton University Press · Zbl 0063.01014
[2] Erdelyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., Tables of integral transforms. vol. I, (1954), McGraw Hill · Zbl 0055.36401
[3] Hull, J. C., Risk management and financial institutions, (2012), Wiley
[4] Khmaladze, E. V., Martingale approach in the theory of goodness-of-fit tests, Theory Probab. Appl., 26, 240-257, (1981) · Zbl 0481.60055
[5] Khmaladze, E. V., Unitary transformations, empirical processes and distribution free testing, Bernoulli J., (2014), (forthcoming)
[6] McNeil, A. J.; Frey, R.; Embrechts, P., Quantitative risk management, (2005), Princeton University Press · Zbl 1089.91037
[7] Shorack, G. R.; Wellner, J. A., Empirical processes with applications to statistics, (1986), Wiley · Zbl 1170.62365
[8] Smith, R. L.; Naylor, J. C., A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution, Appl. Stat., 36, 358-369, (1987)
[9] Taylor, S. J., Asset price dynamics, volatility and prediction, (2005), Princeton University Press
[10] Vo, L.H., Roberts, L.A., 2014. On long memory behaviour and predictability of financial markets. Working Paper 5, School of Economics and Finance, Victoria University, Wellington.
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