×

Optimal statistical decisions about some alternative financial models. (English) Zbl 1360.62504

Summary: We study Neyman-Pearson testing and Bayesian decision making based on observations of the price dynamics \((X_{t}:t\in [0,T])\) of a financial asset, when the hypothesis is the classical geometric Brownian motion with a given constant growth rate and the alternative is a different random diffusion process with a given, possibly price-dependent, growth rate. Examples of asset price observations are introduced and used throughout the paper to demonstrate the applicability of the theory. By a rigorous mathematical approach, we obtain exact formulae and bounds for the most common statistical characteristics of testing and decision making, such as the power of test (type II error probability), the Bayes factor and its moments (power divergences), and the Bayes risk or Bayes error. These bounds can be much more easily evaluated than the exact formulae themselves and, consequently, they are useful for practical applications. An important theoretical conclusion of this paper is that for the class of alternatives considered neither the risk nor the errors converge to zero faster than exponentially in the observation time \(T\). We illustrate in concrete decision situations that the actual rate of convergence is well approximated by the bounds given in the paper.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
62C10 Bayesian problems; characterization of Bayes procedures
62P20 Applications of statistics to economics
91G70 Statistical methods; risk measures
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alwan, L. C.; Ebrahimi, N.; Soofi, E. S., Information theoretic framework for process control, European Journal of Operational Research, 111, 526-542 (1998) · Zbl 0937.90016
[2] Bera, A. K.; Bilias, Y., The MM, ME, ML, EL, EF and GMM approaches to estimation: a synthesis, Journal of Econometrics, 107, 51-86 (2002) · Zbl 1088.62505
[3] Berry, D. A.; Chaloner, K. M.; Geweke, J. K., Bayesian Analysis in Statistics and Econometrics—Essays in Honor of Arnold Zellner (1996), Wiley: Wiley New York, Chichester
[4] Black, F.; Scholes, M., The pricing of options and corporate liabilities, Journal of Political Economy, 81, 637-654 (1973) · Zbl 1092.91524
[5] Cressie, N.; Read, T. R.C., Multinomial goodness-of-fit tests, Journal of the Royal Statistical Society Series B, 46, 440-464 (1984) · Zbl 0571.62017
[6] Duffie, D., Dynamic Asset Pricing Theory (1996), Princeton University Press: Princeton University Press Princeton
[7] Ebrahimi, N.; Maasoumi, E.; Soofi, E. S., Measuring informativeness of data by entropy and variance, (Slottje, D. J., Advances in Econometrics, Income Distribution and Scientific Methodology—Essays in Honor of Camilo Dagum (1999), Physica: Physica Heidelberg, New York), 61-77 · Zbl 0964.62107
[8] Girsanov, I. V., On transforming a certain class of stochastic processes by absolutely continuous substitution of measures, Theory of Probability and its Applications, 5, 285-301 (1960) · Zbl 0100.34004
[9] Golan, A., Information and entropy econometrics—editor’s view, Journal of Econometrics, 107, 1-15 (2002) · Zbl 1088.62519
[10] Golan, A.; Judge, G. G.; Miller, D. J., Maximum Entropy Econometrics: Robust Estimation with Limited Data (1996), Wiley: Wiley Chichester, New York · Zbl 0884.62126
[11] Imbens, G. W.; Spady, R. H., Confidence intervals in generalized method of moments models, Journal of Econometrics, 107, 87-98 (2002) · Zbl 1030.62021
[12] Imbens, G. W.; Spady, R. H.; Johnson, P., Information theoretic approaches to inference in moment condition models, Econometrica, 66, 333-357 (1998) · Zbl 1055.62512
[13] Karatzas, I.; Shreve, S. E., Brownian Motion and Stochastic Calculus (1991), Springer: Springer Berlin, Heidelberg, New York · Zbl 0734.60060
[14] Kitamura, Y.; Stutzer, M., An information-theoretic alternative to generalized method of moments estimation, Econometrica, 65, 861-874 (1997) · Zbl 0894.62011
[15] Krafft, O.; Plachky, D., Bounds for the power of likelihood ratio tests and their asymptotic properties, The Annals of Mathematical Statistics, 41, 1646-1654 (1970) · Zbl 0214.18003
[16] Liese, F., Hellinger integrals of Gaussian processes with independent increments, Stochastics, 6, 81-96 (1982) · Zbl 0476.60041
[17] Liese, F., Hellinger integrals of diffusion processes, Statistics, 17, 63-78 (1986) · Zbl 0598.60042
[18] Liese, F.; Vajda, I., Convex Statistical Distances (1987), Teubner: Teubner Leipzig · Zbl 0656.62004
[19] Maasoumi, E., A compendium to information theory in economics and econometrics, Econometric Reviews, 12, 2, 137-181 (1993) · Zbl 0769.62003
[20] Maasoumi, E.; Racine, J., Entropy and predictability of stock market returns, Journal of Econometrics, 107, 291-312 (2002) · Zbl 1043.62090
[21] Merton, R. C., Continuous-time Finance (1992), Blackwell: Blackwell Cambridge, MA
[22] Morales, D.; Pardo, L.; Vajda, I., Rényi statistics in directed families of exponential experiments, Statistics, 34, 151-174 (2000) · Zbl 0998.62003
[23] Morales, D.; Pardo, L.; Pardo, M. C.; Vajda, I., Rényi statistics for testing composite hypothesis in general exponential models, Statistics, 38, 133-147 (2004)
[24] Novikov, A. A., On an identity for stochastic integrals, Theory of Probability and its Applications, 17, 717-720 (1972) · Zbl 0284.60054
[25] Polson, N. G.; Roberts, G. O., Bayes factors for discrete observations from diffusion processes, Biometrika, 81, 11-26 (1994) · Zbl 0815.62069
[26] Read, T. R.C.; Cressie, N. A.C., Goodness-of-Fit Statistics for Discrete Multivariate Data (1988), Springer: Springer New York · Zbl 0663.62065
[27] Roberts, G. O.; Stramer, O., On inference for partially observed nonlinear diffusion models using the Metropolis-Hastings algorithm, Biometrika, 88, 603-621 (2001) · Zbl 0985.62066
[28] Samuelson, P. A., Rational theory of warrant pricing, Industrial Management Review, 6, 13-31 (1965)
[29] Sivaganesan, S.; Lingham, R. T., On the asymptotic stability of the intrinsic and fractional Bayes factors for testing some diffusion models, Annals of the Institute of Statistical Mathematics, 54, 500-516 (2002) · Zbl 1023.62038
[30] Stummer, W., The Novikov and entropy conditions of multidimensional diffusion processes with singular drift, Probability Theory and Related Fields, 97, 515-542 (1993) · Zbl 0794.60055
[31] Stummer, W., On a statistical information measure of diffusion processes, Statistics & Decisions, 17, 359-376 (1999) · Zbl 0985.62007
[32] Stummer, W., On a statistical information measure for a generalized Samuelson-Black-Scholes model, Statistics & Decisions, 19, 289-314 (2001) · Zbl 0985.62008
[33] Stummer, W.; Sturm, K. Th., On exponentials of additive functionals of Markov processes, Stochastic Processes and their Applications, 85, 45-60 (2000) · Zbl 0996.60090
[34] Stutzer, M., A Bayesian approach to diagnosis of asset pricing models, Journal of Econometrics, 68, 367-397 (1995) · Zbl 0825.62958
[35] Tsallis, C., Possible generalization of Boltzmann-Gibbs statistics, Journal of Statistical Physics, 52, 479-487 (1988) · Zbl 1082.82501
[36] Vajda, I., Limit theorems for total variation of Cartesian product measures, Studia Scientiarum Mathematicarum Hungarica, 6, 317-333 (1971) · Zbl 0243.62034
[37] Vajda, I., Distances and discrimination rates for stochastic processes, Stochastic Processes and their Applications, 35, 47-57 (1990) · Zbl 0701.62084
[38] Zellner, A., An Introduction to Bayesian Inference in Econometrics (1996), Wiley: Wiley New York · Zbl 0868.62090
[39] Zellner, A., Bayesian Analysis in Econometrics and Statistics—The Zellner View and Papers (1997), Elgar: Elgar Cheltenham-Lyme
[40] Zellner, A.; Tobias, J., Further results on Bayesian method of moments analysis of the multiple regression model, International Economic Review, 42, 121-140 (2001)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.