arXiv subjects

Kei Takeuchi

Publications and source records attributed to Kei Takeuchi.

10 recordsLinked to original sources

arXiv2014

Conformal Geometry of Sequential Test in Multidimensional Curved Exponential Family

This article presents a differential geometrical method for analyzing sequential test procedures. It is based on the primal result on the conformal geometry of statistical manifold developed in Kumon, Takemura and Takeuchi (2011). By introducing curvature-type random variables, the condition is first clarified for a statistical manifold to be an exponential family under an appropriate sequential test procedure. This result is further elaborated for investigating the efficient sequential test in a multidimensional curved exponential family. The theoretical results are numerically examined by using von Mises-Fisher and hyperboloid models.

math.ST↗

arXiv2012

Bayesian logistic betting strategy against probability forecasting

We propose a betting strategy based on Bayesian logistic regression modeling for the probability forecasting game in the framework of game-theoretic probability by Shafer and Vovk (2001). We prove some results concerning the strong law of large numbers in the probability forecasting game with side information based on our strategy. We also apply our strategy for assessing the quality of probability forecasting by the Japan Meteorological Agency. We find that our strategy beats the agency by exploiting its tendency of avoiding clear-cut forecasts.

math.PR↗

arXiv2011

Approximations and asymptotics of upper hedging prices in multinomial models

We give an exposition and numerical studies of upper hedging prices in multinomial models from the viewpoint of linear programming and the game-theoretic probability of Shafer and Vovk. We also show that, as the number of rounds goes to infinity, the upper hedging price of a European option converges to the solution of the Black-Scholes-Barenblatt equation.

q-fin.PR↗

arXiv2011

Conformal geometry of statistical manifold with application to sequential estimation

We present a geometrical method for analyzing sequential estimating procedures. It is based on the design principle of the second-order efficient sequential estimation provided in Okamoto, Amari and Takeuchi (1991). By introducing a dual conformal curvature quantity, we clarify the conditions for the covariance minimization of sequential estimators. These conditions are further elabolated for the multidimensional curved exponential family. The theoretical results are then numerically examined by using typical statistical models, von Mises-Fisher and hyperboloid models.

math.ST↗

arXiv2010

New procedures for testing whether stock price processes are martingales

We propose procedures for testing whether stock price processes are martingales based on limit order type betting strategies. We first show that the null hypothesis of martingale property of a stock price process can be tested based on the capital process of a betting strategy. In particular with high frequency Markov type strategies we find that martingale null hypotheses are rejected for many stock price processes.

q-fin.ST↗

arXiv2010

A new formulation of asset trading games in continuous time with essential forcing of variation exponent

We introduce a new formulation of asset trading games in continuous time in the framework of the game-theoretic probability established by Shafer and Vovk (Probability and Finance: It's Only a Game! (2001) Wiley). In our formulation, the market moves continuously, but an investor trades in discrete times, which can depend on the past path of the market. We prove that an investor can essentially force that the asset price path behaves with the variation exponent exactly equal to two. Our proof is based on embedding high-frequency discrete-time games into the continuous-time game and the use of the Bayesian strategy of Kumon, Takemura and Takeuchi (Stoch. Anal. Appl. 26 (2008) 1161--1180) for discrete-time coin-tossing games. We also show that the main growth part of the investor's capital processes is clearly described by the information quantities, which are derived from the Kullback--Leibler information with respect to the empirical fluctuation of the asset price.

q-fin.TR↗

arXiv2009

Sequential optimizing strategy in multi-dimensional bounded forecasting games

We propose a sequential optimizing betting strategy in the multi-dimensional bounded forecasting game in the framework of game-theoretic probability of Shafer and Vovk (2001). By studying the asymptotic behavior of its capital process, we prove a generalization of the strong law of large numbers, where the convergence rate of the sample mean vector depends on the growth rate of the quadratic variation process. The growth rate of the quadratic variation process may be slower than the number of rounds or may even be zero. We also introduce an information criterion for selecting efficient betting items. These results are then applied to multiple asset trading strategies in discrete-time and continuous-time games. In the case of continuous-time game we present a measure of the jaggedness of a vector-valued continuous process. Our results are examined by several numerical examples.

math.PR↗

arXiv2008

Capital process and optimality properties of a Bayesian Skeptic in coin-tossing games

We study capital process behavior in the fair-coin game and biased-coin games in the framework of the game-theoretic probability of Shafer and Vovk (2001). We show that if Skeptic uses a Bayesian strategy with a beta prior, the capital process is lucidly expressed in terms of the past average of Reality's moves. From this it is proved that the Skeptic's Bayesian strategy weakly forces the strong law of large numbers (SLLN) with the convergence rate of O(\sqrt{\log n/n})$ and if Reality violates SLLN then the exponential growth rate of the capital process is very accurately described in terms of the Kullback divergence between the average of Reality's moves when she violates SLLN and the average when she observes SLLN. We also investigate optimality properties associated with Bayesian strategy.

math.ST↗

arXiv2008

Multistep Bayesian strategy in coin-tossing games and its application to asset trading games in continuous time

We study multistep Bayesian betting strategies in coin-tossing games in the framework of game-theoretic probability of Shafer and Vovk (2001). We show that by a countable mixture of these strategies, a gambler or an investor can exploit arbitrary patterns of deviations of nature's moves from independent Bernoulli trials. We then apply our scheme to asset trading games in continuous time and derive the exponential growth rate of the investor's capital when the variation exponent of the asset price path deviates from two.

q-fin.TR↗

arXiv2006

Game-theoretic versions of strong law of large numbers for unbounded variables

We consider strong law of large numbers (SLLN) in the framework of game-theoretic probability of Shafer and Vovk (2001). We prove several versions of SLLN for the case that Reality's moves are unbounded. Our game-theoretic versions of SLLN largely correspond to standard measure-theoretic results. However game-theoretic proofs are different from measure-theoretic ones in the explicit consideration of various hedges. In measure-theoretic proofs existence of moments are assumed, whereas in our game-theoretic proofs we assume availability of various hedges to Skeptic for finite prices.

math.PR↗