Masayuki Kumon

Differential geometry of edgeworth expansions in curved exponential family

SummaryIn order to construct a higher-order asymptotic theory of statistical inference, it is useful to know the Edgeworth expansions of the distributions of related statistics. Based on the differential-geometrical method, the Edgeworth expansions are performed up to the third-order terms for the joint distribution of any efficient estimators and complementary (approximate) ancillary statistics in the case of curved exponential family. The marginal and conditional distributions are also obtained. The roles and meanings of geometrical quantities are elucidated by the geometrical interpretation of the Edgeworth expansions. The results of the present paper provide an indispensable tool for constructing the differential-geometrical theory of statistics.

On a simple strategy weakly forcing the strong law of large numbers in the bounded forecasting game

In the framework of the game-theoretic probability of Shafer and Vovk it is of basic importance to construct an explicit strategy weakly forcing the strong law of large numbers in the bounded forecasting game. We present a simple finite-memory strategy based on the past average of Reality’s moves, which weakly forces the strong law of large numbers with the convergence rate of

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

{O(\sqrt{\log n/n})}

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

. Our proof is very simple compared to a corresponding measure-theoretic result of Azuma (Tôhoku Mathematical Journal, 19, 357–367, 1967) on bounded martingale differences and this illustrates effectiveness of game-theoretic approach. We also discuss one-sided protocols and extension of results to linear protocols in general dimension.

Conformal Geometry of Statistical Manifold with Application to Sequential Estimation

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: Its 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 investors 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.

Identification of Transfer Function Matrix Using Higher-Order Spectra

We consider strong law of large numbers (SLLN) in the framework of game-theoretic probability of Shafer and Vovk (Shafer, G. and Vovk, V. 2001, Probability and Finance: Its Only a Game! (New York: Wiley)). We prove several versions of SLLN for the case that Realitys 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 is assumed, whereas in our game-theoretic proofs we assume availability of various hedges to Skeptic for finite prices.

Conformal geometry of sequential test in multidimensional curved exponential family

Abstract We study capital process behavior in the fair-coin and biased-coin games in the framework of the game-theoretic probability of Shafer and Vovk [11]. 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 Realitys moves. From this it is proved that the Skeptics Bayesian strategy weakly forces the strong law of large numbers (SLLN) with the convergence rate of 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 Realitys moves when she violates SLLN and the average when she observes SLLN. We also investigate optimality properties associated with Bayesian strategy.

Discussion on “Sequential Estimation for Time Series Models” by T. N. Sriram and Ross Iaci

We study multistep Bayesian betting strategies in coin-tossing games in the framework of game-theoretic probability of Shafer and Vovk [12]. We show that by a countable mixture of these strategies, a gambler or an investor can exploit arbitrary patterns of deviations of natures 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 investors capital when the variation exponent of the asset price path deviates from two.