Kei Takeuchi

A Uniformly Asymptotically EfFicient Estimator of a Location Parameter

Abstract Suppose that a sample of size n from a continuous and symmetric population with an unknown parameter is given. We consider a fictitious random subsample of size k drawn from the original sample and construct the best linear estimator based on the subsample. Applying the Rao-Blackwell type argument, we get an estimator which uses the information contained in the whole sample and is supposed to be uniformly efficient for a wide class of distributions. Monte Carlo experiments established that this estimator is highly efficient for small samples of size 10 to 20.

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: 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.

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 (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.

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

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.

Third order asymptotic efficiency of the sequential maximum likelihood estimation procedure

Under suitable regularity conditions, the third order asymptotic bounds for distributions of regular estimators are obtained. It is shown that the modified maximum likelihood estimation procedure combined with appropriate stopping rule is uniformly third order asymptotically efficient in the sense that its asymptotic distribution attains the bound uniformly in stopping rules up to the third order.

On sum of 0–1 random variables I. Univariate case

SummaryDistribution of sum of 0–1 random variables is considered. No assumption is made on the independence of the 0–1 variables. Using the notion of “central binomial moments” we derive distributional properties and the conditions of convergence to standard distributions in a clear and unified manner.

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 [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.

Discretized likelihood methods—asymptotic properties of discretized likelihood estimators (DLE's)

AbstractSuppose thatX1,X2, ...,Xn, ... is a sequence of i.i.d. random variables with a densityf(x, θ). Letcn be a maximum order of consistency. We consider a solution

Bhattacharyya bound of variances of unbiased estimators in nonregular cases

\hat \theta _n