Joint Statistical Papers of Akahira and Takeuchi

The following sections are included:

• Foreword

• Introduction

• Contents

Prediction sufficiency (adequacy), as it is usually defined in terms of conditional expectations, does imply "real" prediction sufficiency; i.e. sufficiency in terms of risk functions. The converse holds provided we permit the loss to depend on the unknown parameter. This is no longer true if we insist on loss functions which do not involve the unknown parameter. Conditional independence still holds but ordinary sufficiency may fail. If, however, we require equivalence of risk functions, then ordinary sufficiency and, consequently, prediction sufficiency follows.

Suppose that X_i's(i=1, 2,…, n) are independently and identically distributed with the density f(x, θ, ξ), where θ is a real valued parameter and ξ is a real (vector) valued parameter. We consider a (sequence of) estimator(s) which is k-th order asymptotically median unbiased, and define k-th order asymptotic efficiency. We have a formula for the distribution of the second order asymptotically efficient estimator and show that a modified maximum likelihood estimator is second order asymptotically efficient.

Gram-Charlier-Edgeworth type expansion of the sum of i. i. d. random variables without higher order moments are given. Exact formulas for the asymptotic expansion of the density for special cases are given and it is shown that the results are generalized to the case when the density function is approximated by a rational function.

Gram-Charlier-Edgeworth type expansions of the sums of independent random variables without higher order moments are given. It is shown that the asymptotic distribution functions for some cases are stable laws with fractional characteristic exponents, and the asymptotic expansion of the density function for one case is also given.

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The asymptotic expansions of the distributions of the sums of independent identically distributed random variables are given by Edgeworth type expansions when moments do not necessarily exist, but when the density can be approximated by rational functions.

Gram-Charlier-Edgeworth type expansion of the sums of i.i.d. random variables without higher order moments was discussed in the previous papers. In this paper it is shown that the expansion is extended to the multidimensional cases.

For multiparameter exponential family distributions the asymptotic expansion of the distribution of extended regular (ER) estimators is obtained up to the order n^-1. And it is shown that the (modified) maximum likelihood estimator attains the third order asymptotic efficiency among ER best asymptotically normal (BAN) estimators.

For all symmetric loss functions, the generalized Bayes estimator is second order asymptotically efficient in the class A₂ of the all second order asymptotically median unbiased (AMU) estimators and third order asymptotically efficient in the restricted class D of estimators. When the loss function is not symmetric, the generalized Bayes estimator is second order asymptotically efficient in the class A₂ but not third order asymptotically efficient in the restricted wider class C than the class D.

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Suppose that X_i (i = 1, 2,…, n) are distributed according to a two-sided truncated normal distribution with unknown mean θ. It is shown that the maximum probability estimator (MPE) of Weiss and Wolfowitz is asymptotically inadmissible and has smaller concentration of probability than the asymptotically efficient estimator .

Suppose that X₁, X₂,⋯, X_n,⋯ is a sequence of i.i.d. random variables with a density f(x, θ). Let c_n be a maximum order of consistency. We consider a solution of the discretized likelihood equation

where a_n(θ, r) is chosen so that is asymptotically median unbiased (AMU). Then the solution is called a discretized likelihood estimator (DLE). In this paper it is shown in comparison with DLE that a maximum likelihood estimator (MLE) is second order asymptotically efficient but not third order asymptotically efficient in the regular case. Further it is seen that the asymptotic efficiency (including higher order cases) may be systematically discussed by the discretized likelihood methods.

The higher order asymptotic efficiency of the generalized Bayes estimator is discussed in multiparameter cases.

For all symmetric loss functions, the generalized Bayes estimator is second order asymptotically efficient in the class A₂ of the all second order asymptotically median unbiased (AMU) estimators and third order asymptotically efficient in the restricted class D of estimators.

In the asymptotic theory of statistical estimation the authors tried to compare the regular versus non-regular situation to clarify the significance and implication of each of the regularity conditions. This paper summarizes the results far obtained.

Let (𝒳, 𝒜) be a measurable space and 𝒫 a family of probability measures on 𝒜. Let ℬ and 𝒞 be sub σ-algebras of 𝒜 and ℬ₀ a sub σ-algebra of ℬ. It is shown that if ℬ₀ is prediction sufficient (adequate) for ℬ with respect to 𝒞 and 𝒫, and 𝒮 is sufficient for with respect to 𝒫 then 𝒮 is sufficient for ℬ^v𝒞 with respect to 𝒫; that if 𝒫 is homogeneous and (ℬ₀; ℬ, 𝒞) is Markov for 𝒫, and is sufficient for ℬ^V𝒞 with respect to 𝒫, then ℬ₀ is sufficient for ℬ with respect to 𝒫; and by example that the Markov property is necessary for the latter proposition to hold.

In the previous paper it was shown that the maximum likelihood estimator (MLE) was third order asymptotically efficient in multiparameter exponential cases. In this paper it is shown that the result is extended to more general cases. The concept of asymptotic completeness of an estimator is introduced and it is shown that the MLE is higher order asymptotically complete in the appropiate classes.

The concept of asymptotic deficiency is extended to the case when a common parameter θ is estimated from m sets of independent samples each of size n, and the asymptotic deficiencies of some asymptotically efficient estimators relative to the maximum likelihood estimator based on the pooled sample are discussed in the presence of nuisance parameters.

The problem to estimate a common parameter for the pooled sample from the uniform distributions is discussed in the presence of nuisance parameters. The maximum likelihood estimator (MLE) and others are compared and it is shown that the MLE based on the pooled sample is not (asymptotically) efficient.

In the case when the Fisher information is infinity, it is shown that the locally minimum variance of unbiased estimators is equal to zero. Some examples are also given.

Bhattacharyya bound is generalized to nonregular cases when the support of the density depends on the parameter, while it is differentiable several times with respect to the parameter within the support. Some example is discussed, where it is shown that the bound is sharp.

Minimizing is discussed under the unbiasedness condition: and the condition (A):f_i(x) (i = 1,…,p) are linearly independent, , and implies a_k+1 = …=a_p=0.

Assume that X(τ) is a continuous time simple Markov process with a parameter θ. The problem is to choose observation points τ₀ < τ₁ < … < τ_T which provide with the maximum possible information on θ. Suppose that the observation points are equally spaced, that is, for t = 1, …,T, τ_t - τ_t-1 = s is constant. Then the optimum value for s is obtained.

In this papar, the case when the order of consistency depends on the parameter is discussed, and in the simple unstable process the asymptotic means and variances of the log-likelihood ratio test statistic are obtained under the null and the alternative hypotheses. Further its asymptotic distribution is also discussed.

In this paper a definition of asymptotic expectation is given and its fundamental properties are discussed.

The exact forms of the locally minimum variance unbiased estimators and their variances are given in the case of a discontinuous density function.

In this paper we introduce the concept of one-directionality which includes both cases of location (and scale) parameter and selection parameter and also other cases, and establish some theorems for sharp lower bounds and for the existence of zero variance unbiased estimator for this class of non-regular distributions.

Under suitable regularity conditions, the Bhattacharyya type bound for asymptotic variances of estimation procedures is obtained. It is also shown that the modified maximum likelihood estimation procedure attains the bound if the stopping rule is properly determined.

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We consider the estimation problem of a location parameter θ on a sample of size n from a two-sided Weibull type density f(x - θ) = C(α) exp (-|x-θ|^α) for -∞< x < ∞, -∞< θ < ∞ and 1 < α < 3/2, where C(α) = α/{2Γ(1/α)}. Then the bound for the distribution of asymptotically median unbiased estimators is obtained up to the 2α-th order, i.e., the order n^-(2α-1)/2. The asymptotic distribution of a maximum likelihood estimator (MLE) is also calculated up to the 2α-th order. It is shown that the MLE is not 2α-th order asymptotically efficient. The amount of the loss of asymptotic information of the MLE is given.

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.

For semiparametric models, it is shown that, under fairly regularity conditions, the asymptotic deficiency of the maximum likelihood estimator or any regular best asymptotically normal estimator is infinity.

Fisher (1934) derived the loss of information of the maximum likelihood estimator (MLE) of the location parameter in the case of the double exponential distribution. Takeuchi & Akahira (1976) showed that the MLE is not second order asymptotically efficient. This paper extends these results by obtaining the (asymptotic) losses of information of order statistics and related estimators, and by comparing them via their asymptotic distributions up to the second order.

In this paper we consider the sampling properties of the bootstrap process, that is, the empirical process obtained from a random sample of size n (with replacement) of a fixed sample of size n of a continuous distribution. The cumulants of the bootstrap process are given up to the order n^-1 and their unbiased estimation is discussed. Furthermore, it is shown that the bootstrap process has an asymptotic minimax property for some class of distributions up to the order n^-1/2.

In the presence of nuisance parameters, the Bhattacharyya type bound for the asymptotic variance of estimation procedures is obtained. It is shown that the modified maximum likelihood (ML) estimation procedures together with any stopping rule does not attain the bound. Further it is shown that the modified ML estimation procedure with the appropriate stopping rule is second order asymptotically efficient in some class of estimation procedures in the sense that it attains the lower bound for the asymptotic variance in the class.

We consider the estimation problem on a location parameter of the density function with a support of a finite interval and contact of the power α-1 at both endpoints, where 1 < α < 2. Then the bound for the asymptotic distribution of asymptotically median unbiased estimators of θ based on a random sample from the density f₀(x-θ) is obtained. It is also shown that the bias-adjusted maximum likelihood estimator is not asymptotically efficient in the sense that its asymptotic distribution does not uniformly attain the bound.

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In the case of sequential Bernoulli trials, a sufficient condition for a parametric function to be unbiasedly estimable is given and the existence of a discontinuous unbiasedly estimable function is shown using non-randomized sample size procedures.

Usually confidence interval is defined as an interval with preassigned confidence level 1 - α for all the value of parameters. More generally, however we may consider interval estimation procedures with confidence coefficient varying according to the value of the unknown parameter, and associated procedure to estimate the actual level. Such a consideration leads to more general procedures including conditional procedures given the ancillary.

The Bhattacharyya type bound for the variance of unbiased estimators of a location parameter of the double exponential distribution is obtained, and also the loss of information of the maximum likelihood estimator based on the distribution rounded off is discussed.

In the paper, we consider the following problem: Let {π_k} be a sequence satisfying 0≤π_k≤1{k= 1,…, N) and . Then, is there an unordered sampling design such that, for each k=1,…N, the inclusion probability of unit k is equal to π_k? It is shown that it can be solved by the straightforward application of the Minkowski-Farkas theorem.

For a family of one-parameter discrete exponential type distributions, the higher order approximation of randomized confidence intervals derived from the optimum test is discussed. Indeed, it is shown that they can be asymptotically constructed by means of the Edgeworth expansion. The usefulness is seen from the numerical results in the case of Poisson and binomial distributions.

For a random sample (X₁,…, X_n) of size n from a two-sided Gamma type distribution, it is shown that the largest order of consistency is equal to n². In a problem of testing the hypothesis H : θ = θ₀ against K : θ = θ₀ + tn^-2, a test with the rejection region {min_1≤i≤n|X_i| > kn^-2} is given as one with the order, where t ≠ 0 and k is a positive constant. The power function of the test is also asymptotically given.

For a sum of not identically but independently distributed discrete random variables, its higher order large-deviation approximations are given. They are compared with the normal and Edgeworth type approximations in various cases. Consequently, the large-deviation approximations give sufficiently accurate results.

For a family of uniform distributions, it is shown that for any small ε > 0 the average mean squared error (MSE) of any estimator in the interval of θ values of length ε and centered at θ₀ can not be smaller than that of the midrange up to the order o(n^-2) as the size n of sample tends to infinity. The asymptotic lower bound for the average MSE is also shown to be sharp.

The following sections are included:

• Permissions