Asymptotic Expansions for General Statistical Models

Pfanzagl, J.; Wefelmeyer, Wolfgang

doi:10.1007/978-1-4615-6479-9

Cited by 79 publications

(63 citation statements)

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“…Extensions to more complicated models were made by Takeuchi and Morimune (1985) and Taniguchi (1986Taniguchi ( , 1987. Thorough theory of asymptotic efficiency up to the third-order in estimation problems is described by the well-written text books by Akahira and Takeuchi (1981), Pfanzagl (1985), Taniguchi (1991), and Ghosh (1994). In the discussion of second-or third-order optimality in estimation, the main ideas are bias-adjustments and Edgeworth expansions.…”

Section: Brief Historical Reviewmentioning

confidence: 99%

More Higher-Order Efficiency: Concentration Probability

Kano

1998

Journal of Multivariate Analysis

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Based on concentration probability of estimators about a true parameter, thirdorder asymptotic efficiency of the first-order bias-adjusted MLE within the class of first-order bias-adjusted estimators has been well established in a variety of probability models. In this paper we consider the class of second-order biasadjusted Fisher consistent estimators of a structural parameter vector on the basis of an i.i.d. sample drawn from a curved exponential-type distribution, and study the asymptotic concentration probability, about a true parameter vector, of these estimators up to the fifth-order. In particular, (i) we show that third-order efficient estimators are always fourth-order efficient; (ii) a necessary and sufficient condition for fifth-order efficiency is provided; and finally (iii) the MLE is shown to be fifth-order efficient. Academic PressAMS 1991 subject classifications: 62F12.

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Section: Brief Historical Reviewmentioning

confidence: 99%

More Higher-Order Efficiency: Concentration Probability

Kano

1998

Journal of Multivariate Analysis

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“…He discussed a more general nonparametric model of dispersion based on a more general ordering of scale (cf. [65,66]). In line with [4], we focus on the scale ordering proposed by [62].…”

Section: Proofmentioning

confidence: 99%

Cumulative Paired φ-Entropy

Klein

Mangold

Doll

2016

Entropy

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Abstract:A new kind of entropy will be introduced which generalizes both the differential entropy and the cumulative (residual) entropy. The generalization is twofold. First, we simultaneously define the entropy for cumulative distribution functions (cdfs) and survivor functions (sfs), instead of defining it separately for densities, cdfs, or sfs. Secondly, we consider a general "entropy generating function" φ, the same way Burbea et al. (IEEE Trans. Inf. Theory 1982, 28, 489-495) and Liese et al. (Convex Statistical Distances; Teubner-Verlag, 1987) did in the context of φ-divergences. Combining the ideas of φ-entropy and cumulative entropy leads to the new "cumulative paired φ-entropy" (CPE φ ). This new entropy has already been discussed in at least four scientific disciplines, be it with certain modifications or simplifications. In the fuzzy set theory, for example, cumulative paired φ-entropies were defined for membership functions, whereas in uncertainty and reliability theories some variations of CPE φ were recently considered as measures of information. With a single exception, the discussions in the scientific disciplines appear to be held independently of each other. We consider CPE φ for continuous cdfs and show that CPE φ is rather a measure of dispersion than a measure of information. In the first place, this will be demonstrated by deriving an upper bound which is determined by the standard deviation and by solving the maximum entropy problem under the restriction of a fixed variance. Next, this paper specifically shows that CPE φ satisfies the axioms of a dispersion measure. The corresponding dispersion functional can easily be estimated by an L-estimator, containing all its known asymptotic properties. CPE φ is the basis for several related concepts like mutual φ-information, φ-correlation, and φ-regression, which generalize Gini correlation and Gini regression. In addition, linear rank tests for scale that are based on the new entropy have been developed. We show that almost all known linear rank tests are special cases, and we introduce certain new tests. Moreover, formulas for different distributions and entropy calculations are presented for CPE φ if the cdf is available in a closed form.

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“…However , it was not clear what the minimum of loss of information meant, and comparison of asymptotic variance or mean squared error up to the second order failed to establish uniform superiority of the MLE or any other estimators. This impasse was broken through by introducing the median-bias correction and considering the concentration probability of the estimator around the true value up to the order n-1 (see also [5], [8], Pfanzagl and Wefelmeyer (1985), Ghosh (1994)). …”

Section: Third Order Asymptotic Efficiencymentioning

confidence: 99%

Joint Statistical Papers of Akahira and Takeuchi

Bosch¹

2003

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Asymptotic Expansions for General Statistical Models

Cited by 79 publications

References 0 publications

More Higher-Order Efficiency: Concentration Probability

More Higher-Order Efficiency: Concentration Probability

Cumulative Paired φ-Entropy

Joint Statistical Papers of Akahira and Takeuchi

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