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

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Abstract

Suppose thatX1,X2, ...,X n , ... is a sequence of i.i.d. random variables with a densityf(x, θ). Letc n be a maximum order of consistency. We consider a solution\(\hat \theta _n \) of the discretized likelihood equation

$$\sum\limits_{i = 1}^n {\log f(X_i ,\hat \theta _n + rc_n^{ - 1} ) - } \sum\limits_{i = 1}^n {\log f(X_i ,\hat \theta _n ) = a_n (\hat \theta _n ,r)} $$

wherea n (θ,r) is chosen so that\(\hat \theta _n \) is asymptotically median unbiased (AMU). Then the solution\(\hat \theta _n \) 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.

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Authors and Affiliations

  1. University of Electro-Communications, University of Tokyo, Tokyo, Japan

    Masafumi Akahira & Kei Takeuchi

Authors
  1. Masafumi Akahira
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  2. Kei Takeuchi
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Additional information

The results of this paper have been presented by the first author at the meeting on “Asymptotic Methods of Statistics” at the Mathematical Institute in Oberwolfach of West Germany, November 1977.

Partially supported by Sakkokai Foundation.

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Akahira, M., Takeuchi, K. Discretized likelihood methods—asymptotic properties of discretized likelihood estimators (DLE's). Ann Inst Stat Math 31, 39–56 (1979). https://doi.org/10.1007/BF02480264

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  • DOI: https://doi.org/10.1007/BF02480264

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