Maximum likelihood estimation for a one-sided truncated family of distributions
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Maximum likelihood estimation for a one‑sided truncated family of distributions Masafumi Akahira1 Received: 16 May 2020 / Accepted: 15 October 2020 © Japanese Federation of Statistical Science Associations 2020
Abstract For a one-sided truncated family of distributions with an interest parameter 𝜃 and a truncation parameter 𝛾 as a nuisance parameter, we consider the maximum like𝛾 lihood estimators (MLEs) 𝜃̂ML and 𝜃̂ML of 𝜃 for known 𝛾 and unknown 𝛾 , respec𝛾 tively. In this paper, the stochastic expansions of 𝜃̂ML and 𝜃̂ML are derived, and their second-order asymptotic variances are obtained. The second-order asymptotic loss 𝛾 of a bias-adjusted 𝜃̂ML∗ relative to 𝜃̂ML is also given. The results are a generalization of those for a one-sided truncated exponential family of distributions. Examples on a one-sided truncated Cauchy distribution, a general truncated exponential family, etc. are also given. Keywords Truncated family · Truncated exponential family · Interest parameter · Truncation parameter · Maximum likelihood estimator · Stochastic expansion · Asymptotic variance · Second-order asymptotic loss · Truncated Cauchy distribution Mathematics Subject Classification 62F10 · 62F12
1 Introduction In the presence of nuisance parameters, the asymptotic deficiencies of some asymptotically efficient estimators relative to the maximum likelihood estimator (MLE) of an interest parameter based on the pooled samples were obtained by Akahira and Takeuchi (1982) and Akahira (1986) under suitable regularity conditions from the viewpoint of higher order asymptotics. On the other hand, in the elimination of nuisance parameters, the contents of marginalization and conditionality arguments were reexamined by Basu (1977) from the Bayesian point of view. In particular, the conditional likelihood method is useful for eliminating nuisance parameters. The * Masafumi Akahira [email protected] 1
Institute of Mathematics, University of Tsukuba, Ibaraki 305‑8571, Japan
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Vol.:(0123456789)
Japanese Journal of Statistics and Data Science
consistency, asymptotic normality and asymptotic efficiency of the maximum conditional likelihood estimator (MCLE) were discussed by Andersen (1970), Huque and Katti (1976), Bar-Lev and Reiser (1983), Bar-Lev (1984), Liang (1984), and others. Further, in higher order asymptotics, asymptotic properties of the MCLE were also discussed by Cox and Reid (1987) and Ferguson (1992) in the regular case. For a truncated exponential family (TEF) of distributions which is regarded as a typical non-regular case, we consider a problem of estimating a natural parameter 𝜃 in the presence of a truncation parameter 𝛾 as a nuisance parameter. The consistency and asymptotic normality of the MLE and the MCLE of 𝜃 are shown to hold when 𝛾 is unknown, and their estimators have the same asymptotic distribution and coincide with that of the MLE of 𝜃 when 𝛾 is known [see Bar-Lev (1984)]. Further, in higher order asymptotics, the bias-adjusted MLE and the MCLE of 𝜃 when 𝛾 is unknown are shown to be second-
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