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Estimation for a density ratio has an important role in statistical inference. We can use the estimator for testing homogeneity of two samples, detecting change point etc. Let f(x) and g(x) denote pro...bability density functions and g(x0) ≠ 0 (x0 ∈ R). There are several ways to estimate the density ratio f(x0)/g(x0). In this paper we discuss a kernel estimation that is a popular method in nonparametric statistical inference. A naive estimator is constituted from separate estimators of f(x0) and g(x0), which we call an indirect estimator. Another estimator is proposed by Ćwik and Mielniczuk (1989), which we call a direct estimator. Extending Ćwik and Mielniczuk (1989)'s method, we propose a new direct estimator, and derive an asymptotic mean squared error. We also prove central limit theorem of the new estimator, and compare mean squared errors of the proposed estimator and the direct estimator by simulation.続きを見る
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