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Population Monte Carlo algorithm in high dimensions. (English) Zbl 1221.65028

For sampling from a multi-dimensional target distribution \(\pi\) , the population Monte Carlo (PMC) algorithm, is an iterative importance sampling such that the updated step is re-sampled from the previous sample by using the importance weight (normalized ratio of the previous importance function valued at the entries of the previous sample) and the updated importance function is constructed by this resample values distributed by their importance weights. Considering one step, in the present paper, the updated importance function \(Q\) is given, which is independent of the above iterative importance weight in PMC scheme. The main result in this paper is, when \(\pi\) (up to a constant), \(Q\) and the initial distribution are assumed to be D-dimensional normal distributions with different parameters, under a mild condition among the parameters, the Monte Carlo estimator of a linear function of a \(\pi\) distributed vector is asymptotically approximated by \(N(0,\Phi)\) in the sense of distribution with \(\Phi\) explicitly given as \(D\) goes to infinite. Optimal asymptotic variance is also discussed in some special case.

MSC:

65C60 Computational problems in statistics (MSC2010)
62D05 Sampling theory, sample surveys
62E20 Asymptotic distribution theory in statistics
65C05 Monte Carlo methods
62J10 Analysis of variance and covariance (ANOVA)
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