Filtering the Wright-Fisher diffusion

Chaleyat-Maurel, Mireille; Genon-Catalot, Valentine

doi:10.1051/ps:2008006

Chaleyat-Maurel, Mireille ; Genon-Catalot, Valentine

ESAIM: Probability and Statistics, Tome 13 (2009), pp. 197-217.

Résumé

We consider a Wright-Fisher diffusion $(x (t))$ whose current state cannot be observed directly. Instead, at times $t_{1}$ < $t_{2}$ < ..., the observations $y (t_{i})$ are such that, given the process $(x (t))$ , the random variables $(y (t_{i}))$ are independent and the conditional distribution of $y (t_{i})$ only depends on $x (t_{i})$ . When this conditional distribution has a specific form, we prove that the model $((x (t_{i}), y (t_{i}))$ , $i$ $\geq$ 1) is a computable filter in the sense that all distributions involved in filtering, prediction and smoothing are exactly computable. These distributions are expressed as finite mixtures of parametric distributions. Thus, the number of statistics to compute at each iteration is finite, but this number may vary along iterations.

DOI : 10.1051/ps:2008006

Classification : Primary 93E11, 60G35, secondary, 62C10
Mots-clés : stochastic filtering, partial observations, diffusion processes, discrete time observations, hidden Markov models, prior and posterior distributions

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     author = {Chaleyat-Maurel, Mireille and Genon-Catalot, Valentine},
     title = {Filtering the {Wright-Fisher} diffusion},
     journal = {ESAIM: Probability and Statistics},
     pages = {197--217},
     publisher = {EDP-Sciences},
     volume = {13},
     year = {2009},
     doi = {10.1051/ps:2008006},
     mrnumber = {2518546},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps:2008006/}
}

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Chaleyat-Maurel, Mireille; Genon-Catalot, Valentine. Filtering the Wright-Fisher diffusion. ESAIM: Probability and Statistics, Tome 13 (2009), pp. 197-217. doi : 10.1051/ps:2008006. http://www.numdam.org/articles/10.1051/ps:2008006/

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