Deconvolution estimation of onset of pregnancy with replicate observations

Abstract : Except in the specific case of in vitro fertilization, the precise date of onset of pregnancy is unknown. In clinical practice, the date of pregnancy may only be estimated, and most commonly from ultrasound biometric measurements of the embryo. Denoting $X$ the interval between last menstrual period and true onset of pregnancy and $Y$ the interval between last menstrual period and the date estimated by ultrasound, we wish to estimate the density $\fx$ of $X$. Only noisy observations $Y_{j}=X_{j}+\varepsilon_{j}, j=1,\dots,n$ are observed and the density $f_\varepsilon$ of $\varepsilon$ is unknown. Because the noise itself cannot be sampled for the estimation of its density, we consider the specific setting of replicate noisy observations $Y_{-j1}=X_{-j}+\varepsilon_{-j1}$ and $Y_{-j2}=X_{-j}+\varepsilon_{-j2}$, $j=1,\dots,M$. We suggest an adaptive non-parametric estimator of $\fx$ built following a deconvolution device. Convergence rates are studied and compared to other settings that do not involve replicates. Lastly, we estimate the density $\fx$ in spontaneous pregnancies with an estimation of the noise obtained from replicate observations in twin pregnancies.
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Scandinavian Journal of Statistics, Wiley, 2014, 41 (2), pp.325-345. <10.1111/sjos.12029>
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Fabienne Comte, Adeline Samson, Julien Stirnemann. Deconvolution estimation of onset of pregnancy with replicate observations. Scandinavian Journal of Statistics, Wiley, 2014, 41 (2), pp.325-345. <10.1111/sjos.12029>. <hal-00588235v3>

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