Abstract
In this paper, we consider the observation of $n$ i.i.d. mixed Poisson processes with random intensity having an unknown density $f$ on $\mathbb{R}^{+}$. For fixed observation time $T$, we propose a nonparametric adaptive strategy to estimate $f$. We use an appropriate Laguerre basis to build adaptive projection estimators. Non-asymptotic upper bounds of the $\mathbb{L}^{2}$-integrated risk are obtained and a lower bound is provided, which proves the optimality of the estimator. For large $T$, the variance of the previous method increases, therefore we propose another adaptive strategy. The procedures are illustrated on simulated data.
Highlights
Consider n independent Poisson processes (Nj(t), j = 1, . . . , n) with unit intensity and n i.i.d. positive random variables (Cj, j = 1, . . . , n)
We assume that the random variables Cj have an unknown density f on (0, +∞) and our concern is the nonparametric estimation of f from the observation of a nsample (Xj(T ), j = 1, . . . , n) for a given value T
We investigate this subject for large n and both for fixed T and large T with two different methods
Summary
As it is always the case in nonparametric estimation[1], we must link the bias term f − fl 2 with regularity properties of function f In our context, these should be expressed in relation with the rate of decay of the coefficients (θk(f ))k≥0. The number of coefficients in the projection estimators does not depend on the regularity space In this sense, the above methods are adaptive. When the function under estimation has stronger regularity properties than considered in lower bounds, we show that the rate can be improved (polynomial instead of logarithmic). This justifies the proposal of an adaptive procedure, see Theorem 2.2, which is non asymptotic
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