Due to its low computational cost, Lasso is an attractive regularization
method for high-dimensional statistical settings. In this paper, we consider
multivariate counting processes depending on an unknown function parameter to
be estimated by linear combinations of a fixed dictionary. To select
coefficients, we propose an adaptive $\ell_1$-penalization methodology, where
data-driven weights of the penalty are derived from new Bernstein type
inequalities for martingales. Oracle inequalities are established under
assumptions on the Gram matrix of the dictionary. Nonasymptotic probabilistic
results for multivariate Hawkes processes are proven, which allows us to check
these assumptions by considering general dictionaries based on histograms,
Fourier or wavelet bases. Motivated by problems of neuronal activity inference,
we finally carry out a simulation study for multivariate Hawkes processes and
compare our methodology with the adaptive Lasso procedure proposed by Zou in
(J. Amer. Statist. Assoc. 101 (2006) 1418-1429). We observe an excellent
behavior of our procedure. We rely on theoretical aspects for the essential
question of tuning our methodology. Unlike adaptive Lasso of (J. Amer. Statist.
Assoc. 101 (2006) 1418-1429), our tuning procedure is proven to be robust with
respect to all the parameters of the problem, revealing its potential for
concrete purposes, in particular in neuroscience.

本文提出了一种自适应的 L1 - 罚函数定量分析方法，通过基于概率的非渐进性结果和新的 Bernstein 型不等式，以估计数量过程中的未知函数参数的线性组合形式对固定字典进行了选择，检验了不同字典假设下的巨额矩阵，并通过机械活动推断问题的仿真研究与自适应 Lasso 方法进行了比较。