We study a penalized log-wavelet density estimation method using an \(\ell _2\) -type group penalization. The underlying density function is approximated by sequences exponential families based on the wavelet basis. The \(\ell _2\) -type group penalty is adopted to identify the relevant frequency component and prevent unnecessary local behaviors of the estimator. The proposed estimator is a valid density function in the sense that it is positive and integrates to one. Theoretical properties of the estimator are studied when the quality of fit is measured by the Kullback-Leibler divergence (relative entropy). We establish a nonasymptotic oracle inequality, and use it to prove minimax adaptation and resolution identification consistency properties. The proposed method is implemented with the alternating direction method of multipliers algorithm. Numerical studies are conducted to complement the theoretical results.
