This paper considers nonparametric estimations of a density function in a two-class mixture model. A linear wavelet estimator and an adaptive wavelet estimator are constructed. Upper bound estimations over $L^p(1\le p<+\mathrm{\infty })$ risk of those wavelet estimators are proved in Besov spaces. When $\tilde{p}\ge p\ge 1$, the convergence rate of adaptive wavelet estimator is the same as the linear estimator up to a $\ln n$ factor. The adaptive wavelet estimator can get better than the linear estimator in the case of $1\le \tilde{p}<p$. Finally, some numerical experiments are presented to validate the theoretical results.

This paper studies nonparametric estimations of the derivatives $r^{(m)}(x)$ of the variance function in a heteroscedastic model. Using a wavelet method, a linear estimator and an adaptive nonlinear estimator are constructed. The convergence rates under $L^{\tilde{p}}(1\le \tilde{p}<\mathrm{\infty })$ risk of those two wavelet estimators are considered with some mild assumptions. A simulation study is presented to validate the performances of the wavelet estimators.