= Wavelet regression estimator
{title2=$\widehat m_J=P_J\widetilde Y_n$}
At design sites $i/n$, let $\widetilde Y_n$ be the step function taking response $Y_i$ on $((i-1)/n,i/n]$. Project it onto an <interval-adapted wavelet basis> through resolution $J$. Each coefficient is $\widehat c_\lambda=\sum_iY_i\int_{I_i}b_\lambda(t)dt$. Its <expectation> is the corresponding coefficient of the step function of sampled regression means. For a <Haar wavelet> basis, dyadic sample size and resolution at most the sampling scale, this is exactly $n^{-1}\sum_iY_i b_\lambda(i/n)$ with right-closed pointwise versions. Independent errors of <variance> $\sigma^2$ give coefficient variance at most $\sigma^2/n$: <Cauchy-Schwarz inequality> on each sampling cell yields $\sum_i(\int_{I_i}b_\lambda)^2\leq\|b_\lambda\|_2^2/n$.
Back to article page
