\begin{algorithm}\small
\caption{Approximating $\hat\sigma^2$}
\algorithmicrequire  positive integers $N_\sigma$ and $N_\sigma^\prime$

\begin{algorithmic}[1]
\For{$l$ in 1 to $N_\sigma$}
\State draw subsample $\{ \iota_1,\dots, \iota_{n - D_\sigma} \}$ of size $n - D_\sigma$  without replacement from $\{ 1, \dots, n \}$
\For{$l^\prime$ in 1 to $N_\sigma^\prime$}
\State draw subsample $\{ \iota_1^\ast,\dots, \iota_{s}^\ast \}$ of size $s$ without replacement from $\{ \iota_1,\dots, \iota_{n - D_\sigma} \}$
\State apply Algorithm 1 to $\{ \iota_1^\ast,\dots, \iota_{s}^\ast \}$ and record $\mathcal{L}^{\ast}_{(l^\prime)}$
\EndFor 
\State  compute $\{ \tilde{\omega}_1 (x), \dots ,\tilde{\omega}_n (x) \}$ by applying Eq.\ (\ref{eq:wei}) to $\{\mathcal{L}^{\ast}_{(1)} ,\dots, \mathcal{L}^{\ast}_{(N_\sigma^\prime)}  \}$ 
\State $\hat{\boldsymbol{\mu}}_{(l)} \gets \text{Average}( \{ \tilde{\omega}_1 (x)\boldsymbol{\phi} (y_1)  , \dots ,\tilde{\omega}_n (x) \boldsymbol{\phi} (y_n) \} )$
\EndFor
\State $\hat{\sigma}^{2\ast}   \gets (n - D_\sigma) [\hat{\mathcal{T}}_x (y)  \times \text{CovarianceMatrix}( \{  \hat{\boldsymbol{\mu}}_{(1)},\dots, \hat{\boldsymbol{\mu}}_{(N_\sigma)}\} ) \times \hat{\mathcal{T}}_x (y)^\tau ]/ D_\sigma$. 
\end{algorithmic}
\end{algorithm}

\begin{algorithm}
\small
\caption{Approximating $\hat\sigma^2$}
\algorithmicrequire positive integers $N_\sigma$ and $N_\sigma^\prime$
\begin{algorithmic}
[1]
\For{$l$ in 1 to $N_\sigma$}
\State draw subsample $\{ \iota_1,\dots, \iota_{n - D_\sigma} \}$ of size $n - D_\sigma$ without replacement from $\{ 1, \dots, n \}$
\For{$l^\prime$ in 1 to $N_\sigma^\prime$}
\State draw subsample $\{ \iota_1^\ast,\dots, \iota_{s}^\ast \}$ of size $s$ without replacement from $\{ \iota_1,\dots, \iota_{n - D_\sigma} \}$
\State apply Algorithm 1 to $\{ \iota_1^\ast,\dots, \iota_{s}^\ast \}$ and record $\mathcal{L}^{\ast}_{(l^\prime)}$
\EndFor
\State compute $\{ \tilde{\omega}_1 (x), \dots ,\tilde{\omega}_n (x) \}$ by applying Eq.\ (\ref{eq:wei}) to $\{\mathcal{L}^{\ast}_{(1)} ,\dots, \mathcal{L}^{\ast}_{(N_\sigma^\prime)} \}$
\State $\hat{\boldsymbol{\mu}}_{(l)} \gets \text{Average}( \{ \tilde{\omega}_1 (x)\boldsymbol{\phi} (y_1) , \dots ,\tilde{\omega}_n (x) \boldsymbol{\phi} (y_n) \} )$
\EndFor
\State $\hat{\sigma}^{2\ast} \gets (n - D_\sigma) [\hat{\mathcal{T}}_x (y) \times \text{CovarianceMatrix}( \{ \hat{\boldsymbol{\mu}}_{(1)},\dots, \hat{\boldsymbol{\mu}}_{(N_\sigma)}\} ) \times \hat{\mathcal{T}}_x (y)^\tau ]/ D_\sigma$.
\end{algorithmic}
\end{algorithm}
{
    "title": "nonparametric estimation of conditional densities by generalized random   forests",
    "id": "2309.13251",
    "abstract": "considering a continuous random variable y together with a continuous random vector x, i propose a nonparametric estimator f^(.|x) for the conditional density of y given x=x. this estimator takes the form of an exponential series whose coefficients t = (t1,...,tj) are the solution of a system of nonlinear equations that depends on an estimator of the conditional expectation e[p(y)|x=x], where p(.) is a j-dimensional vector of basis functions. a key feature is that e[p(y)|x=x] is estimated by generalized random forest (athey, tibshirani, and wager, 2019), targeting the heterogeneity of t across x. i show that f^(.|x) is uniformly consistent and asymptotically normal, while allowing j to grow to infinity. i also provide a standard error formula to construct asymptotically valid confidence intervals. results from monte carlo experiments and an empirical illustration are provided.",
    "categories": "econ.em",
    "doi": "",
    "created": "2023-09-23",
    "updated": "2024-01-10",
    "authors": [
        "federico zincenko"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2309.13251"
}
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