Merge pull request #26 from DoubleML/m-math-in-docu

math / latex in docu parts

Author: MalteKurz

DESCRIPTION

@@ -7,11 +7,15 @@ Authors@R: c(
     person("Victor", "Chernozhukov", role="aut"),
     person("Malte S.", "Kurz", email = "malte.simon.kurz@uni-hamburg.de", role=c("aut", "cre")),
     person("Martin", "Spindler", email="martin.spindler@gmx.de", role="aut"))
-Description: The R package DoubleML provides an implementation of the double / debiased machine learning framework of Chernozhukov et al. (2018) for
-    partially linear regression models, partially linear instrumental variable regression models, interactive regression models and interactive instrumental
-    variable regression models. DoubleML allows estimation of the nuisance parts in these models by machine learning methods and computation of the Neyman
-    orthogonal score functions. DoubleML is built on top of mlr3 and the mlr3 ecosystem. The object-oriented implementation of DoubleML that is based on 
-    the R6 package for R is very flexible. 
+Description: The R package DoubleML provides an implementation of the double / 
+    debiased machine learning framework of Chernozhukov et al. (2018) for 
+    partially linear regression models, partially linear instrumental variable 
+    regression models, interactive regression models and interactive 
+    instrumental variable regression models. DoubleML allows estimation of the 
+    nuisance parts in these models by machine learning methods and computation 
+    of the Neyman orthogonal score functions. DoubleML is built on top of mlr3 
+    and the mlr3 ecosystem. The object-oriented implementation of DoubleML that
+    is based on the R6 package for R is very flexible. 
 License: MIT + file LICENSE
 URL: https://github.com/DoubleML/doubleml-for-r/
 Encoding: UTF-8

R/datasets.R

@@ -315,7 +315,7 @@ make_plr_turrell2018 = function(n_obs = 100, dim_x = 20, theta = 0.5, return_typ
 #' 
 #' with 
 #' 
-#' \eqn{\left(\begin{matrix} \varepsilon_i \\ u_i \\ \zeta_i \\ x_i \end{matrix} \right) \sim \mathcal{N}\left(0, \left(\begin{matrix} 1 & 0.6 & 0 & 0 \\ 0.6 & 1 & 0 & 0 \\ 0 & 0 & 0.25 I_{p_n^z} & 0 \\ 0 & 0 & 0 & \Sigma \end{matrix} \right) \right)}
+#' \eqn{\left(\begin{array}{c} \varepsilon_i \\ u_i \\ \zeta_i \\ x_i \end{array} \right) \sim \mathcal{N}\left(0, \left(\begin{array}{cccc} 1 & 0.6 & 0 & 0 \\ 0.6 & 1 & 0 & 0 \\ 0 & 0 & 0.25 I_{p_n^z} & 0 \\ 0 & 0 & 0 & \Sigma \end{array} \right) \right)}
 #' 
 #' where \eqn{\Sigma} is a \eqn{p_n^x \times p_n^x} matrix with entries \eqn{\Sigma_{kj} = 0.5^{|j-k|}} and \eqn{I_{p_n^z}} is the \eqn{p^z_n \times p^z_n} identity matrix. \eqn{\beta=\gamma} iis a \eqn{p^x_n}-vector with entries \eqn{\beta_j = \frac{1}{j^2}}, \eqn{\delta} is a \eqn{p^z_n}-vector with entries \eqn{\delta_j = \frac{1}{j^2}} and \eqn{\Pi = (I_{p_n^z}, O_{p_n^z \times (p_n^x - p_n^z)})}.
 #' 
@@ -483,9 +483,9 @@ make_irm_data = function(n_obs = 500, dim_x = 20, theta = 0, R2_d = 0.5, R2_y =
 #'
 #' \eqn{y_i = \theta d_i + x_i' \beta + u_i,}
 #' 
-#' \eqn{Z \sim \text{Bernoulli}(0.5)} and 
+#' \eqn{Z \sim \textstyle{Bernoulli} (0.5)} and 
 #' 
-#' \eqn{\left(\begin{matrix} u_i \\ v_i \end{matrix} \right) \sim \mathcal{N}\left(0, \left(\begin{matrix} 1  0.3 \\ 0.3  1 \end{matrix} \right) \right).}
+#' \eqn{\left(\begin{array}{c} u_i \\ v_i \end{array} \right) \sim \mathcal{N}\left(0, \left(\begin{array}{cc} 1 & 0.3 \\ 0.3 & 1 \end{array} \right) \right).}
 #' 
 #' The covariates :\eqn{x_i \sim \mathcal{N}(0, \Sigma)}, where  \eqn{\Sigma} is a matrix with entries
 #' \eqn{\Sigma_{kj} = 0.5^{|j-k|}} and \eqn{\beta} is a `dim_x`-vector with entries \eqn{\beta_j=\frac{1}{j^2}}.

R/double_ml_iivm.R

@@ -14,17 +14,17 @@
 #' 
 #' \eqn{Z = m_0(X) + V}, 
 #' 
-#' with \eqn{\mathbb{E}[\zeta|X,Z]=0} and \eqn{\mathbb{E}[V|X] = 0}. \eqn{Y} is the outcome variable, \eqn{D \in \{0,1\}} is the binary treatment variable and \eqn{Z \in \{0,1\}} is a binary instrumental variable. Consider the functions \eqn{g_0}, \eqn{r_0} and \eqn{m_0}, where \eqn{g_0} maps the support of \eqn{(Z,X)} to \eqn{\mathbb{R}} and \eqn{r_0} and \eqn{m_0}, respectively, map the support of \eqn{(Z,X)} and \eqn{X} to \eqn{(\epsilon, 1-\epsilon)} for some \eqn{\epsilon \in (1, 1/2)}, such that 
+#' with \eqn{E[\zeta|X,Z]=0} and \eqn{E[V|X] = 0}. \eqn{Y} is the outcome variable, \eqn{D \in \{0,1\}} is the binary treatment variable and \eqn{Z \in \{0,1\}} is a binary instrumental variable. Consider the functions \eqn{g_0}, \eqn{r_0} and \eqn{m_0}, where \eqn{g_0} maps the support of \eqn{(Z,X)} to \eqn{R} and \eqn{r_0} and \eqn{m_0}, respectively, map the support of \eqn{(Z,X)} and \eqn{X} to \eqn{(\epsilon, 1-\epsilon)} for some \eqn{\epsilon \in (1, 1/2)}, such that 
 #' 
 #' \eqn{Y = g_0(D,X) + \zeta,}
 #'  
 #' \eqn{D = r_0(D,X) + U,} 
 #' 
 #' \eqn{Z = m_0(X) + V,} 
 #' 
-#' with \eqn{\mathbb{E}[\zeta|Z,X]=0}, \eqn{\mathbb{E}[U|Z,X]=0} and \eqn{\mathbb{E}[V|X]=0}. The target parameter of interest in this model is the local average treatment effect (LATE), 
+#' with \eqn{E[\zeta|Z,X]=0}, \eqn{E[U|Z,X]=0} and \eqn{E[V|X]=0}. The target parameter of interest in this model is the local average treatment effect (LATE), 
 #' 
-#' \eqn{\theta_0 = \frac{\mathbb{E}[g_0(1,X)] - \mathbb{E}[g_0(0,X)]}{\mathbb{E}[r(1,X)] - \mathbb{E}[r(0,X)]}.}
+#' \eqn{\theta_0 = \frac{E[g_0(1,X)] - E[g_0(0,X)]}{E[r(1,X)] - E[r(0,X)]}.}
 #' 
 #' 
 #' @usage NULL
@@ -65,17 +65,17 @@ DoubleMLIIVM =R6:: R6Class("DoubleMLIIVM", inherit = DoubleML, public = list(
   #' @param ml_g ([`LearnerRegr`][mlr3::LearnerRegr], `character(1)`) \cr
   #' An object of the class [mlr3 regression learner][mlr3::LearnerRegr] to pass a learner, possibly with specified parameters, for example `lrn(regr.cv_glmnet, s = "lambda.min")`.  
   #' Alternatively, a `character(1)` specifying the name of a [mlr3 regression learner][mlr3::LearnerRegr] that is available in [mlr3](https://mlr3.mlr-org.com/index.html) or its extension packages [mlr3learners](https://mlr3learners.mlr-org.com/) or [mlr3extralearners](https://mlr3extralearners.mlr-org.com/), for example `"regr.cv_glmnet"`. \cr
-  #' `ml_g` refers to the nuisance function \eqn{g_0(Z,X) = \mathbb{E}[Y|X,Z]}.
+  #' `ml_g` refers to the nuisance function \eqn{g_0(Z,X) = E[Y|X,Z]}.
   #' 
   #' @param ml_m ([`LearnerClassif`][mlr3::LearnerClassif], `character(1)`) \cr
   #' An object of the class [mlr3 classification learner][mlr3::LearnerClassif] to pass a learner, possibly with specified parameters, for example `lrn(classif.cv_glmnet, s = "lambda.min")`. 
   #' Alternatively, a `character(1)` specifying the name of a [mlr3 classification learner][mlr3::LearnerClassif] that is available in [mlr3](https://mlr3.mlr-org.com/index.html) or its extension packages [mlr3learners](https://mlr3learners.mlr-org.com/) or [mlr3extralearners](https://mlr3extralearners.mlr-org.com/), for example `"regr.cv_glmnet"`. \cr
-  #' `ml_m` refers to the nuisance function \eqn{m_0(X) = \mathbb{E}[Z|X]}.
+  #' `ml_m` refers to the nuisance function \eqn{m_0(X) = E[Z|X]}.
   #' 
   #' @param ml_r ([`LearnerClassif`][mlr3::LearnerClassif], `character(1)`) \cr
   #' An object of the class [mlr3 classification learner][mlr3::LearnerClassif] to pass a learner, possibly with specified parameters, for example `lrn(classif.cv_glmnet, s = "lambda.min")`.
   #' Alternatively, a `character(1)` specifying the name of a [mlr3 classification learner][mlr3::LearnerClassif] that is available in [mlr3](https://mlr3.mlr-org.com/index.html) or its extension packages [mlr3learners](https://mlr3learners.mlr-org.com/) or [mlr3extralearners](https://mlr3extralearners.mlr-org.com/), for example `"regr.cv_glmnet"`. \cr
-  #' `ml_r` refers to the nuisance function \eqn{r_0(Z,X) = \mathbb{E}[D|X,Z]}.
+  #' `ml_r` refers to the nuisance function \eqn{r_0(Z,X) = E[D|X,Z]}.
   #' 
   #' @param n_folds (`integer(1)`)\cr
   #' Number of folds. Default is `5`. 

R/double_ml_irm.R

@@ -14,13 +14,13 @@
 #' 
 #' \eqn{D = m_0(X) + V}, 
 #' 
-#' with \eqn{\mathbb{E}[U|X,D]=0} and \eqn{\mathbb{E}[V|X] = 0}. \eqn{Y} is the outcome variable and \eqn{D \in \{0,1\}} is the binary treatment variable. We consider estimation of the average treamtent effects when treatment effects are fully heterogeneous. Target parameters of interest in this model are the average treatment effect (ATE), 
+#' with \eqn{E[U|X,D]=0} and \eqn{E[V|X] = 0}. \eqn{Y} is the outcome variable and \eqn{D \in \{0,1\}} is the binary treatment variable. We consider estimation of the average treamtent effects when treatment effects are fully heterogeneous. Target parameters of interest in this model are the average treatment effect (ATE), 
 #' 
-#' \eqn{\theta_0 = \mathbb{E}[g_0(1,X) - g_0(0,X)]} 
+#' \eqn{\theta_0 = E[g_0(1,X) - g_0(0,X)]} 
 #' 
 #' and the average treament effect on the treated (ATTE), 
 #' 
-#' \eqn{\theta_0 = \mathbb{E}[g_0(1,X) - g_0(0,X)|D=1]}.
+#' \eqn{\theta_0 = E[g_0(1,X) - g_0(0,X)|D=1]}.
 #' 
 #' @usage NULL
 #' 
@@ -55,12 +55,12 @@ DoubleMLIRM = R6::R6Class("DoubleMLIRM", inherit = DoubleML, public = list(
   #' @param ml_g ([`LearnerRegr`][mlr3::LearnerRegr], `character(1)`) \cr
   #' An object of the class [mlr3 regression learner][mlr3::LearnerRegr] to pass a learner, possibly with specified parameters, for example `lrn(regr.cv_glmnet, s = "lambda.min")`.
   #' Alternatively, a `character(1)` specifying the name of a [mlr3 regression learner][mlr3::LearnerRegr] that is available in [mlr3](https://mlr3.mlr-org.com/index.html) or its extension packages [mlr3learners](https://mlr3learners.mlr-org.com/) or [mlr3extralearners](https://mlr3extralearners.mlr-org.com/), for example `"regr.cv_glmnet"`.  \cr
-  #' `ml_g` refers to the nuisance function \eqn{g_0(X) = \mathbb{E}[Y|X,D]}.
+  #' `ml_g` refers to the nuisance function \eqn{g_0(X) = E[Y|X,D]}.
   #' 
   #' @param ml_m ([`LearnerClassif`][mlr3::LearnerClassif], `character(1)`) \cr
   #' An object of the class [mlr3 classification learner][mlr3::LearnerClassif] to pass a learner, possibly with specified parameters, for example `lrn(classif.cv_glmnet, s = "lambda.min")`.
   #' Alternatively, a `character(1)` specifying the name of a [mlr3 classification learner][mlr3::LearnerClassif] that is available in [mlr3](https://mlr3.mlr-org.com/index.html) or its extension packages [mlr3learners](https://mlr3learners.mlr-org.com/) or [mlr3extralearners](https://mlr3extralearners.mlr-org.com/), for example `"regr.cv_glmnet"`. \cr
-  #' `ml_m` refers to the nuisance function \eqn{m_0(X) = \mathbb{E}[D|X]}.
+  #' `ml_m` refers to the nuisance function \eqn{m_0(X) = E[D|X]}.
   #' 
   #' @param n_folds (`integer(1)`)\cr
   #' Number of folds. Default is `5`. 

R/double_ml_pliv.R

@@ -50,7 +50,7 @@ DoubleMLPLIV = R6::R6Class("DoubleMLPLIV", inherit = DoubleML, public = list(
   #' @param ml_g ([`LearnerRegr`][mlr3::LearnerRegr], `character(1)`) \cr
   #' An object of the class [mlr3 regression learner][mlr3::LearnerRegr] to pass a learner, possibly with specified parameters, for example `lrn(regr.cv_glmnet, s = "lambda.min")`. 
   #' Alternatively, a `character(1)` specifying the name of a [mlr3 regression learner][mlr3::LearnerRegr] that is available in [mlr3](https://mlr3.mlr-org.com/index.html) or its extension packages [mlr3learners](https://mlr3learners.mlr-org.com/) or [mlr3extralearners](https://mlr3extralearners.mlr-org.com/), for example `"regr.cv_glmnet"`.  \cr
-  #' `ml_g` refers to the nuisance function \eqn{g_0(X) = \mathbb{E}[Y|X]}.
+  #' `ml_g` refers to the nuisance function \eqn{g_0(X) = E[Y|X]}.
   #' 
   #' @param ml_m ([`LearnerRegr`][mlr3::LearnerRegr], `character(1)`) \cr
   #' An object of the class [mlr3 regression learner][mlr3::LearnerRegr] to pass a learner, possibly with specified parameters, for example `lrn(regr.cv_glmnet, s = "lambda.min")`. 

R/double_ml_plr.R

@@ -13,7 +13,7 @@
 #' 
 #' \eqn{D = m_0(X) + V,}
 #' 
-#' with \eqn{\mathbb{E}[\zeta|D,X]=0} and \eqn{\mathbb{E}[V|X] = 0}. \eqn{Y} is the outcome variable variable and \eqn{D} is the policy variable of interest. The high-dimensional vector \eqn{X = (X_1, \ldots, X_p)} consists of other confounding covariates, and \eqn{\zeta} and \eqn{V} are stochastic errors.  
+#' with \eqn{E[\zeta|D,X]=0} and \eqn{E[V|X] = 0}. \eqn{Y} is the outcome variable variable and \eqn{D} is the policy variable of interest. The high-dimensional vector \eqn{X = (X_1, \ldots, X_p)} consists of other confounding covariates, and \eqn{\zeta} and \eqn{V} are stochastic errors.  
 #'
 #' @usage NULL
 #' 
@@ -40,11 +40,11 @@ DoubleMLPLR = R6::R6Class("DoubleMLPLR", inherit = DoubleML, public = list(
   #' @param ml_g ([`LearnerRegr`][mlr3::LearnerRegr], `character(1)`,) \cr
   #' An object of the class [mlr3 regression learner][mlr3::LearnerRegr] to pass a learner, possibly with specified parameters, for example `lrn(regr.cv_glmnet, s = "lambda.min")`. 
   #' Alternatively, a `character(1)` specifying the name of a [mlr3 regression learner][mlr3::LearnerRegr] that is available in [mlr3](https://mlr3.mlr-org.com/index.html) or its extension packages [mlr3learners](https://mlr3learners.mlr-org.com/) or [mlr3extralearners](https://mlr3extralearners.mlr-org.com/), for example `"regr.cv_glmnet"`. \cr
-  #' `ml_g` refers to the nuisance function \eqn{g_0(X) = \mathbb{E}[Y|X]}.
+  #' `ml_g` refers to the nuisance function \eqn{g_0(X) = E[Y|X]}.
   #' 
   #' @param ml_m ([`LearnerRegr`][mlr3::LearnerRegr], `character(1)`,) \cr
   #' An object of the class [mlr3 regression learner][mlr3::LearnerRegr] to pass a learner, possibly with specified parameters, for example `lrn(regr.cv_glmnet, s = "lambda.min")`. Alternatively, a `character(1)` specifying the name of a [mlr3 regression learner][mlr3::LearnerRegr] that is available in [mlr3](https://mlr3.mlr-org.com/index.html) or its extension packages [mlr3learners](https://mlr3learners.mlr-org.com/) or [mlr3extralearners](https://mlr3extralearners.mlr-org.com/), for example `"regr.cv_glmnet"`. \cr
-  #' `ml_m` refers to the nuisance function \eqn{m_0(X) = \mathbb{E}[D|X]}.
+  #' `ml_m` refers to the nuisance function \eqn{m_0(X) = E[D|X]}.
 
   #' 
   #' @param n_folds (`integer(1)`)\cr