# BENK: The Beran Estimator with Neural Kernels for Estimating the Heterogeneous Treatment Effect
## 1 Introduction  
Survival analysis is an important and fundamental tool for modeling applications when using time-to-event data [1], which can be encountered in medicine, reliability, safety, finance, etc. This is a reason why many machine learning models have been developed to deal with time-to-event data and to solve the corresponding problems in the framework of survival analysis [2]. The crucial peculiarity of time-to-event data is that a training set consists of censored and uncensored observations. When time-to-event exceeds the duration of an observation, we have a censored observation. When an event is observed, i.e., time-to-event coincides with the duration of the observation, we deal with an uncensored observation. Many survival models are able to cover various cases of time-to-event probability distributions and their parameters [2]. One of the important models is the Cox proportional hazards model [3], which can be regarded as a semi-parametric regression model. There are also many parametric and nonparametric models. When considering machine learning survival models, it is important to point out that, in contrast to other machine learning models, their outcomes are functions, for instance, survival functions, hazard functions or cumulative hazard functions. For instance, the well-known effective model called the random survival forest (RSF) [4] predicts survival functions (SFs) or cumulative hazard functions. An important area of survival model application is the problem of treatment effect estimation, which is often solved in the framework of machine learning problems [5]. The treatment effect shows how a treatment may be efficient depending on characteristics of a patient. The problem is solved by dividing patients into two groups called treatment and control, such that patients from the different groups can be compared. One of the popular measures of efficient treatment that is used in machine learning models is the average treatment effect (ATE) [6], which is estimated on the basis of observed data about patients, such as the mean difference between outcomes of patients from the treatment and control groups. Due to the difference between characteristics of patients and their responses to a particular treatment, the treatment effect is measured using the conditional average treatment effect (CATE), which is defined as the mean difference between outcomes of patients from the treatment and control groups, conditional on a patient feature vector [7]. In fact, most methods of CATE estimation are based on constructing two regression models for controls and treatments. However, two difficulties in CATE estimation can be met. The first one is that the treatment group is usually very small. Therefore, many machine learning models cannot be accurately trained on the small datasets. The second difficulty is fundamental. Each patient cannot be simultaneously in the treatment and control groups, i.e., we either observe the patient outcome under the treatment or control, but never both [8]. Nevertheless, to overcome these difficulties, many methods for estimating CATE have been proposed and developed due to the importance of the problem in many areas [9; 10; 11; 12; 13]. One of the approaches for constructing regression models for controls and treatments is the application of the Nadaraya-Watson kernel regression [14; 15], which uses standard kernel functions, for instance, the Gaussian, uniform or Epanechnikov kernels. In order to avoid selecting a standard kernel, [PERSON] et al. [16] proposed to implement kernels and the whole Nadaraya-Watson kernel regression by using a set of identical neural subnetworks with shared parameters, with a specific way of the network training. The corresponding method called TNW-CATE (Trainable Nadaraya-Watson regression for CATE) is based on an important assumption that domains of the feature vectors from the treatment and control groups are similar.