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Importance sampling is a well developed method in statistics. Given a random variable $X$, the problem of estimating its expected value $\mu$ is addressed. The standard approach is to use the sample mean as an estimator $\bar x$. In…

In this paper, we consider the nonasymptotic sequential estimation of means of random variables bounded in between zero and one. We have rigorously demonstrated that, in order to guarantee prescribed relative precision and confidence level,…

We consider the problem of low probability estimation: given a machine learning model and a formally-specified input distribution, how can we estimate the probability of a binary property of the model's output, even when that probability is…

We study convex empirical risk minimization for high-dimensional inference in binary models. Our first result sharply predicts the statistical performance of such estimators in the linear asymptotic regime under isotropic Gaussian features.…

We aim at estimating a function $\lambda:[0,1]\to \mathbb {R}$, subject to the constraint that it is decreasing (or increasing). We provide a unified approach for studying the $\mathbb {L}_p$-loss of an estimator defined as the slope of a…

For some estimations and predictions, we solve minimization problems with asymmetric loss functions. Usually, we estimate the coefficient of regression for these problems. In this paper, we do not make such the estimation, but rather give a…

In this paper we develop rate--optimal estimation procedures in the problem of estimating the $L_p$--norm, $p\in (0, \infty)$ of a probability density from independent observations. The density is assumed to be defined on $R^d$, $d\geq 1$…

Inverse probability weighted estimators are the oldest and potentially most commonly used class of procedures for the estimation of causal effects. By adjusting for selection biases via a weighting mechanism, these procedures estimate an…

We consider the problem of estimating a continuous distribution function $F$, as well as meaningful functions $\tau(F)$ under a large class of loss functions. We obtain best invariant estimators and establish their minimaxity for H\"{o}lder…

Relative error estimation has been recently used in regression analysis. A crucial issue of the existing relative error estimation procedures is that they are sensitive to outliers. To address this issue, we employ the $\gamma$-likelihood…

Consider estimating the n by p matrix of means of an n by p matrix of independent normally distributed observations with constant variance, where the performance of an estimator is judged using a p by p matrix quadratic error loss function.…

The deployment of safe and trustworthy machine learning systems, and particularly complex black box neural networks, in real-world applications requires reliable and certified guarantees on their performance. The conformal prediction…

Many results have been proved for various nuclear norm penalized estimators of the uniform sampling matrix completion problem. However, most of these estimators are not robust: in most of the cases the quadratic loss function and its…

The fate of scientific hypotheses often relies on the ability of a computational model to explain the data, quantified in modern statistical approaches by the likelihood function. The log-likelihood is the key element for parameter…

Naive approaches to amortized inference in probabilistic programs with unbounded loops can produce estimators with infinite variance. This is particularly true of importance sampling inference in programs that explicitly include rejection…

Say $X_1,X_2,\ldots$ are independent identically distributed Bernoulli random variables with mean $p$. This paper builds a new estimate $\hat p$ of $p$ that has the property that the relative error, $\hat p /p - 1$, of the estimate does not…

Many statistical applications involve models for which it is difficult to evaluate the likelihood, but from which it is relatively easy to sample. Approximate Bayesian computation is a likelihood-free method for implementing Bayesian…

We describe the minimax reconstruction rates in linear ill-posed equations in Hilbert space when smoothness is given in terms of general source sets. The underlying fundamental result, the minimax rate on ellipsoids, is proved similarly to…

AIMS. The maximum-likelihood method is the standard approach to obtain model fits to observational data and the corresponding confidence regions. We investigate possible sources of bias in the log-likelihood function and its subsequent…

We consider random discrepancy under weighted importance sampling of a class of stratified input. We give the expected $L_p-$discrepancy($2\leq p<\infty$) upper bound in weighted form under a class of stratified sampling. This result…