Searcharxiv⌕ Search

arXiv subjects

Caterina May

Publications and source records attributed to Caterina May.

9 recordsLinked to original sources
arXiv2024

Optimal design of experiments for functional linear models with dynamic factors

In this work we build optimal experimental designs for precise estimation of the functional coefficient of a function-on-function linear regression model where both the response and the factors are continuous functions of time. After obtaining the variance-covariance matrix of the estimator of the functional coefficient which minimizes the integrated sum of square of errors, we extend the classical definition of optimal design to this estimator, and we provide the expression of the A-optimal and of the D-optimal designs. Examples of optimal designs for dynamic experimental factors are then computed through a suitable algorithm, and we discuss different scenarios in terms of the set of basis functions used for their representation. Finally, we present an example with simulated data to illustrate the feasibility of our methodology.

stat.ME↗
arXiv2023

Maximum Likelihood Estimation under the Emax Model: Existence, Geometry and Efficiency

This study focuses on the estimation of the Emax dose-response model, a widely utilized framework in clinical trials, agriculture, and environmental experiments. Existing challenges in obtaining maximum likelihood estimates (MLE) for model parameters are often ascribed to computational issues but, in reality, stem from the absence of a MLE. Our contribution provides a new understanding and control of all the experimental situations that practitioners might face, guiding them in the estimation process. We derive the exact MLE for a three-point experimental design and we identify the two scenarios where the MLE fails. To address these challenges, we propose utilizing Firth's modified score, providing its analytical expression as a function of the experimental design. Through a simulation study, we demonstrate that, in one of the problematic cases, the Firth modification yields a finite estimate. For the remaining case, we introduce a design-augmentation strategy akin to a hypothesis test.

stat.ME↗
arXiv2020

Designing experiments for estimating an appropriate outlet size for a silo type problem

The problem of jam formation during the discharge by gravity of granular material through a two-dimensional silo has a number of practical applications. In many problems the estimation of the minimum outlet size which guarantees that the time to the next jamming event is long enough is crucial. Assuming that the time is modeled by an exponential distribution with two unknown parameters, this goal translates to the optimal estimation of a non-linear transformation of the parameters. We obtain $c$-optimum experimental designs with that purpose, applying the graphic Elfving method. Since the optimal designs depend on the nominal values of the parameters, a sensitivity study is additionally provided. Finally, a simulation study checks the performance of the approximations made, first with the Fisher Information matrix, then with the linearization of the function to be estimated. The results are useful for experimenting in a laboratory and translating then the results to a larger scenario. Apart from the application a general methodology is developed in the paper for the problem of precise estimation of a one-dimensional parametric transformation in a non-linear model.

stat.ME↗
arXiv2019

The Effects of Adaptation on Inference for Non-Linear Regression Models with Normal Errors

This work studies the properties of the maximum likelihood estimator (MLE) of a non-linear model with Gaussian errors and multidimensional parameter. The observations are collected in a two-stage experimental design and are dependent since the second stage design is determined by the observations at the first stage; the MLE maximizes the total likelihood. Differently from the most of the literature, the first stage sample size is small, and hence asymptotic approximation is used only in the second stage. It is proved that the MLE is consistent and that its asymptotic distribution is a specific Gaussian mixture, via stable convergence. Finally, a simulation study is provided in the case of a dose-response Emax model.

math.ST↗
arXiv2016

Best estimation of functional linear models

Observations which are realizations from some continuous process are frequent in sciences, engineering, economics, and other fields. We consider linear models, with possible random effects, where the responses are random functions in a suitable Sobolev space. The processes cannot be observed directly. With smoothing procedures from the original data, both the response curves and their derivatives can be reconstructed, even separately. From both these samples of functions, just one sample of representatives is obtained to estimate the vector of functional parameters. A simulation study shows the benefits of this approach over the common method of using information either on curves or derivatives. The main theoretical result is a strong functional version of the Gauss-Markov theorem. This ensures that the proposed functional estimator is more efficient than the best linear unbiased estimator based only on curves or derivatives.

math.ST↗
arXiv2014

KL-optimum designs: theoretical properties and practical computation

In this paper some new properties and computational tools for finding KL-optimum designs are provided. KL-optimality is a general criterion useful to select the best experimental conditions to discriminate between statistical models. A KL-optimum design is obtained from a minimax optimization problem, which is defined on a infinite-dimensional space. In particular, continuity of the KL-optimality criterion is proved under mild conditions; as a consequence, the first-order algorithm converges to the set of KL-optimum designs for a large class of models. It is also shown that KL-optimum designs are invariant to any scale-position transformation. Some examples are given and discussed, together with some practical implications for numerical computation purposes.

stat.ME↗
arXiv2011

A functional equation whose unknown is P([0,1]) valued

We study a functional equation whose unknown maps a Euclidean space into the space of probability distributions on [0,1]. We prove existence and uniqueness of its solution under suitable regularity and boundary conditions, we show that it depends continuously on the boundary datum, and we characterize solutions that are diffuse on [0,1]. A canonical solution is obtained by means of a Randomly Reinforced Urn with different reinforcement distributions having equal means. The general solution to the functional equation defines a new parametric collection of distributions on [0,1] generalizing the Beta family.

math.PR↗
arXiv2009

Asymptotics in response-adaptive designs generated by a two-color, randomly reinforced urn

This paper illustrates asymptotic properties for a response-adaptive design generated by a two-color, randomly reinforced urn model. The design considered is optimal in the sense that it assigns patients to the best treatment, with probability converging to one. An approach to show the joint asymptotic normality of the estimators of the mean responses to the treatments is provided in spite of the fact that allocation proportions converge to zero and one. Results on the rate of convergence of the number of patients assigned to each treatment are also obtained. Finally, we study the asymptotic behavior of a suitable test statistic.

math.ST↗