Robust Universal Inference

Learning and making inference from a finite set of samples are among the fundamental problems in science. In most popular applications, the paradigmatic approach is to seek a model that best explains the data. This approach has many desirable properties when the number of samples is large. However, in many practical setups, data acquisition is costly and only a limited number of samples is available. In this work, we study an alternative approach for this challenging setup. Our framework suggests that the role of the train-set is not to provide a single estimated model, which may be inaccurate due to the limited number of samples. Instead, we define a class of “reasonable” models. Then, the worst-case performance in the class is controlled by a minimax estimator with respect to it. Further, we introduce a robust estimation scheme that provides minimax guarantees, also for the case where the true model is not a member of the model class. Our results draw important connections to universal prediction, the redundancy-capacity theorem, and channel capacity theory. We demonstrate our suggested scheme in different setups, showing a significant improvement in worst-case performance over currently known alternatives.

Minimax Estimation of the Scale Parameter of Laplace Distribution under Modified Linear Exponential (MLINEX) Loss Function

The main objective of this paper is to find the minimax estimator of the scale parameter of Laplace distribution under MLINEX loss function by applying the theorem of Lehmann (1950). The estimator is then compared with classical estimator like moment estimator with respect to mean square errors (MSEs) through R- Code simulation. The result has shown that the minimax estimator under MLINEX loss function is better than moment estimator for all sample sizes. Finally, mean square errors of different estimators corresponding to sample size are presented graphically.

قياس كفاءة أسلوب بيز مع طرائق أخرى لتقدير معلمة القياس لتوزيع ويبل ذي المعلمتين

يتناول هذا البحث مقارنة عدة مقدرات لمعلمة القياس لتوزيع ويبل ذي المعلمتين,حيثمعلمة الشكل و معلمة القياس وتحت افتراض إنمعلومة. تم تقدير المعلمة بطريقة الإمكان الأعظم وطريقة مقدر (Minimax estimator) الذي يعتمد على طريقة بيز في التقدير وتحت دالة خسارة تربيعية ولابد من إيجاد المقدر الذي يجعل اكبر خسارة متوقعة هي اقل ما يمكن،ومقدر الوسيط والمقدر الرابع هو مقدر وايت المعتمد على الانحدار ومقدر طريقة المربعات الصغرى .وسوف تتم المقارنة بين المقدرات باعتبار إنثابت مفروض وتجرى المقارنة بين المقدرات الأربعة للمعلمةبواسطة المحاكاة وباعتماد المقياس الإحصائي متوسط مربعات الخطأ mse كأساس في المقارنة وتحت حجوم العيناتوقيم مختلفة للمعلمات والتي أظهرت بان الطريقة minimax المعتمدة على دالة الخسارة التربيعية mom كانت أفضل وأكفاء طريقة لأنها حققت اقل MSE لجميع الحالات وأحجام العينات المستخدمة في البحث.

A SIMPLE MINIMAX ESTIMATOR FOR QUANTUM STATES

Quantum tomography requires repeated measurements of many copies of the physical system, all prepared by a source in the unknown state. In the limit of very many copies measured, the often-used maximum-likelihood (ML) method for converting the gathered data into an estimate of the state works very well. For smaller data sets, however, it often suffers from problems of rank deficiency in the estimated state. For many systems of relevance for quantum information processing, the preparation of a very large number of copies of the same quantum state is still a technological challenge, which motivates us to look for estimation strategies that perform well even when there is not much data. After reviewing the concept of minimax state estimation, we use minimax ideas to construct a simple estimator for quantum states. We demonstrate that, for the case of tomography of a single qubit, our estimator significantly outperforms the ML estimator for small number of copies of the state measured. Our estimator is always full-rank, and furthermore, has a natural dependence on the number of copies measured, which is missing in the ML estimator.