A Bayesian Look at Classical Estimation: The Exponential Distribution

In classical estimation usually unbiased estimators are used. This is mainly because the bias term in classical biased estimators in general depends on the parameter to be estimated. However, recently a considerable amount of research has been spent on improving unbiased estimators by introducing a bias, e.g. based on a minimax optimization strategy. In this work we follow this idea of introducing a bias, but we describe a different strategy for optimizing the estimators' performance. Although we stick to classical estimation, we show that the Bayesian linear minimum mean square error estimator can be brought into the same algebraic form as the resulting biased estimator improving the best linear unbiased estimator. This not only emphasizes the fact that this approach leads to betters estimators than the minimax approach on average over all parameters, but also can be seen as another way of reducing the gap between classical and Bayesian estimation.

In the United States, a course in mathematical statistics is a part of the undergraduate curriculum in many departments of mathematics. The traditional course in mathematical statistics, it seems, has not really ‘caught on’ with large numbers of students enrolled in it. In this paper an alternative approach to the course is presented. The approach was implemented in an experimental course at Salisbury State College. Observations concerning the advantages and disadvantages of the method are given. Instructors of courses in mathematical statistics might find the approach appropriate for clarifying concepts in the subject.

In this chapter we present a model for using case studies for the core of an undergraduate theoretical statistics course. These case studies have more depth than most examples found in typical mathematical statistics texts. Each case study provides a real-world question to be addressed as well as background information on the question, data to address the problem, and suggestions for investigating the problem. An important goal is to encourage and develop statistical thinking. This chapter provides both a sample case study on quality control and analysis of variance and a directed project on experimental design that includes a discussion of how we have changed the class to fit these activities into the course. In essence, the case studies become the centerpiece of the course, and as a result, the curriculum, lectures, and assignments are significantly different from a traditional mathematical statistics course.

The purpose of this research is to analyze the difficulty of students in solving problems related to mathematical creative thinking ability comprehensively as a result of the implementation of PACE (Project, Activity, Cooperative learning, Exercise) model and conventional learning in Mathematical Statistics Course. This research was a qualitative research. The subjects in this research were students of Mathematics Education Program who took Mathematical Statistics Course at one of the private universities in East Jakarta. This research used purposive sampling and various instruments. They were test of mathematical prior knowledge (MPK), test of mathematical creative thinking ability (MCTA), observation sheet, interview sheet, and researcher. The data of this research was collected using triangulation techniques. The result of this research show that students are still experiencing difficulties based on MPK level and overall in solving problems related to mathematical creative thinking ability in both learning (PACE model and conventional). However, the difficulty of students who have obtained PACE Model learning is lower than students who have obtained conventional learning. The most of difficulty of students in both learning (PACE model and conventional) is in indicator of ‘originality’. Keywords: mathematical creative thinking ability, Mathematical Statistics, PACE model

Considerable research suggests that traditional assessments in higher education are outdated, and newer, more authentic assessment strategies should be pursued. One such form of authentic assessment is a portfolio of learning evidence, and another is the oral interview or oral assessment. However, the portfolio and interview are not only methods of assessment, but also primary drivers of learning and creative thinking. In statistics and data science education, the portfolio of learning evidence and interview assessment have not been implemented widely, although some authors have explored the implications of assessments. This mixed-methods study examines the interplay between the use of portfolios and interview assessments in a mathematical statistics course and Fink’s framework for significant learning. The study finds that there are indeed numerous advantages that can be gained from using these novel methods instead of traditional learning and teaching methods.

Statistical power is an important topic taught in most graduate-level and undergraduate-level mathematical statistics courses, but it is often difficult to understand conceptually. Visualizing the power curve, sampling distributions, and how they interact can help students more easily conceptualize power, but the creation of such visuals can be difficult and time-consuming. Interactive web applications provide a way for students to dynamically visualize power, and many web applications for understanding power exist. However, most of these applications assume samples are drawn from a Normal population, concern only the sample mean, and/or were created for introductory classes. In this paper, we describe a web application suitable for undergraduate-level and graduate-level mathematical statistics courses that was created to allow users to visualize the complex relationships underlying power for multiple different statistics and population distributions (available at https://powerapp.shinyapps.io/powerapp/); source code is provided for instructors who wish to modify the application. Our experience implementing this application across two different semesters is also discussed, and example activities are provided in the appendix.

Probability theory and mathematical statistics is an important basic course of science and technology, economics and management, and has close contact with industry, agriculture, military and science and technology, so it is particularly important for students to study probability theory and mathematical statistics. The learning effect of students largely depends on the teaching methods of teachers. Under the background of the Internet, the emerging development of Internet education, in order to conform to the development of The Times and improve the quality of teaching, the innovative teaching methods of probability theory and mathematical statistics should also be studied. Based on the analysis of the characteristics and advantages of Internet education, combined with the characteristics of probability theory and mathematical statistics and the shortcomings of current teaching, this paper designs an innovative teaching method based on the application of flipped classroom teaching mode under the Internet. Taking the rain classroom as an example, the teaching method is designed from three links: preview before class, teaching in class and feedback after class. Under the background of Internet education, it is of great significance to improve the teaching quality of probability theory and mathematical statistics courses.

Probability theory and mathematical statistics are fundamental courses for various majors in science and engineering. In response to the current teaching situation, we should integrate theory with practice, implement teaching reform, and carry out teaching innovation. The article carries out blended teaching with deep integration of online and offline modes and within and outside of class, constructing innovative measures of “four integrations and four reshaping.” The article conducts diversified evaluations to stimulate learning motivation and help achieve talent cultivation goals. Through the close integration of probability theory and mathematical statistics course teaching with professional education and practical application, the “three-in-one” teaching goal of value shaping, ability cultivation, and knowledge exploration is achieved. The fundamental task of “cultivating morality and talents” is implemented.

To analyze the performance of various estimators, this paper compares Bayesian and classical estimate approaches across a variety of loss functions. The work focuses on informative versus non-informative priors in Bayesian estimation. The calculations are based on a number of loss functions, including the Square Error Loss Function (SELF), Quadratic Loss Function (QLF), Precautionary Loss Function (PLF), and Entropy Loss Function (ELF). The methodology uses simulation techniques as well as real-world datasets to test the performance and robustness of the proposed estimators. The Bayesian estimators, classical estimators, and related Bayes hazards using empirical analysis and extensive simulations to computed. The findings show that the Minimum Mean Square Error estimator (MiniMSE) is more accurate and reliable than other estimators in a variety of settings. Furthermore, for both types of priors, it is demonstrated that the Bayes risk under the Quadratic Loss Function (QLF) is the lowest among all loss functions considered, including SELF, PLF, and ELF. This suggests that, when compared to other loss functions, QLF is a more effective and balanced criterion for making estimation decisions. The study’s findings provide vital new insights into how to choose the optimum estimate techniques and loss functions for statistical decision theory and practical data processing.

In this department The American Statistician publishes articles, reviews, and notes of interest to teachers of the first mathematical statistics course and of applied statistics courses. The department includes the Accent on Teaching Materials section; suitable contents for the section are described under the section heading. Articles and notes for the department, but not intended specifically for the section, should be useful to a substantial number of teachers of the indicated types of courses or should have the potential for fundamentally affecting the way in which a course is taught. This article examines the usefulness of multimedia technology for teaching statistics, with attention to both promises and pitfalls. We suggest some principles for the design and use of multimedia, and we offer opinions on the role of human teachers in a multimedia educational environment.

In this department The American Statistician publishes articles, reviews, and notes of interest to teachers of the first mathematical statistics course and of applied statistics courses. The department includes the Accent on Teaching Materials section; suitable contents for the section are described under the section heading. Articles and notes for the department, but not intended specifically for the section, should be useful to a substantial number of teachers of the indicated types of courses or should have the potential for fundamentally affecting the way in which a course is taught. When the residuals from a generalized linear model are to be plotted against the fitted values, the plot can be made to resemble that for a normal error model by transforming the fitted values to the constant-information scale.

In this paper, a minimum mean square error-optimized mixture-based estimator is used for packet video error concealment. At the same time as on-line computational complexity is reduced, performance in peak signal-to-noise ratio (PSNR) is increased in comparison with a Gaussian mixture-based estimator that obtains its parameters through probability density estimation by means of the expectation maximization algorithm. Moreover, our method increases performance in PSNR compared to several other previous error concealment algorithms.

<p><em>This qualitative descriptive study aims to describe the implementation of mathematics learning based on Lesson Study using the Reciprocal Teaching model in the Mathematical Statistics course at FKIP Untan and ISBI Singkawang. The research focuses on improving students’ ability to write definitions and theorems symbolically through collaborative interactions among lecturers, students, and classroom contexts. The research procedure followed the Lesson Study cycle (plan–do–see), with data collected through interviews, observations, field notes, and assessments. Data were analyzed using the Miles and Huberman model, involving data reduction, presentation, and conclusion drawing. The results show that implementing Lesson Study with Reciprocal Teaching fosters dialogic, collaborative, and continuous learning. In the plan stage, collaborative planning produced a learning design incorporating the strategies of predicting, questioning, clarifying, and summarizing, aligned with students’ characteristics. The do stage revealed active student engagement in group discussions, SPSS practice, and symbolic writing of concepts, although some misconceptions regarding probability, distribution, and variance were still observed. The see stage provided a space for collaborative reflection to identify strengths—such as increased participation, mathematical communication, and conceptual understanding and weaknesses, including uneven participation and limited time allocation. The study concludes that integrating Lesson Study with the Reciprocal Teaching model effectively enhances students’ ability to write definitions and theorems symbolically while strengthening their understanding of Mathematical Statistics concepts. This model is recommended as an innovative strategy for preparing prospective teachers capable of managing reflective, interactive, and meaningful learning.</em></p><p><strong>Pembelajaran Matematika Berbasis <em>Lesson Study</em> Dengan Menggunakan Model <em>Reciprocal Teaching</em> Pada Mata Kuliah Statistika Matematika</strong></p><p><strong>ABSTRAK<br /></strong>Penelitian deskriptif kualitatif ini bertujuan untuk mendeskripsikan implementasi pembelajaran matematika berbasis Lesson Study dengan menggunakan model Reciprocal Teaching pada mata kuliah Statistika Matematika di FKIP Untan dan ISBI Singkawang. Fokus penelitian diarahkan pada peningkatan kemampuan mahasiswa dalam menuliskan definisi dan teorema secara simbolik melalui interaksi kolaboratif antara dosen, mahasiswa, dan konteks kelas. Prosedur penelitian mengikuti siklus Lesson Study (plan–do–see) dengan teknik pengumpulan data berupa wawancara, observasi, catatan lapangan, dan asesmen. Analisis data dilakukan dengan model Miles dan Huberman melalui reduksi, penyajian, serta penarikan kesimpulan. Hasil penelitian menunjukkan bahwa implementasi Lesson Study dengan Reciprocal Teaching mampu menciptakan pembelajaran yang dialogis, kolaboratif, dan berkesinambungan. Pada tahap plan, perencanaan kolaboratif menghasilkan rancangan pembelajaran dengan strategi predicting, questioning, clarifying, dan summarizing yang sesuai dengan karakteristik mahasiswa. Tahap do memperlihatkan keterlibatan aktif mahasiswa dalam diskusi kelompok, praktik penggunaan SPSS, dan penulisan konsep secara simbolik, meskipun masih ditemukan miskonsepsi terkait peluang, distribusi, dan variansi. Tahap see memberikan ruang refleksi kolaboratif untuk mengidentifikasi kelebihan berupa peningkatan partisipasi, komunikasi matematis, dan pemahaman konseptual, serta kelemahan berupa distribusi partisipasi yang belum merata dan keterbatasan waktu. Kesimpulan penelitian menegaskan bahwa integrasi Lesson Study dengan Reciprocal Teaching efektif meningkatkan kemampuan menulis definisi dan teorema dalam bentuk simbolik, sekaligus memperkuat pemahaman konsep Statistika Matematika. Model ini direkomendasikan sebagai strategi inovatif untuk menyiapkan calon guru yang mampu mengelola pembelajaran reflektif, interaktif, dan bermakna.</p><p><strong>Kata Kunci</strong><strong> </strong>:<br />Lesson Study; Model Reciprocal Teaching; Statistika Matematika</p>

The aims of this study are to: a) develop Socrates-based mathematical statistics teaching materials; b) testing the quality of teaching materials in terms of validity, effectiveness, and practicality. The research was carried out in the even semester of 2021/2022 in the fourth semester of the mathematical statistics course for all classes. This study uses a development method using a 4-D model consisting of 4 main stages, namely define, design, develop, disseminate. The product developed is in the form of teaching materials that have been tested for feasibility with content validity, effectiveness tests by analyzing student learning outcomes after the teaching materials are used and practicality tests through student response analysis. The collection techniques and instruments are: 1) indirect communication techniques with the instruments being expert validation questionnaires and response questionnaires for instrument validation and student responses; 2) the measurement technique with the instrument is a test of learning outcomes for the effectiveness of the product; 3) direct communication techniques with interview sheets as instruments to find out the advantages and disadvantages of teaching materials when used for students. The data analysis techniques are: a) describe or describe each step of development with a 4-D model; b) expert validation analysis on the feasibility of teaching materials, namely qualitative descriptive, the results of the study are stated in the form of logical analysis; c) calculation of the validity of the assessment results and validation of the consistency of the experts using Aiken's formulation. The overall research results are: 1) the development of Socrates-based mathematical statistics teaching materials is carried out according to the main 4-D stages in sequence; 2) Socrates-based mathematical statistics teaching materials have content validity and reliability with very high criteria; 3) student learning outcomes after using Socrates-based mathematics teaching materials are classified as high criteria. So it can be concluded that the teaching materials developed are effective.

In estimating the parameters of the logistic curve when the response at any given dose is binomial, the method of “maximum likelihood” and the method of “minimum logit chi-square” have both been suggested. Computational difficulties formerly associated with the maximum likelihood solution have now largely disappeared. In a comparison of the two methods it is shown that the maximum likelihood estimates, but not the minimum logit chi-square estimates, are sufficient estimators for the parameters. Particular attention is paid to samples where all but one of the responses are zero or 100 per cent. Also, the effects of using the “2n-rule” for handling the minimum logit chi-square estimation problem for cases of zero or 100 per cent kill in any class are examined. The two sets of estimators are also surveyed from the viewpoint of consistency and some remarks made on the criterion of minimum mean square error estimators when sufficient estimators exist.