Course Name: Statistical Inference (SI) and Simulations
Participants: BSc Mathematics and Data Science students of Sorbonne University
Faculty: Dr. Tanujit Chakraborty
Timeline: September to December every year | Sessions: 60 Sessions
Email: tanujit.chakraborty@sorbonne.ae
Course Introduction: The course is a classical advanced course in (parametric) Mathematical Statistics. The general theory of point estimation is presented, within which the notions of unbiasedness, consistency, and optimality are studied. Particular attention is given to maximum likelihood estimation. The next topics are confidence intervals and hypothesis testing. The course ends with an account of the linear model. In the framework of the course, a series of computer labs will be run on the following topics: simulations of random variables (by inversion, rejection sampling, and by the Box-Muller transform) and Monte-Carlo methods in Statistics.
Course Objectives: The course will help the students by:
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Proving the Glivenko-Cantelli Theorem and assessing its fundamental character within Mathematical Statistics.
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Determining the laws of order statistics and of functions of order statistics.
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Inferring the laws of the main statistics built from a Gaussian random sample, including the chi-squared statistic and the T-statistic.
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Articulate the notion of a dominated statistical model as well as that of an unbiased and consistent estimator. Construct the minimum variance unbiased estimator by the Rao-Blackwell/Lehmann-Scheffé procedure and efficient estimators in the sense of Cramér-Rao.
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Prove efficiency properties of Maximum Likelihood Estimators.
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Construct exact and asymptotic confidence intervals, likelihood ratio tests, and goodness-of-fit tests (chi-squared and Kolmogorov-Smirnov).
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Construct least-squares estimators in the framework of the general linear model and prove their efficiency properties.
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Test generalized linear hypotheses on parameters of linear models.
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Represent ANOVA as a linear model and perform statistical inferences in these models.
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Computer Lab Classes using RStudio.
Evaluation Components: The evaluation components for the Statistical Inference (SI) course will be as follows:
- Mid-Term Examination: 25%; 2) Capstone Project (Report and Presentation): 25%; 2) End Term Test (Theory): 50%.
Textbooks
• Rice, John A. (2006). Mathematical statistics and data analysis, Cengage Learning (Easy-to-Follow Textbook)
• Casella, George, and Roger L. Berger (2002). Statistical inference, Cengage Learning (Another Textbook)
• Wasserman, Larry (2004). All of statistics: a concise course in statistical inference, Springer. (Must Try!)
Some Very Interesting Papers For Reading:
I would recommend that all participants go through these research articles (mostly non-mathematical) along with the course. Please click on the paper name to view these outstanding and interesting papers:
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Probability - Fair coins tend to land on the same side they started: Evidence from 350,757 flips (2024): https://arxiv.org/pdf/2310.04153
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Information Geometry - The Many Faces of Information Geometry (2022): https://www.ams.org/journals/notices/202201/rnoti-p36.pdf
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Statistics Vs Data Science - The science of statistics versus data science: What is the future? (2021): https://www.sciencedirect.com/science/article/pii/S0040162521005448
Main References:
[1] Rice, John A. Mathematical statistics and data analysis. Cengage Learning, 2006. (Download Link: https://korivernon.com/documents/MathematicalStatisticsandDataAnalysis3ed.pdf)
[2] Course Notes of Stanford University: https://web.stanford.edu/class/stats200/lectures.html
[3] Course Notes of UCL: https://github.com/FJRubio67/StatisticalInference
[4] Course Notes of Columbia University: http://www.stat.columbia.edu/~bodhi/Talks/Lecture-Notes-Sp-2020.pdf
[5] Course Notes of Penn State University: https://online.stat.psu.edu/stat415/
Disclaimer: These notes closely follow the material in the textbooks cited in the documents and several publicly available online materials, including courses from Stanford University, UCL, PSU, and Columbia University, with some additions by the author. These notes should not be distributed or used for commercial purposes.