Preface

Welcome

Welcome to the notes for Statistical Inference. The course is part of the MSc in Statistics for Data Science from Carlos III University of Madrid.

The course is designed to have, roughly, one session per each main topic in the syllabus. The schedule is tight due to time constraints, which will inevitably make the exposition of certain methods somewhat superficial. Nevertheless, the course and exercises will hopefully give you a respectable panoramic view of the fundamentals of (mostly univariate) statistical inference. A broad view of the syllabus is:

Credits

These notes are possible thanks to the existence of incredible software by Xie (2016), Xie (2020), Allaire et al. (2020), Xie and Allaire (2020), and R Core Team (2020). Also, certain hacks to improve the design layout have been possible due to the outstanding work by Úcar (2018). The icons used in the notes were designed by madebyoliver, freepik, and roundicons from Flaticon.

More general and complete monographs on statistical inference include the books by Casella and Berger (2002), Silvey (1975), Lehmann and Casella (1998), Lehmann and Romano (2005), and Shao (2003).

Last but not least, the notes have benefited from contributions from the following people:

Arconada Manteca, Lúa (fixed one typo).
Askey, Jonathan (fixed one typo).
Blanco Blanco, Germán (fixed one typo).
Díaz Pérez, Álvaro (fixed one typo).
Diego Velarde, Alba (fixed one typo).
Fernández de Marcos Giménez de los Galanes, Alberto (fixed three typos)
Fernández Martín, Manuel (fixed one typo).
Fernández Morín, María (fixed two typos).
Gianelli Nan, Sofía (fixed three typos).
Goñi Artieda, Javier (fixed one typo).
Hackett, Nicole (fixed one typo).
Levenfeld Sabau, Gabriela Hortensia (fixed two typos).
Lodato Olano, Paolo Salvatore (fixed two typos).
Montero de Espinosa Reina, Juan Antonio (fixed four typos).
Morilla Ortega, Francisco Jesús (fixed two typos).
Novillo Correas, Álvaro (fixed one typo).
Pérez Gómez, Eloy (fixed one typo).
Pérez Peralta, Arturo (fixed one typo).
Pérez Piedra, Pilar (fixed one typo).
Qian, Jingye (fixed seven typos).
Rab, Ismael Ahmad (fixed three typos).
Sánchez Polo, Jorge (fixed one typo).
Serrano Ortega, Diego (fixed four typos).
Smith, Kendal Raymond William (performed a thorough proofreading fixing eighty typos and issues).
Velázquez Jamart, Rolando (fixed four typos).
Vidal Pereira, Tomás (fixed one typo).

Contributions

Contributions, reporting of typos, and feedback on the notes are very welcome. Just send an email to edgarcia@est-econ.uc3m.es and give me a good reason for writing your name in the list of contributors!

License

All the material in these notes is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Public License (CC BY-NC-ND 4.0). You may not use this material except in compliance with the aforementioned license. The human-readable summary of the license states that:

You are free to:
- Share – Copy and redistribute the material in any medium or format.
Under the following terms:
- Attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- NonCommercial – You may not use the material for commercial purposes.
- NoDerivatives – If you remix, transform, or build upon the material, you may not distribute the modified material.

Citation

You may use the following \(\mathrm{B{\scriptstyle{IB}} \! T\!_{\displaystyle E} \! X}\) entry when citing these notes:

@book{Molina2026,
    title        = {A First Course on Statistical Inference},
    author       = {Molina Peralta, I. and Garc\'ia-Portugu\'es, E.},
    year         = {2026},
    note         = {Version 2.4.5. ISBN 978-84-09-29680-4},
    url          = {https://egarpor.github.io/SI-UC3M/}
}

You may also want to use the following template:

Molina Peralta, I. and García-Portugués, E. (2026). A First Course on Statistical Inference. Version 2.4.5. ISBN 978-84-09-29680-4. Available at https://egarpor.github.io/SI-UC3M/.

References

Allaire, J. J., Y. Xie, J. McPherson, J. Luraschi, K. Ushey, A. Atkins, H. Wickham, J. Cheng, W. Chang, and R. Iannone. 2020. rmarkdown: Dynamic Documents for R. https://github.com/rstudio/rmarkdown.

Casella, G., and R. L. Berger. 2002. Statistical Inference. Duxbury Advanced Series. Duxbury-Thomson Learning.

Lehmann, E. L., and G. Casella. 1998. Theory of Point Estimation. Second. Springer Texts in Statistics. New York: Springer-Verlag. https://doi.org/10.1007/b98854.

Lehmann, E. L., and J. P. Romano. 2005. Testing Statistical Hypotheses. Third. Springer Texts in Statistics. New York: Springer. https://doi.org/10.1007/0-387-27605-X.

R Core Team. 2020. R: A Language and Environment for Statistical Computing. Vienna. https://www.R-project.org/.

Shao, J. 2003. Mathematical Statistics. Second. Springer Texts in Statistics. New York: Springer-Verlag. https://doi.org/10.1007/b97553.

Silvey, S. D. 1975. Statistical Inference. Monographs on Statistical Subjects. New York: Chapman; Hall. https://doi.org/10.1201/9780203738641.

Úcar, I. 2018. “Energy Efficiency in Wireless Communications for Mobile User Devices.” PhD thesis, Universidad Carlos III de Madrid. https://enchufa2.github.io/thesis/.

Xie, Y. 2016. Bookdown: Authoring Books and Technical Documents with R Markdown. The r Series. Boca Raton: CRC Press. https://bookdown.org/yihui/bookdown/.

———. 2020. knitr: A General-Purpose Package for Dynamic Report Generation in R. https://CRAN.R-project.org/package=knitr.

Xie, Y., and J. J. Allaire. 2020. tufte: Tufte’s Styles for R Markdown Documents. https://CRAN.R-project.org/package=tufte.