This book is dedicated to the mathematical modeling of statistical phenomena. It attempts to fill a gap: most textbooks on mathematical statistics only treat the case of sampling, and yet, in applications, the observed variables are very often correlated – examples are numerous in physics, chemistry, biology, demography, economics and finance. It is for this reason that we have given an important place to process statistics.
This book is divided into three parts: Mathematical Statistics, Statistics for Stochastic Processes, and a supplement on probability. The first part begins with decision theory, to develop the classical theory of estimation and tests. It then adopts an asymptotic viewpoint, in particular, in a non-parametric framework. In the second part, we first study the statistics of discrete-time stationary processes, and then the statistics of Poisson processes. The third part is dedicated to continuous-time processes: we define the Itô integral and give some results for the statistics of second-order process and for diffusion. Statistical prediction is addressed in the last chapter. Finally, the principles of probability are described, based on measure theory.
The mathematical level is that of a master’s degree in applied mathematics: I have based the material on some lectures given at UPMC (Paris 6) in France, particularly [BOS 78] and [BOS 93].
A certain number of exercises, some original, illustrate the contents of this textbook. I would like to thank the teachers who contributed in putting them together, notably Gérard Biau, Delphine Blanke, Jérôme Dedecker, and Florence Merlevède.
I would also like to thank François-Xavier Lejeune for his excellent write-up of the manuscript, and for his contribution to the development of this book.
Finally, I would like to thank Maximilian Lock for all his work on this book.