# Mathematical Statistics (2023/2024)

Course code
cod wi: dt001090
Name of lecturer
Catia Scricciolo
Coordinator
Catia Scricciolo
Number of ECTS credits allocated
5
SECS-S/01 - STATISTICS
Language of instruction
Italian
Location
VERONA
Period
Anno accademico 2023/2024 Dottorato di Ricerca dal Oct 1, 2023 al Sep 30, 2024.

#### Learning outcomes

Introduce students to the theory of nonparametric estimation through models and examples.

#### Syllabus

Introduction to the problem of nonparametric estimation and overview of the course topics:
a. methods of construction of estimators,
b. statistical properties of estimators (convergence and rates of convergence),
c. study of optimality of estimators.
Examples of nonparametric problems and models:
- estimation of a probability density,
- nonparametric regression,
- Gaussian white noise model.
Distances/divergences between probability measures:
- Hellinger and total variation distances,
- Scheffè’s theorem and Le Cam’s inequalities,
- Kullback-Leibler and χ2-divergences,
- link inequalities among distances and divergences.
Estimation of the distribution function: definition of the empirical distribution function and consistency.
Estimation of a probability density:
- definition of the Parzen–Rosenblatt kernel density estimator in the uni- and multidimensional cases, examples of kernels,
- definition of the mean squared error (MSE) of kernel estimators at a point and decomposition into the sum of the variance and the squared bias,
- upper bound on the point-wise variance,
- upper bound on the point-wise bias under regularity conditions on the density and the kernel: definitions of Hölder classes and higher order kernels,
- upper bound on the supremum point-wise MSE of kernel estimators,
- mean integrated squared error (MISE): decomposition into the
sum of the integrated variance and the squared bias,
- control of the variance term,
- control of the bias term on Nikol’ski and Sobolev classes of regular densities, upper bound on the MISE for densities in
Sobolev classes.
Fourier analysis of kernel density estimators:
- preliminary facts on Fourier transforms (FT’s),
- the empirical characteristic function: unbiasedness of the FT for the distribution function, expression of the variance,
- expression of the exact MISE of kernel density estimators,
- control of the bias term over Sobolev classes of densities,
- discussion of the local condition around zero on the FT of the
kernel.
Nonparametric regression:
- nonparametric regression with fixed or random design,
- nonparametric regression with random design and the Nadaraya-Watson (N-W) estimator,
- derivation of the expression of the N-W estimator from kernel
density estimators,
- the N-W estimator as a linear nonparametric regression
estimator,
- asymptotic analysis of the N-W estimator,
- nonparametric regression with fixed (regular) design,
- definition of projection (or orthogonal series) estimators,
- the trigonometric basis as an example of orthonormal basis,
- Sobolev classes and ellipsoids,
- bias and MSE of the coefficient estimators,
- control of the residuals by the condition that the vector of
coefficients belongs to a Sobolev ellipsoid, decomposition of
the MISE of the projection estimator and optimal choice of the
cut-off point,
- upper bound on the MISE for the projection estimator,
- connection between the Gaussian white noise model and
nonparametric regression.
Lower bounds on the minimax risk:
- minimax risk associated with a statistical model and a semi-metric,
- definition of an optimal rate of convergence,
- a general reduction scheme for proving lower bounds,
- main theorem on lower bounds based on many hypotheses using the Kullback-Leibler divergence,
- example of lower bound on the minimax L2-risk for the Hölder class in nonparametric regression estimation with fixed
design.

#### Assessment methods and criteria

There is both the possibility of taking a written assessment test in classical form with questions related to topics covered in lectures and the possibility of writing a report on findings from the recent literature on nonparametric statistical inference.

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