Vladimir Spokoiny - Basics of Modern Mathematical Statistics

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Springer-Verlag Berlin Heidelberg 2015

Vladimir Spokoiny and Thorsten Dickhaus Basics of Modern Mathematical Statistics Springer Texts in Statistics 10.1007/978-3-642-39909-1_1

1. Basic Notions

Vladimir Spokoiny 1 and Thorsten Dickhaus 2

(1)

Weierstrass Institute (WIAS), Berlin, Germany

(2)

Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, D-10117 Berlin, Germany

The starting point of any statistical analysis is data , also called observations or a sample . A statistical model is used to explain the nature of the data. A standard approach assumes that the data is random and utilizes some probabilistic framework. On the contrary to probability theory, the distribution of the data is not known precisely and the goal of the analysis is to infer on this unknown distribution.

The parametric approach assumes that the distribution of the data is known up to the value of a parameter from some subset of a finite-dimensional space . In this case the statistical analysis is naturally reduced to the estimation of the parameter : as soon as is known, we know the whole distribution of the data. Before introducing the general notion of a statistical model, we discuss some popular examples.

1.1 Example of a Bernoulli Experiment

Let be a sequence of binary digits zero or one. We distinguish between deterministic and random sequences. Deterministic sequences appear, e.g., from the binary representation of a real number, or from digitally coded images, etc. Random binary sequences appear, e.g., from coin throw, games, etc. In many situations incomplete information can be treated as random data: the classification of healthy and sick patients, individual vote results, the bankruptcy of a firm or credit default, etc.

Basic assumptions behind a Bernoulli experiment are:

• the observed data Y i are independent and identically distributed.
• each Y i assumes the value one with probability .

The parameter completely identifies the distribution of the data . Indeed, for every i n and y { 0,1},

and the independence of the Y i s implies for every sequence that

(1.1)

To indicate this fact, we write in place of .

Equation () can be rewritten as

where

The value s n is often interpreted as the number of successes in the sequence .

Probabilistic theory focuses on the probabilistic properties of the data under the given measure . The aim of the statistical analysis is to infer on the measure for an unknown based on the available data . Typical examples of statistical problems are:

Estimate the parameter , i.e. build a function of the data into [0,1] which approximates the unknown value as well as possible;

Build a confidence set for , i.e. a random (data-based) set (usually an interval) containing with a prescribed probability;

Testing a simple hypothesis that coincides with a prescribed value 0, e.g. 0=12;

Testing a composite hypothesis that belongs to a prescribed subset of the interval [0,1].

Usually any statistical method is based on a preliminary probabilistic analysis of the model under the given .

Theorem 1.1.1.

Let be i.i.d. Bernoulli with the parameter . Then the mean and the variance of the sum satisfy

Exercise 1.1.1.

Prove this theorem.

This result suggests that the empirical mean is a reasonable estimate of . Indeed, the result of the theorem implies

The first equation means that is an unbiased estimate of , that is, for all . The second equation yields a kind of concentration (consistency) property of : with n growing, the estimate

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