Appunti di Statistica con esercizi

Name: Appunti di Statistica con esercizi
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Appunti completi di Statistica. Argomenti: inferenza, stimatori, verosimiglianza, problemi regolari ed irregolari, score function, fisher information, famiglia esponenziale, proprietà …

Esame Numerical and statistical methods for finance

Facoltà Economia

Dal corso del Prof. Favaro Stefano

Università Università degli studi di Torino

A.A. 2022-2023

107 pagine

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SCHEMES - STATISTICS

INTRODUCTION

In Statistic we always start from data and we try to get some informations in order to learn something about the phenomenon. ⇒ we want to MAKE INFERENCE

(x₁, x₂, ..., x_n), n ≥ 1 ⇒ SAMPLE (realizations)
(X₁, X₂, ..., X_n), n ≥ 1 ⇒ RANDOM SAMPLE (random variables)

We assume that the random sample comes from the model X_n∼F_X(., θ), θ∈Θ. It means that we assume that the random variables X₁, X₂, ..., X_n are INDEPENDENT and IDENTICALLY DISTRIBUTED, following a model parameterized by θ.

NOTICE THAT if I fix θ I know everything about the model.

What do we want to do?

→ Starting from data we want to estimate the parameter θ, in order to be able to make inference on the population.

EXAMPLE:

X_n∼N(μ,σ²) , (μ,σ²)∈ℝ x ℝ⁺

In this example the parameters are two: θ = (μ, σ²)So, θ is bi-dimensional (and in general it can be n-dimensional)

Now, let's consider a random sample (X₁, ..., X_n) from X_n∼F_X(., θ) , θ∈Θand suppose that the sample mean X̄_n = 1/n ∑ X_i is an estimator for θ.

But, is it a good estimator? Could we have a better estimator?

We'll try to answer to this kind of questions.

First of all: What is an estimator?

An ESTIMATOR θ̂ = f(X₁, ..., X_n) is a function of the random variables X₁, ..., X_n
We want to study his distributional properties
Examples of estimators: 1) SAMPLE MEAN 2) SAMPLE VARIANCE

The best we can do is to find the distribution of the estimator,but it is almost impossible.

EXCEPTION:

Since the Gaussian model is a particular case we are able to find the distribution of the sample mean and the sample variance.

So, let's assume that we know the model X_n∼F_X(., θ), θ∈Θ and the estimator θ̂ = f(X₁, ..., X_n).

We need to know if the estimator is good, that means if it has some specific desirable properties.

In particular, we'll introduce tools and methods that allow us to get estimators with good properties in order to be a good estimator.

SCHEMES – STATISTICS

INTRODUCTION

In Statistic we always start from data and we try to get some informations in order to learn something about the phenomenon. ⇒ we want to MAKE INFERENCE

(x₁, x₂, ..., x_n), n ≥ 1 ⇒ SAMPLE (realizations)
(X₁, X₂, ..., X_n), n ≥ 1 ⇒ RANDOM SAMPLE (random variables)

We assume that the random sample comes from the model X_n~F(.,θ), θ∈Θ. It means that we assume that the random variables X₁, X₂, ..., X_n are INDEPENDENT and IDENTICALLY DISTRIBUTED, following a model parameterized by θ.

NOTICE THAT if I fix θ I know everything about the model.

What do we want to do ? ⇒ Starting from data, we want to estimate the parameter θ, in order to be able to make inference on the population.

EXAMPLE: X_n~N(μ, σ²) , (μ, σ²) ∈ ℝ⁺

In this example the parameters are two : θ^′ = (μ, σ²)So, θ^′ is bi-dimensional (and in general it can be n-dimensional)

Now, let’s consider a random sample (X₁,...,X_n ) from X_n~F(.,θ), θ∈Θ and suppose that the sample mean X_n = ¹/_n ∑_iⁿ Xi is an estimator for θ.

But, is it a good estimator? Could we have a better estimator?

We’ll try to answer to this kind of questions.

First of all : What is an estimator?

An ESTIMATOR Θ̂ = f(X₁,...,X_n) is a function of the random variables X₁,...,X_n
We want to study his distributional properties
Examples of estimators : 1) SAMPLE MEAN 2) SAMPLE VARIANCE

The best we can do is to find the distribution of the estimator, but it is almost impossible.

EXCEPTION: Since the Gaussian model is a particular case we are able to find the distribution of the sample mean and the sample variance.

So, let’s assume that we know the model X_n~F(.,θ), θ∈Θ and the estimator Θ̂ = f(X₁,...,X_n).

We need to know if the estimator is good, that means if it has some specific desirable p

Anteprima

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