Estadística II: Inferencia Estadística en Una Población, Apuntes de Estadística

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Course - Statistics II

Statistical inference in one population

Edoardo MENGA [email protected]

Lecture 10/09/

• From population to random samples
• Inference
• Point estimator and its properties
• Exercises

A ‘sample’ statistic is a random variable because it depends on the sample choice. Different ‘random’ choice of the sample will lead to different sample statistics! 𝑛 1 , 𝑥 1 , 𝑛 2 , 𝑥 2 , 𝑛 3 , 𝑥 3 , 𝑛 4 , 𝑥 4 , The Sampling distribution of the sample means is called the sample mean.

The sample mean is a random variable, it can be used as a point estimator of the population mean. A specific value of an estimator is called an estimate (it is a single number!). 𝐸 𝑋 = 𝐸( 1 𝑛 1 𝑛 𝑋𝑖) = 𝐸(𝑋) The expected value of the sample mean is the population mean The variance of the sample mean is the population variance reduced by a factor n 𝑉𝑎𝑟 𝑋 = 𝑉𝑎𝑟( 1 𝑛 1 𝑛 𝑋𝑖) = 1 𝑛

What are Point Estimators? Point estimators are functions that are used to find an approximate value of a population parameter from random samples of the population. They use the sample data of a population to calculate a point estimate or a statistic that serves as the best estimate of an unknown parameter of a population. 𝜽 Unbiasedness (^) 𝑏𝑖𝑎𝑠 𝜽 = 𝐸(𝜽) - 𝜽 A point estimator is said to be an unbiased estimator of a population parameter ϴ if its expected value is equal to that parameter. Efficiency The unbiased estimator with the smallest variance is called the most efficient estimator (minimum variance unbiased estimator) 𝑟𝑒𝑙. 𝑒𝑓𝑓. 𝜽𝟏, 𝜽𝟐 =

𝑀𝑆𝐸 𝜽 = 𝐸[(𝜽 − 𝜽)

𝟐 ] = 𝑉𝑎𝑟( 𝜽)+[𝑏𝑖𝑎𝑠 𝜽 ] 2

• The mean squared error of an unbiased estimator equals its variance.
• An estimator with a smaller MSE is better.
• The minimum variance unbiased estimator has the smallest variance/MSE among all estimators General criterion If the optimal estimator does not exist?
  • Maximum likelihood estimator
  • Method of moments

Sum n terms of an arithmetic series

• End of the first part of lesson