Point Point EstimationEstimation

Notes of STAT 6205 by Dr. Fan

OverviewOverview• Section 6.1• Point estimation• Maximum likelihood estimation• Methods of moments• Sufficient statistics

o Definitiono Exponential familyo Mean square error (how to choose an estimator)

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Big Picture• Goal: To study the unknown distribution of a

population• Method: Get a representative/random sample and

use the information obtained in the sample to make statistical inference on the unknown features of the distribution

• Statistical Inference has two parts: o Estimation (of parameters)o Hypothesis testing

• Estimation:o Point estimation: use a single value to estimate a parametero Interval estimation: find an interval covering the unknown parameter

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Point EstimatorPoint Estimator• Biased/unbiased: an estimator is called unbiased

if its mean is equal to the parameter of estimate; otherwise, it is biased

• Example: X_bar is unbiased for estimating mu

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Maximum Likelihood Estimation

• Given a random sample X1, X2, …, Xn from a distribution f(x; ) where is a (unknown) value in the parameter space

• Likelihood function vs. joint pdf

• Maximum Likelihood Estimator (m.l.e.) of , denoted as is the value which maximizes the likelihood function, given the sample X1, X2, …, Xn.

n

iixfxL

1

);();(

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Examples/Exercises• Problem 1: To estimate p, the true probability of heads

up for a given coin.• Problem 2: Let X1, X2, …, Xn be a random sample

from a Exp(mu) distribution. Find the m.l.e. of mu.• Problem 3: Let X1, X2, …, Xn be a random sample

from a Weibull(a=3,b) distribution. Find the m.l.e. of b.

• Problem 4: Let X1, X2, …, Xn be a random sample from a N(,^2) distribution. Find the m.l.e. of and .

• Problem 5: Let X1, X2, …, Xn be a random sample from a Weibull(a,b) distribution. Find the m.l.e. of a and b.

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Method of MomentsMethod of Moments• Idea: Set population moments = sample

moments and solve for parameters

• Formula: When the parameter is r-dimensional, solve the following equations for :

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n

i

ki

k ,...r,knXXE1

21for /)(

Examples/ExercisesExamples/ExercisesGiven a random sample from a population

•Problem 1: Find the m.m.e. of for a Exp() population.

•Exercise 1: Find the m.m.e. of and for a N(^2) population.

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Sufficient StatisticsSufficient Statistics• Idea: The “sufficient” statistic contains all

information about the unknown parameter; no other statistic can provide additional information as to the unknown parameter.

• If for any event A, P[A|Y=y] does not depend on the unknown parameter, then the statistic Y is called “sufficient” (for the unknown parameter).

• Any one-to-one mapping of a sufficient statistic Y is also sufficient.

• Sufficient statistics do not need to be estimators of the parameter.

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Sufficient StatisticsSufficient Statistics

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Examples/ExercisesExamples/ExercisesLet X1, X2, …, Xn be a random sample from f(x)

Problem: Let f be Poisson(a). Prove that1.X-bar is sufficient for the parameter a2.The m.l.e. of a is a function of the sufficient statistic

Exercise: Let f be Bin(n, p). Prove that X-bar is sufficient for p (n is known). Hint: find a sufficient statistic Y for p and then show that X-bar is a function of Y

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Exponential FamilyExponential Family

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Examples/ExercisesExamples/Exercises

Example 1: Find a sufficient statistic for p for Bin(n, p)

Example 2: Find a sufficient statistic for a for Poisson(a)

Exercise: Find a sufficient statistic for for Exp()

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Joint Sufficient StatisticsJoint Sufficient Statistics

Example: Prove that X-bar and S^2 are joint sufficient statistics for and of N(, ^2)

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Application of SufficienceApplication of Sufficience

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ExampleExampleConsider a Weibull distribution with parameter(a=2, b)

1)Find a sufficient statistic for b

2)Find an unbiased estimator which is a function of the sufficient statistic found in 1)

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Good Estimator?Good Estimator?• Criterion: mean square error

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ExampleExample• Which of the following two estimator of variance

is better?

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