Statistical Estimation Ch05 Statistical Estimation

CHAPTER CONTENTS CHAPTER CONTENTS 5.1 Introduction 220 5.2 The Methods of Finding Point Estimators 221 5.3 Some Desirable Properties of Point Estimators 245 5.4 A Method of Finding the Confidence Interval: Pivotal Method 261 5.5 One Sample Confidence Intervals 269 5.6 A Confidence Interval for the Population Variance 284 5.7 Confidence Interval Concerning Two Population Parameters 289 5.8 Chapter Summary 298 5.9 Computer Examples 299 Projects for Chapter 5 303

Unknown population parameters 5.1 Introduction Unknown population parameters To estimate: point estimation interval estimation How much money do I have in my pocket? 1000 $ (700, 1200)

5.2The Methods of Finding Point Estimators

Pdf or pmf of the population(?) (1, . . ., l) X1, . . ., Xn independent and identically distributed (iid) random variables (in statistical language, a random sample) f (x, 1, . . ., l) Pdf or pmf of the population(?) (1, . . ., l) the unknown population parameters Point estimation： to determine statistics gi(X1, . . ., Xn), i = 1, . . ., l, which can be used to estimate the value of each of the parameters

the method of maximum likelihood Bayes’ method Capital letters such as X and S2 to represent the estimators; Lowercase letters such as x and s2 to represent the estimates. Three of the more popular methods of estimation the method of moments This chapter the method of maximum likelihood Bayes’ method Chapter 11

Unbiased Bias consistency The estimator are said to satisfy the consistency property if the sample estimator has a high probability of being close to the population value  for a large sample size. efficiency smaller variance

5.2.1 THE METHOD OF MOMENTS : the kth population moment about the origin of a random variable X, : the kth sample moment of the random variable X

5.2.2 THE METHOD OF MAXIMUM LIKELIHOOD Even though the method of moments is intuitive and easy to apply, it usually does not yield “good” estimators. The method of maximum likelihood is intuitively appealing, because we attempt to find the values of the true parameters that would have most likely produced the data that we in fact observed. For most cases of practical interest, the performance of MLEs is optimal for large enough data. This is one of the most versatile methods for fitting parametric statistical models to data.

Maximum likelihood estimates give the parameter values for which the observed sample is most likely to have been generated.

At times, the MLEs may be hard to calculate At times, the MLEs may be hard to calculate. It may be necessary to use numerical methods to approximate values of the estimate.

5.3 Some Desirable Properties of Point Estimators

5.3.1 UNBIASED ESTIMATORS

The sample mean is always an unbiased estimator of the population mean.

Sample variance Population variance: Size of population = N Elements of population: X1, X2, ,… , XN

Unbiased estimators need not be unique. If we have two unbiased estimators, there are infinitely many unbiased estimators. It is better to have an estimator that has low bias as well as low variance.

For unbiased estimators,

5.3.2 SUFFICIENCY*

5.4 A Method of Finding the Confidence Interval: Pivotal Method

5.5 One Sample Confidence Intervals

5.6 A Confidence Interval for the Population Variance

5.7 Confidence Interval Concerning Two Population Parameters

5.8 Chapter Summary

5.9 Computer Examples (Optional)

Projects for Chapter 5