1. Statistical Estimation
Ch05
Statistical Estimation

2. CHAPTER CONTENTS
CHAPTER CONTENTS
5.1 Introduction
5.2 The Methods of Finding Point Estimators
5.3 Some Desirable Properties of Point Estimators
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
Projects for Chapter

3. 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)

4. 5.2The Methods of Finding Point Estimators

5. 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

6. 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

7. 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

8. 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

16. 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.

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

27. 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.

31. 5.3 Some Desirable Properties of Point Estimators

32. 5.3.1 UNBIASED ESTIMATORS

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

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

38. 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.

39. For unbiased estimators,

40. 5.3.2 SUFFICIENCY*

46. 5.4 A Method of Finding the Confidence Interval: Pivotal Method

49. 5.5 One Sample Confidence Intervals

51. 5.6 A Confidence Interval for the Population Variance

54. 5.7 Confidence Interval Concerning Two Population Parameters

57. 5.8 Chapter Summary

60. 5.9 Computer Examples (Optional)

61. Projects for Chapter 5