1. Chapter 7 Point Estimation
Is there a point to all of this
Chapter 7B

2. This is shaping up to be one terrific class.
Today in Prob/Stat
This is shaping up to be one terrific class.

3. 7-3 General Concepts of Point Estimation
7-3.3 Standard Error: Reporting a Point Estimate
Definition

4. 7-3.3 Standard Error: Reporting a Point Estimate

5. This is how you will compute it
Comment on Notation
Based on review of statistics texts, typical notation does not distinguish between a point estimate definition and its evaluation as a number. Examples:
Both of these co-exist in texts
Similarly, you will see both of these in the same text
These are the numbers.
This is how you will compute it

6. Example 7-5

7. Example 7-5 (continued)

8. Mean Square Error (MSE)
A measure of the worth of an estimator
Definition:
The MSE assesses the quality of the estimator
in terms of its variation and unbiasedness

9. MSE cont’d

10. Problem 7-14
Which is the better estimate of m

11. Problem 7-18
Pick the best of the three estimators of q:

12. 7-4 Methods of Point Estimation
Definition

13. The Method of Moments Solve simultaneously
for the p unknown parameters

14. Method of Moments – Normal Probability Distribution

15. Example 7-7 is wrong

16. A Method of Moments Moment
For the exponential distribution:
Since E[X] = 1/

17. More Method of Moments Moments The Rectangular Distribution

18. Many More Method of Moments Moments

19. 7-4.2 Method of Maximum Likelihood
Definition

20. Point Estimation Methods
Max Likelihood Methods – given the observed sample, what is the best set of parameters for the assumed distribution?
The max likelihood estimator is the value of q that maximizes L(q). Maximizing L(q) is equivalent to maximizing the natural log of L(q). Using the log generally gives a simpler function form to maximize.

21. Maximum Likelihood Estimators
The likelihood function:
maximize the log of the likelihood function:
The function that is to be maximized is called the likelihood function since it provides the probability (likelihood) of generating the actual sample. For a continuous distribution, the density function evaluated at the ith failure time, ti, is used in place of the probability mass function. For censored data, the probability that no failures will occur (i.e. reliability) before the censored time must be included in the likelihood function.
Solve k equations for k unknowns

22. MLE – Geometric Distribution
Let X = a discrete random variable, the number of trials
to obtain the first success.
Prob{X=x} = f(x) = (1-p)x-1 p, x = 1, 2, ... , n
Understanding the MLE concept is easier with a discrete probability distribution. A function is constructed that provides the probability of obtaining the sample x1, x2 , … xn as a function of the parameter(s). This function is then maximized with respect to the parameter(s) in order to maximize the probability of obtaining the observed sample. In other words, we assign a value to the parameter that maximizes the probability of generating the sample that was actually observed.

23. MLE - Geometric
In general, a optimization problem must be solved to find the MLE.

24. Example MLE of Geometric
The following data was collected on the number of
production runs which resulted in a failure which
stopped the production line: 5, 8, 2, 10, 7, 1, 2, 5.
Therefore, X = the number of production runs necessary
to obtain a failure.
An example to find the MLE for a discrete geometric distribution.
Mean = 1/p = 40/8 = 5
Pr{X = 3} = .82 (0.2) = 0.128

25. Exponential MLE
The above derives the MLE for the exponential when Type II censored data is present.

26. More Exponential MLE
Continuing the derivation. For Type I data, replace tr with t*. The MLE for the lambda is simply the total time on test divided into the number of failures. The MLE for the MTTF is the total time on test, T, divided by r, the number of failures.

27. Example 7-12

28. Example 7-12 (continued)

29. Complications in Using Maximum Likelihood Estimation
It is not always easy to maximize the likelihood function because the equation(s) obtained may be difficult to solve.
It may not always be possible to use calculus methods directly to determine the maximum of L().

30. Weibull MLE
Deriving the Weibull MLEs under Type II testing.

31. More Weibull MLE
see problem 7-37

32. Properties of the MLE 1. MLE’s are invariant: 2. MLE’s are Consistent:
3. MLE’s are (best) asymptotically normal:
4. Required for certain tests such as the Chi-Square GOF test.
5. Has an intuitive appeal.

33. The Invariance Property

34. Example 7-13

35. Now begins a case study in point estimation…
No one is to be admitted once the case study begins
You must take your seats – there can be no standing during this part of the presentation
Not for the faint-hearted – participate at your own risk
No refunds once this part of the presentation begins

36. MoM and MLE and the Rectangular (uniform) Distribution
Let X1, X2, …,Xn be a random sample from a rectangular distribution where b is unknown.
method of moments:

37. MoM and MLE and the Rectangular (uniform) Distribution
maximum likelihood estimator:

38. Compare MoM and MLE for the rectangular

39. Let’s Find another Unbiased Estimator

40. The overachieving student would use this one.
What is the best?
The following estimator has about 1/2 the MSE that the MLE has:
The overachieving student would use this one.

41. A Very Good Summary
Statistic is a point estimator of a population parameter.
(S2, s2, s2 ) <=> (Statistic, point estimate, parameter)
Bias, variance, mean square error are important properties of estimators.
Relative Efficiency of estimators is ratio of their MSE
Central Limit Theorem – allows the use of normal distr
Parameter Estimation
Method of Moments
MLE
MLE have some desirable properties.
Asymptotically unbiased (examples – uniform distribution, normal distribution s2)
Asymptotic normal distributions.
Competitively small variance.
Invariance property.

42. Next Week – Inferential Statistics at its best
Statistical Intervals
- confidence
- tolerance
- prediction