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

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

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

7-3.3 Standard Error: Reporting a Point Estimate

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

Example 7-5

Example 7-5 (continued)

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

MSE cont’d

Problem 7-14 Which is the better estimate of m

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

7-4 Methods of Point Estimation Definition

The Method of Moments Solve simultaneously for the p unknown parameters

Method of Moments – Normal Probability Distribution

Example 7-7 is wrong

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

More Method of Moments Moments The Rectangular Distribution

Many More Method of Moments Moments

7-4.2 Method of Maximum Likelihood Definition

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.

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

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.

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

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

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

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.

Example 7-12

Example 7-12 (continued)

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().

Weibull MLE Deriving the Weibull MLEs under Type II testing.

More Weibull MLE see problem 7-37

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.

The Invariance Property

Example 7-13

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

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:

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

Compare MoM and MLE for the rectangular

Let’s Find another Unbiased Estimator

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.

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.

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