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7. Sampling Distributions and Point Estimation of Parameters. CHAPTER OUTLINE. 7-1 Point Estimation 7-2 Sampling Distributions and the Central Limit Theorem 7-3 General Concepts of Point Estimation 7-3.1 Unbiased Estimators 7-3.2 Variance of a Point Estimator - PowerPoint PPT Presentation
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1Chapter 7 Title and Outline
7 Sampling Distributions and Point Estimation of Parameters
7-1 Point Estimation7-2 Sampling Distributions and the Central Limit Theorem7-3 General Concepts of Point Estimation 7-3.1 Unbiased Estimators 7-3.2 Variance of a Point Estimator 7-3.3 Standard Error: Reporting a Point Estimate
7-3.4 Mean Square Error of an Estimator7-4 Methods of Point Estimation 7-4.1 Method of Moments 7-4.2 Method of Maximum Likelihood 7-4.3 Bayesian Estimation of Parameters
CHAPTER OUTLINE
Chapter 7 Learning Objectives 2
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Learning Objectives for Chapter 7After careful study of this chapter, you should be able to do the
following:1. Explain the general concepts of estimating the parameters of a
population or a probability distribution.2. Explain the important role of the normal distribution as a sampling
distribution.3. Understand the central limit theorem.4. Explain important properties of point estimators, including bias,
variances, and mean square error.5. Know how to construct point estimators using the method of moments,
and the method of maximum likelihood.6. Know how to compute and explain the precision with which a parameter
is estimated.7. Know how to construct a point estimator using the Bayesian approach.
Sec 7-1 Point Estimation 3
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Point Estimation
• A point estimate is a reasonable value of a population parameter.
• Data collected, X1, X2,…, Xn are random variables.
• Functions of these random variables, x-bar and s2, are also random variables called statistics.
• Statistics have their unique distributions that are called sampling distributions.
Sec 7-1 Point Estimation 4
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Point Estimator
A point estimate of some population parameter θ
is a single numerical value .
The statistic pis called the oint esti r.mato
As an example,suppose the random variable is normally distributed with an unknown mean μ. The sample mean is a point estimator of the unknown
population mean μ. That is, μ . After the sample ha
X
X
1 2 3 4
s been selected,
the numerical value is the point estimate of μ.Thus if 25, 30, 29, 31, the point estimate of μ is
25 30 29 31 28.754
xx x x x
x
Sec 7-1 Point Estimation 5
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Some Parameters & Their Statistics
• There could be choices for the point estimator of a parameter.• To estimate the mean of a population, we could choose the:
– Sample mean.– Sample median.– Average of the largest & smallest observations of the sample.
• We need to develop criteria to compare estimates using statistical properties.
Parameter Measure Statisticμ Mean of a single population x-barσ2 Variance of a single population s 2
σ Standard deviation of a single population sp Proportion of a single population p -hat
μ1 - μ2 Difference in means of two populations x bar1 - x bar2
p 1 - p 2 Difference in proportions of two populations p hat1 - p hat2
Sec 7-2 Sampling Distributions and the Central Limit Theorem 6
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Some Definitions
• The random variables X1, X2,…,Xn are a random sample of size n if:a) The Xi are independent random variables.b) Every Xi has the same probability distribution.
• A statistic is any function of the observations in a random sample.
• The probability distribution of a statistic is called a sampling distribution.
Sec 7-2 Sampling Distributions and the Central Limit Theorem 7
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Sampling Distribution of the Sample Mean
• A random sample of size n is taken from a normal population with mean μ and variance σ2.
• The observations, X1, X2,…,Xn, are normally and independently distributed.
• A linear function (X-bar) of normal and independent random variables is itself normally distributed.
1 2
2 2 22
2
... has a normal distribution
...with mean
...and variance
n
X
X
X X XX
n
n
n
Sec 7-2 Sampling Distributions and the Central Limit Theorem 8
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Central Limit Theorem
1 2
2
If , ,..., is a random sample of size is taken from a population (either finite or infinite)
with mean and finite variance , and if is the sample mean, then the limiting form of the dist
nX X X n
X
ribution of
(7-1)
as , is th standard normal distribute .ion
XZn
n
Sec 7-2 Sampling Distributions and the Central Limit Theorem 9
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Sampling Distributions of Sample Means
Figure 7-1 Distributions of average scores from throwing dice. Mean = 3.5
n dies var
std dev
a) 1 2.9 1.7b) 2 1.5 1.2c) 3 1.0 1.0d) 5 0.6 0.8e) 10 0.3 0.5
a = 1b = 6
22
2 2
Form
2
ulas
1 112X
XX
b a
b a
n
Sec 7-2 Sampling Distributions and the Central Limit Theorem 10
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 7-1: ResistorsAn electronics company manufactures
resistors having a mean resistance of 100 ohms and a standard deviation of 10 ohms. The distribution of resistance is normal. What is the probability that a random sample of n = 25 resistors will have an average resistance of less than 95 ohms?
10 2.025
95 2002
2.5 0.0062
XX
X
n
X
Figure 7-2 Desired probability is shaded
Answer:
0.0062 = NORMSDIST(-2.5)A rare event at less than 1%.
Sec 7-2 Sampling Distributions and the Central Limit Theorem 11
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 7-2: Central Limit TheoremSuppose that a random variable X has a continuous uniform
distribution:
Find the distribution of the sample mean of a random sample of size n = 40.
1 2, 4 x 60, otherwise
f x
2 22
22
Distribution is normal by the CLT.6 4 5.0
2 26 4
1 312 12
1 3 140 120X
b a
b a
n
Figure 7-3 Distributions of X and X-bar
Sec 7-2 Sampling Distributions and the Central Limit Theorem 12
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Two PopulationsWe have two independent normal populations. What is
the distribution of the difference of the sample means?
1 2 1 2
1 2 1 2
1 2
1 2
2 22 2 2 1 2
1 2
1 2
1 2
The sampling distribution of is:
The distribution of is normal if:(1) and are both greater than 30,regardless of the distributions of
X X X X
X X X X
X X
n n
X Xn n
1 2
1 2
and .(2) and are less than 30,while the distributions are somewhat normal.
X Xn n
Sec 7-2 Sampling Distributions and the Central Limit Theorem 13
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Sampling Distribution of a Difference in Sample Means
• If we have two independent populations with means μ1 and μ2, and variances σ1
2 and σ22,
• And if X-bar1 and X-bar2 are the sample means of two independent random samples of sizes n1 and n2 from these populations:
• Then the sampling distribution of:
is approximately standard normal, if the conditions of the central limit theorem apply.
• If the two populations are normal, then the sampling distribution is exactly standard normal.
1 2 1 2
2 21 2
1 2
(7-4)X X
Z
n n
Sec 7-2 Sampling Distributions and the Central Limit Theorem 14
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 7-3: Aircraft Engine LifeThe effective life of a component
used in jet-turbine aircraft engines is a normal-distributed random variable with parameters shown (old). The engine manufacturer introduces an improvement into the manufacturing process for this component that changes the parameters as shown (new).
Random samples are selected from the “old” process and “new” process as shown.
What is the probability the difference in the two sample means is at least 25 hours?
Old (1) New (2) Diff (2-1)x -bar = 5,000 5,050 50
s = 40 30 50n = 16 25
s / √n = 10 6 11.7z = -2.14
P(xbar2-xbar1 > 25) = P(Z > z) = 0.9840= 1 - NORMSDIST(z)
Process
Calculations
Figure 7-4 Sampling distribution of the sample mean difference.
Sec 7-3 General Concepts of Point Estimation 15
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
General Concepts of Point Estimation
• We want point estimators that are:– Are unbiased.– Have a minimal variance.
• We use the standard error of the estimator to calculate its mean square error.
Sec 7-3.1 Unbiased Estimators 16
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Unbiased Estimators Defined
The point estimator is an for the parameter θ if:
θ (7-5)
If the estimator is not unbiased, then the diff
u
e
n
rence:
biased estimator
E
θ (7-6)
is called the of the estimator .
The mean of the sampling distribution of is equ
bi
al
a
o θ.
s
t
E
Sec 7-3.1 Unbiased Estimators 17
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 7-4: Sample Man & Variance Are Unbiased-1
• X is a random variable with mean μ and variance σ2. Let X1, X2,…,Xn be a random sample of size n.
• Show that the sample mean (X-bar) is an unbiased estimator of μ.
1 2
1 2
...
1 ...
1 ...
n
n
X X XE X E
n
E X E X E Xn
nn n
Sec 7-3.1 Unbiased Estimators 18
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 7-4: Sample Man & Variance Are Unbiased-2
Show that the sample variance (S2) is a unbiased estimator of σ2.
2
2 2 21
1
2 2 2 2
1 1
2 2 2 2
1
2 2 2 2 2 2
1 21 1
1 11 1
11
1 1 11 1
n
ni
i ii
n n
i ii i
n
i
X XE S E E X X XX
n n
E X nX E X nE Xn n
n nn
n n n nn n
Sec 7-3.1 Unbiased Estimators 19
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Other Unbiased Estimators of the Population Mean
• All three statistics are unbiased.– Do you see why?
• Which is best?– We want the most reliable one.
110.4Mean = 11.0410
10.3 11.6Median = 10.952
110.04 8.5 14.1Trimmed mean = 10.818
X
X
i x i x i '1 12.8 8.52 9.4 8.73 8.7 9.44 11.6 9.85 13.1 10.36 9.8 11.67 14.1 12.18 8.5 12.89 12.1 13.1
10 10.3 14.1Σ 110.4
Sec 7-3.2 Variance of a Point Estimate 20
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Choosing Among Unbiased Estimators
Figure 7-5 The sampling distributions of two unbiased estimators.
1 2
1 2
1
Suppose that and are unbiased estimators of θ.
The variance of is less than the variance of .
is preferable.
Sec 7-3.2 Variance of a Point Estimate 21
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Minimum Variance Unbiased Estimators
• If we consider all unbiased estimators of θ, the one with the smallest variance is called the minimum variance unbiased estimator (MVUE).
• If X1, X2,…, Xn is a random sample of size n from a normal distribution with mean μ and variance σ2, then the sample X-bar is the MVUE for μ.
• The sample mean and a single observation are unbiased estimators of μ. The variance of the:– Sample mean is σ2/n– Single observation is σ2
– Since σ2/n ≤ σ2, the sample mean is preferred.
Sec 7-3.3 Standard Error Reporting a Point Estimate 22
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Standard Error of an Estimator
The of an estimator is its standard deviation, given by
.
If the standard error involves unknown parameters tha
s
t can be estimated
tandard error
,
V
substitution of these values into
produces an , denoted by .
Equivalent notation:
If the are ~ , , then
estimated
is norm
standar
ally di
d error
stributed,
and . If is not kno
i
X
s se
X N X
n
wn, then .X
sn
Sec 7-3.3 Standard Error Reporting a Point Estimate 23
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 7-5: Thermal Conductivity
• These observations are 10 measurements of thermal conductivity of Armco iron.
• Since σ is not known, we use s to calculate the standard error.
• Since the standard error is 0.2% of the mean, the mean estimate is fairly precise. We can be very confident that the true population mean is 41.924 ± 2(0.0898).
x i41.6041.4842.3441.9541.8642.1841.7242.2641.8142.04
41.924 = Mean0.284 = Std dev (s )
0.0898 = Std error
Sec 7-3.4 Mean Squared Error of an Estimator 24
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Mean Squared Error
Conclusion: The mean squared error (MSE) of the estimator is equal to the variance of the estimator plus the bias squared. It measures both characteristics.
2
2 2
The mean squared error of an estimator of the parameter θ is defined as:
MSE θ (7-7)
Can be rewritten as θ
E
E E E
2 biasV
Sec 7-3.4 Mean Squared Error of an Estimator 25
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Relative Efficiency
• The MSE is an important criterion for comparing two estimators.
• If the relative efficiency is less than 1, we conclude that the 1st estimator is superior to the 2nd estimator.
1
2
Relative efficieMSE
n E
cyMS
Sec 7-3.4 Mean Squared Error of an Estimator 26
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Optimal Estimator• A biased estimator can be
preferred to an unbiased estimator if it has a smaller MSE.
• Biased estimators are occasionally used in linear regression.
• An estimator whose MSE is smaller than that of any other estimator is called an optimal estimator.
Figure 7-6 A biased estimator has a smaller variance than the unbiased estimator.
Sec 7-4 Methods of Point Estimation 27
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Methods of Point Estimation
• There are three methodologies to create point estimates of a population parameter.– Method of moments– Method of maximum likelihood– Bayesian estimation of parameters
• Each approach can be used to create estimators with varying degrees of biasedness and relative MSE efficiencies.
Sec 7-4.1 Method of Moments 28
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Method of Moments
• A “moment” is a kind of an expected value of a random variable.
• A population moment relates to the entire population or its representative function.
• A sample moment is calculated like its associated population moments.
Sec 7-4.1 Method of Moments 29
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Moments Defined• Let X1, X2,…,Xn be a random sample from the probability
f(x), where f(x) can be either a:– Discrete probability mass function, or– Continuous probability density function
• The kth population moment (or distribution moment) is E(Xk), k = 1, 2, ….
• The kth sample moment is (1/n)ΣXk, k = 1, 2, ….• If k = 1 (called the first moment), then:– Population moment is μ.– Sample moment is x-bar.
• The sample mean is the moment estimator of the population mean.
Sec 7-4.1 Method of Moments 30
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Moment Estimators
1 2
1 2
1 2
Let , ,..., be a random sample from either a probability mass function or a probability density function with unknown parameters θ ,θ ,...,θ .
The , ,..., are found by e
moment estimatqu
ors
n
m
m
X X X
m
ating the first population moments
to the first sample moments and solving the resulting simultaneous equations for the unknown parameters.
mm
Sec 7-4.1 Method of Moments 31
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 7-6: Exponential Moment Estimator-1
• Suppose that X1, X2, …, Xn is a random sample from an exponential distribution with parameter λ.
• There is only one parameter to estimate, so equating population and sample first moments, we have E(X) = X-bar.
• E(X) = 1/λ = x-bar• λ = 1/x-bar is the moment estimator.
Sec 7-4.1 Method of Moments 32
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 7-6: Exponential Moment Estimator-2
• As an example, the time to failure of an electronic module is exponentially distributed.
• Eight units are randomly selected and tested. Their times to failure are shown.
• The moment estimate of the λ parameter is 0.04620.
x i11.96
5.0367.4016.0731.50
7.7311.1022.38
21.646 = Mean0.04620 = λ est
Sec 7-4.1 Method of Moments 33
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 7-7: Normal Moment Estimators
Suppose that X1, X2, …, Xn is a random sample from a normal distribution with parameter μ and σ2. So E(X) = μ and E(X2) = μ2 + σ2.
2 2 2
1 1
22
2 1 12 2
1
22
12 1
1
1 1 and
11
1 (biased)
n n
i ii i
n n
i ini i
ii
n n
i ini i
ii
X X Xn n
X n Xn
X Xn n
X X XX
n n n
Sec 7-4.1 Method of Moments 34
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 7-8: Gamma Moment Estimators-1
222
22
2
2 2
1
2 2
1
Parameters = Statistics
is the mean
is the variance or
1 and now solving for and :
1/
1/
n
ii
n
ii
r E X X
r E X E X
r rE X r
Xrn X X
X
n X X
Sec 7-4.1 Method of Moments 35
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 7-8: Gamma Moment Estimators-2
Using the exponential example data shown, we can estimate the parameters of the gamma distribution.
2 2
22 2
1
22 2
1
21.646 1.291 8 6645.4247 21.6461/
21.646 0.05981 8 6645.4247 21.6461/
n
ii
n
ii
Xrn X X
X
n X X
x i x i2
11.96 143.04165.03 25.3009
67.40 4542.760016.07 258.244931.50 992.2500
7.73 59.752911.10 123.210022.38 500.8644
x-bar = 21.646ΣX 2 = 6645.4247
Sec 7-4.2 Method of Maximum Likelihood 36
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Maximum Likelihood Estimators• Suppose that X is a random variable with probability
distribution f(x:θ), where θ is a single unknown parameter. Let x1, x2, …, xn be the observed values in a random sample of size n. Then the likelihood function of the sample is:
L(θ) = f(x1: θ) ∙ f(x2; θ) ∙…∙ f(xn: θ) (7-9)
• Note that the likelihood function is now a function of only the unknown parameter θ. The maximum likelihood estimator (MLE) of θ is the value of θ that maximizes the likelihood function L(θ).
• If X is a discrete random variable, then L(θ) is the probability of obtaining those sample values. The MLE is the θ that maximizes that probability.
Sec 7-4.2 Method of Maximum Likelihood 37
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 7-9: Bernoulli MLELet X be a Bernoulli random variable. The probability mass
function is f(x;p) = px(1-p)1-x, x = 0, 1 where P is the parameter to be estimated. The likelihood function of a random sample of size n is:
1 21 2
11
1 1 1
1
1
1 1
11
1
1 1 ... 1
1 1
ln ln ln 1
ln0
1
nn
nn
ii ii i
i
x x xxx x
n xx n xx
i
n n
i ii i
nn
iiii
n
ii
L p p p p p p p
p p p p
L p x p n x p
n xxd L pdp p p
xp
n
Sec 7-4.2 Method of Maximum Likelihood 38
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 7-10: Normal MLE for μLet X be a normal random variable with unknown mean μ and
known variance σ2. The likelihood function of a random sample of size n is:
2 2
22
1
2
1
12
22
222
1
21
1
12
1
2
1ln ln 22 2
ln 1 0
(same as moment estimator)
i
n
ii
nx
i
x
n
n
ii
n
ii
n
ii
L e
e
nL x
d Lx
d
xX
n
Sec 7-4.2 Method of Maximum Likelihood 39
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 7-11: Exponential MLELet X be a exponential random variable with parameter λ. The
likelihood function of a random sample of size n is:
1
1
1
1
1
ln ln
ln0
1 (same as moment estimator)
n
ii i
n xx n
i
n
ii
n
ii
n
ii
L e e
L n x
d L n xd
n x X
Sec 7-4.2 Method of Maximum Likelihood 40
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Why Does MLE Work?• From Examples 7-6 & 11 using the 8 data observations, the plot of the
ln L(λ) function maximizes at λ = 0.0462. The curve is flat near max indicating estimator not precise.
• As the sample size increases, while maintaining the same x-bar, the curve maximums are the same, but sharper and more precise.
• Large samples are better
Figure 7-7 Log likelihood for exponential distribution. (a) n = 8, (b) n = 8, 20, 40.
Sec 7-4.2 Method of Maximum Likelihood 41
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 7-12: Normal MLEs for μ & σ2
Let X be a normal random variable with both unknown mean μ and variance σ2. The likelihood function of a random sample of size n is:
2 2
22
1
22
1
12
22
22 22
1
2
21
22
2 421
2
21
1,2
1
2
1ln , ln 22 2
ln , 1 0
ln , 1 02 2
and
i
n
ii
nx
i
x
n
n
ii
n
ii
n
ii
n
ii
L e
e
nL x
Lx
L n x
x XX
n
Sec 7-4.2 Method of Maximum Likelihood 42
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Properties of an MLE
Under very general and non-restrictive conditions,
when the sample size n is large and if is the MLE of the parameter ,
(1) is an approximately unbiased estimator for θ, i.e., θ
(2) The var
E
iance of is nearly as small as the variance that could be obtained with any other estimator, and
(3) has an approximate normal distribution.
Notes:• Mathematical statisticians will often prefer MLEs because of these properties. Properties (1) and (2) state that MLEs are MVUEs.• To use MLEs, the distribution of the population must be known or assumed.
Sec 7-4.2 Method of Maximum Likelihood 43
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Importance of Large Sample Sizes• Consider the MLE for σ2 shown in Example 7-12:
• Since the bias is negative, the MLE underestimates the true variance σ2.
• The MLE is an asymptotically (large sample) unbiased estimator. The bias approaches zero as n increases.
2
2 21
22 2 2 2
1
Then the bias is:
1
n
ii
x XnE
n n
nEn n
44
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Invariance Property
Sec 7-4.2 Method of Maximum Likelihood
1 2
1 2
1 2
1 2
Let , ,..., be the maximum likelihood estimators (MLEs)of the parameters θ ,θ ,...,θ .
Then the MLEs for any function θ ,θ ,...,θ of these parameters
is the same function , ,..., of the
k
k
k
k
h
h
1 2estimators , ,..., k
This property is illustrated in Example 7-13.
Sec 7-4.2 Method of Maximum Likelihood 45
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 7-13: Invariance
For the normal distribution, the MLEs were:
2
21
2 2
2
2
2 1
and
To obtain the MLE of the function , ,
substitute the estimators and into the function :
which is the sample standard deviation not s.
n
ii
n
ii
x XX
n
h
h
x X
n
Sec 7-4.2 Method of Maximum Likelihood 46
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Complications of the MLE Method
The method of maximum likelihood is an excellent technique, however there are two complications:
1. It may not be easy to maximize the likelihood function because the derivative function set to zero may be difficult to solve algebraically.
2. The likelihood function may be impossible to solve, so numerical methods must be used.
The following two examples illustrate.
Sec 7-4.2 Method of Maximum Likelihood 47
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 7-14: Uniform Distribution MLE
Let X be uniformly distributed on the interval 0 to a.
1
11
1 for 0
1 1 for 0
max
nn
ini
nn
i
f x a x a
L a a x aa a
dL a n nada aa x
Calculus methods don’t work here because L(a) is maximized at the discontinuity. Clearly, a cannot be smaller than max(xi), thus the MLE is max(xi).
Figure 7-8 The likelihood function for this uniform distribution
Sec 7-4.2 Method of Maximum Likelihood 48
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 7-15: Gamma Distribution MLE-1Let X1, X2, …, Xn be a random sample from a gamma
distribution. The log of the likelihood function is:
1
1
1 1
1
1
1
ln , ln
ln 1 ln ln
ln , 'ln ln 0
ln ,0
' and ln ln
There is no closed solution
ixr rni
i
n n
i ii i
n
ii
n
ii
n
ii
x eL r
r
nr r x n r x
L r rn x n
r r
L r nr x
rr n x nrx
for and .r
Sec 7-4.2 Method of Maximum Likelihood 49
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 7-15: Gamma Distribution MLE-2
Figure 7-9 Log likelihood for the gamma distribution using the failure time data (n=8). (a) is the log likelihood surface. (b) is the contour plot. The log likelihood function is maximized at r = 1.75, λ = 0.08 using numerical methods. Note the imprecision of the MLEs inferred by the flat top of the function.
7-4.3 Bayesian Estimation of Parameters 50
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Bayesian Estimation of Parameters-1• The moment and likelihood methods interpret probabilities as
relative frequencies and are called objective frequencies.• The Bayesian method combines sample information with prior
information.• The random variable X has a probability distribution of parameter
θ called f(x|θ). θ could be determined by classical methods.• Additional information about θ can be expressed as f(θ), the prior
distribution, with mean μ0 and variance σ02, with θ as the random
variable. Probabilities associated with f(θ) are subjective probabilities.
• The joint distribution is f(x1, x2, …, xn, θ)• The posterior distribution is f(θ|x1, x2, …, xn) is our degree of belief
regarding θ after gathering data
7-4.3 Bayesian Estimation of Parameters 51
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Bayesian Estimation of Parameters-2• Now putting these together, the joint is:– f(x1, x2, …, xn, θ) = f(x1, x2, …, xn |θ) ∙ f(θ)
• The marginal is:
• The desired posterior distribution is:
• And the Bayesian estimator of θ is the expected value of the posterior distribution
1 2θ
1 2
1 2
, ,..., ,θ , for θ discrete
, ,...,, ,..., ,θ θ, for θ continuous
n
n
n
f x x x
f x x xf x x x d
1 21 2
1 2
, ,..., ,θθ | , ,...,
, ,...,n
nn
f x x xf x x x
f x x x
7-4.3 Bayesian Estimation of Parameters 52
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 7-16: Bayes Estimator for a Normal Mean-1
Let X1, X2, …, Xn be a random sample from a normal distribution unknown mean μ and known variance σ2. Assume that the prior distribution for μ is:
2 2 22 20 0 00 0
2 22
20 0
1 1μ2 2
f e e
The joint distribution of the sample is:
2 2
1
2 2 2
1 1
1 2 ( )
1 2 22
1 2 2
22
1, ,..., |2
1
2
n
ii
n n
i ii i
x
n n
x x n
n
f x x x e
e
7-4.3 Bayesian Estimation of Parameters 53
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 7-16: Bayes Estimator for a Normal Mean-2
Now the joint distribution of the sample and μ is:
2 0
2 2 2 20 0
1 2 1 2
22 20
2 22 0 0
2 2 2 2 2 20 0 0
1 11 2 2
1
, ,..., , , ,..., | μ
1
2 2
1 1where 22
& completing the
n n
un
i i
xn n
f x x x f x x x f
e
x xnu
h e
2 20 0
2 2 2 20
22 202 0
2 2 2 2 2 2
0
0 0 0
2
1 11 2
2
1 12 1
1 2 3
square
| , ,..., is the posterior distribution
functi
n x
n
n xn n n
n
i
n
h e
f x x x h e
h
on to collect unneeded components (not )
7-4.3 Bayesian Estimation of Parameters 54
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 7-16: Bayes Estimator for a Normal Mean-3
• After all that algebra, the bottom line is:
• Observations:– Estimator is a weighted average of μ0 and x-bar.– x-bar is the MLE for μ.– The importance of μ0 decreases as n increases.
2 20 0
2 20
1 2 20
2 2 2 20 0
1 1
n xE
n
nV
n n
7-4.3 Bayesian Estimation of Parameters 55
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 7-16: Bayes Estimator for a Normal Mean-4
To illustrate:– The prior parameters: μ0 = 0, σ0
2= 1– Sample: n = 10, x-bar = 0.75, σ2 = 4
2 20 0
2 20
4 10 0 1 0.750.536
1 4 10
n x
n
Chapter 7 Summary 56
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Important Terms & Concepts of Chapter 7
Bayes estimatorBias in parameter estimationCentral limit theoremEstimator vs. estimateLikelihood functionMaximum likelihood estimatorMean square error of an estimatorMinimum variance unbiased
estimatorMoment estimatorNormal distribution as the sampling
distribution of the:– sample mean– difference in two sample
meansParameter estimationPoint estimatorPopulation or distribution momentsPosterior distributionPrior distributionSample momentsSampling distributionAn estimator has a:
– Standard error– Estimated standard error
StatisticStatistical inferenceUnbiased estimator