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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 7-3.3 Standard Error: Reporting a Point Estimate 7-3.4 Mean Square Error of an Estimator 7-4 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 7 Title and Outline 1 Learning Objectives for Chapter 7 After careful study of this chapter, you should be able to do the following: 1. 2. 3. 4. 5. 6. 7. Explain the general concepts of estimating the parameters of a population or a probability distribution. Explain the important role of the normal distribution as a sampling distribution. Understand the central limit theorem. Explain important properties of point estimators, including bias, variances, and mean square error. Know how to construct point estimators using the method of moments, and the method of maximum likelihood. Know how to compute and explain the precision with which a parameter is estimated. Know how to construct a point estimator using the Bayesian approach. Chapter 7 Learning Objectives © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 2 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 3 © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Point Estimator A point estim ate of som e population para m eter θ is a single num erical value . T he statistic is called the p oint esti m ato r. A s an exam ple, suppose the random variable X is norm ally distributed w ith an unknow n m ean μ. T he sam ple m ean is a point estim ator of the unknow n population m ean μ. T hat is, μ X . A fter the sam ple ha s been selected, the num erical value x is the point estim a te of μ. T hus if x1 25, x 2 30, x 3 29, x 4 31, the point estim ate of μ is x 25 30 29 31 28.75 4 Sec 7-1 Point Estimation 4 © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Some Parameters & Their Statistics Parameter μ σ2 σ p μ1 - μ2 p1 - p2 Measure Mean of a single population Variance of a single population Standard deviation of a single population Proportion of a single population Difference in means of two populations Difference in proportions of two populations Statistic x-bar s2 s p -hat x bar1 - x bar2 p hat1 - p hat2 • 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. Sec 7-1 Point Estimation 5 © 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 © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 6 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. X X 1 X 2 ... X n has a norm al distribution n w ith m ean X ... and variance X 2 n 2 2 n ... 2 2 Sec 7-2 Sampling Distributions and the Central Limit Theorem © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 7 Central Limit Theorem If X 1 , X 2 , ..., X n is a random sam ple of size n is taken from a population (either finite o r infinite) w ith m ean and finite variance , and 2 if X is the sam ple m ean, then the lim iting form of the dist ribution of Z X (7-1) n as n , is th e standard norm al distribut ion . Sec 7-2 Sampling Distributions and the Central Limit Theorem © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 8 Sampling Distributions of Sample Means Figure 7-1 Distributions of average scores from throwing dice. Mean = 3.5 F o rm u las ba 2 2 X b a 1 12 X X n 2 2 2 1 a) b) c) d) e) n std dies var dev 1 2.9 1.7 2 1.5 1.2 3 1.0 1.0 5 0.6 0.8 10 0.3 0.5 a= 1 b= 6 Sec 7-2 Sampling Distributions and the Central Limit Theorem © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9 Example 7-1: Resistors An 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? Answer: X X n 10 Figure 7-2 Desired probability is shaded 2 .0 25 X 95 200 2 X 2 .5 0 .0 0 6 2 0.0062 = NORMSDIST(-2.5) A rare event at less than 1%. Sec 7-2 Sampling Distributions and the Central Limit Theorem © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 10 Example 7-2: Central Limit Theorem Suppose that a random variable X has a continuous uniform distribution: 1 2 , 4 x 6 f x 0, otherw ise Find the distribution of the sample mean of a random sample of size n = 40. D istribution is norm al by the C LT . ba 64 2 2 2 b a 2 6 4 12 X 2 5.0 2 n 2 1 3 12 1 3 40 1 120 Figure 7-3 Distributions of X and X-bar Sec 7-2 Sampling Distributions and the Central Limit Theorem © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 11 Two Populations We have two independent normal populations. What is the distribution of the difference of the sample means? T he sam pling distribution of X 1 X 2 is: X 1 X2 X X 1 2 1 2 1 2 X 2 1 X2 2 X1 2 X2 n1 2 2 n2 T he distribution of X 1 X 2 is norm al if: (1) n1 and n 2 are both greater than 30, regardless of the distributions of X 1 and X 2 . (2) n1 and n 2 are less than 30, w hile the distributions are som ew hat norm al. Sec 7-2 Sampling Distributions and the Central Limit Theorem © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 12 Sampling Distribution of a Difference in Sample Means • If we have two independent populations with means μ1 and μ2, and variances σ12 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: Z X 1 X 2 1 2 2 1 n1 (7-4) 2 2 n2 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. Sec 7-2 Sampling Distributions and the Central Limit Theorem © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 13 Example 7-3: Aircraft Engine Life The 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? Figure 7-4 Sampling distribution of the sample mean difference. Process Old (1) New (2) Diff (2-1) x -bar = 5,000 5,050 50 s= 40 30 50 n= 16 25 Calculations s / √n = 10 6 11.7 z= -2.14 P(xbar2-xbar1 > 25) = P(Z > z) = 0.9840 = 1 - NORMSDIST(z) Sec 7-2 Sampling Distributions and the Central Limit Theorem © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 14 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 General Concepts of Point Estimation © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 15 Unbiased Estimators Defined T he point estim ator is an u n biased estim ator for the param eter θ if: E θ (7-5) If the estim ator is not unbiased, then t he diff e rence: E θ (7-6) is called the bi a s of the estim ator . T he m ean of the sam pling distribution of is equ al t o θ . Sec 7-3.1 Unbiased Estimators © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 16 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. EX X 1 X 2 ... X n E n 1 n 1 n E X 1 E X 2 ... E X n ... n n Sec 7-3.1 Unbiased Estimators © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 17 Example 7-4: Sample Man & Variance Are Unbiased-2 Show that the sample variance (S2) is a unbiased estimator of σ2. E S 2 E n X X 2 i 1 n 1 1 n 2 E X i X n 1 i 1 2 2 XX i n 1 n 2 2 2 E X nX E X nE X i i n 1 i 1 n 1 i 1 1 n 2 n 1 i 1 1 1 n 2 n n 1 2 2 n 2 2 2 n 1 2 2 n 1 2 n n 1 Sec 7-3.1 Unbiased Estimators © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 2 18 Other Unbiased Estimators of the Population Mean M ean = X 110.4 11.04 10 M edian = X 10.3 11.6 10.95 2 T rim m ed m ean = 110.04 8.5 14.1 10.81 8 • All three statistics are unbiased. – Do you see why? • Which is best? – We want the most reliable one. Sec 7-3.1 Unbiased Estimators © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. i xi xi' 1 2 3 4 5 6 7 8 9 10 Σ 12.8 9.4 8.7 11.6 13.1 9.8 14.1 8.5 12.1 10.3 110.4 8.5 8.7 9.4 9.8 10.3 11.6 12.1 12.8 13.1 14.1 19 Choosing Among Unbiased Estimators S uppose that 1 and 2 are unbiased estim ato rs of θ. T he variance of 1 is less than the varian ce of 2 . 1 is preferable. Figure 7-5 The sampling distributions of two unbiased estimators. Sec 7-3.2 Variance of a Point Estimate © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 20 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 7-3.2 Variance of a Point Estimate © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 21 Standard Error of an Estimator T he s tandard error of an estim ator is its standard de viation, given by V . If the standard error involves unknow n p aram eters tha t can be estim ated , substitution of these values into produces an estim ated standar d error , denoted by . E quivalent notation: s se If the X i are ~ N , , then X is norm ally di stributed, and X . If is not kno w n, then n X s . n Sec 7-3.3 Standard Error Reporting a Point Estimate © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 22 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). xi 41.60 41.48 42.34 41.95 41.86 42.18 41.72 42.26 41.81 42.04 41.924 = Mean 0.284 = Std dev (s ) 0.0898 = Std error Sec 7-3.3 Standard Error Reporting a Point Estimate © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 23 Mean Squared Error T he m ean squared error of an estim ator of the param eter θ is defined as: M SE E θ 2 (7-7) 2 C an be rew ritten as E E θ E V bias 2 2 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. Sec 7-3.4 Mean Squared Error of an Estimator © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 24 Relative Efficiency • The MSE is an important criterion for comparing two estimators. M SE M SE 1 R elative efficie n cy 2 • If the relative efficiency is less than 1, we conclude that the 1st estimator is superior to the 2nd estimator. Sec 7-3.4 Mean Squared Error of an Estimator © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 25 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 7-3.4 Mean Squared Error of an Estimator © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 26 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 Methods of Point Estimation © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 27 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 © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 28 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 © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 29 Moment Estimators L et X 1 , X 2 , ..., X n be a random sam ple from eithe r a probability m ass function or a probabi lity density function w ith m unknow n param eters θ 1 , θ 2 , ..., θ m . T he m om ent estim at ors 1 , 2 , ..., m are found by e qu ating the first m population m om ents to the first m sam ple m om ents and solving the resulting sim ultaneous equat ions for the unknow n param eters. Sec 7-4.1 Method of Moments © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 30 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 © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 31 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. xi 11.96 5.03 67.40 16.07 31.50 7.73 11.10 22.38 21.646 = Mean 0.04620 = λ est Sec 7-4.1 Method of Moments © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 32 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. X 1 n n Xi and 2 2 i 1 1 n n i 1 Xi X 2 n 2 Xi i 1 n 2 n 1 2 i 1 1 2 Xi n n n 2 n n Xi 1 i 1 2 X i n i 1 n n i 1 Xi n X i i 1 X 2 2 (biased) n Sec 7-4.1 Method of Moments © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 33 Example 7-8: Gamma Moment Estimators-1 P aram eters = S tatistics r r 2 r r 1 2 EX EX 2 EX EX 2 r X is th e m ean 2 is th e varian ce o r an d n o w so lvin g fo r r an d : 2 X n 1 / n Xi X 2 2 i 1 X n 1 / n Xi X 2 2 i 1 Sec 7-4.1 Method of Moments © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 34 Example 7-8: Gamma Moment Estimators-2 xi Using the exponential example data shown, we can estimate the parameters of the gamma distribution. x-bar = 21.646 11.96 5.03 67.40 16.07 31.50 7.73 11.10 22.38 ΣX 2 = 6645.4247 2 X r n 1 / n Xi X 2 2 21.646 2 xi 143.0416 25.3009 4542.7600 258.2449 992.2500 59.7529 123.2100 500.8644 2 1 8 6645.4247 21.646 2 1.29 i 1 X n 1 / n X i X 2 2 21.646 1 8 6645.4247 21.646 2 0.0598 i 1 Sec 7-4.1 Method of Moments © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 35 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 © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 36 Example 7-9: Bernoulli MLE Let 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: L p p x1 1 p 1 x1 p x2 1 p 1 x2 ... p xn 1 p 1 xn n n p xi 1 p 1 xi xi p i 1 n 1 p n xi i 1 i 1 n ln L p x i ln p n i 1 n i 1 x i ln 1 p n n x xi i d ln L p i 1 i 1 0 dp p 1 p n n p x i i 1 n Sec 7-4.2 Method of Maximum Likelihood © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 37 Example 7-10: Normal MLE for μ Let X be a normal random variable with unknown mean μ and known variance σ2. The likelihood function of a random n x 2 1 sample of size n is: L e 2 2 i i 1 2 1 1 2 2 ln L d ln L d n 2 ln 2 n 1 n 2 2 e x i 2 2 2 n xi 2 i 1 n 1 2 2 xi 2 i 1 0 i 1 n x i 1 i X (sam e as m om ent estim ator) n Sec 7-4.2 Method of Maximum Likelihood © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 38 Example 7-11: Exponential MLE Let X be a exponential random variable with parameter λ. The likelihood function of a random sample of size n is: n L n e xi e n xi i 1 i 1 n ln L n ln x i i 1 d ln L d n n x i 0 i 1 n n x i 1 X (sam e as m om ent estim ator) i 1 Sec 7-4.2 Method of Maximum Likelihood © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 39 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 © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 40 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: 1 L , e n i 1 1 2 2 ln L , ln L , 2 2 ln L , 2 2 n n x 2 e 2 2 n xi i 1 Sec 7-4.2 Method of Maximum Likelihood n 1 2 x 2 2 i i 2 0 2 n 1 2 4 xi and 2 0 i 1 n X 2 i 1 n 2 2 i 1 1 n 2 ln 2 2 2 2 1 2 xi 2 2 xi X 2 i 1 n © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 41 Properties of an MLE U nder very general and non-restrictive c onditions, w hen the sam ple size n is large and if is the M LE of the param eter , (1) is an approxim ately unbiased estim ator for θ, i.e., E θ (2) T he var iance of is nearly as sm all as the vari ance that could be obtained w ith any other es tim ator, and (3) has an approxim ate norm al distribu tion. 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 © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 42 Importance of Large Sample Sizes • Consider the MLE for σ2 shown in Example 7-12: n E 2 x i X 2 i 1 n 1 n 2 n T hen the bias is: E 2 2 n 1 n 2 2 2 n • 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. Sec 7-4.2 Method of Maximum Likelihood © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 43 Invariance Property Let 1 , 2 , ..., k be the m axim um likelihood est im ators (M LE s) of the param eters θ 1 , θ 2 , ..., θ k . T hen the M LE s for any function h θ 1 , θ 2 , ..., θ k is the sam e function h 1 , 2 , ..., k of these param eters of the estim ators , 1 2 , ..., k This property is illustrated in Example 7-13. Sec 7-4.2 Method of Maximum Likelihood © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 44 Example 7-13: Invariance For the normal distribution, the MLEs were: n X and 2 x X i 2 i 1 n T o obtain the M LE of the function h , substitute the estim ators and n 2 x i X 2 2 2 , into the function h : 2 i 1 n w hich is not the sam ple standard deviation s. Sec 7-4.2 Method of Maximum Likelihood © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 45 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 © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 46 Example 7-14: Uniform Distribution MLE Let X be uniformly distributed on the interval 0 to a. f x 1 a for 0 x a L a n i 1 dL a da 1 a n a n 1 1 a n na a n for 0 x i a n 1 a m ax x i Figure 7-8 The likelihood function for this uniform distribution 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). Sec 7-4.2 Method of Maximum Likelihood © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 47 Example 7-15: Gamma Distribution MLE-1 Let X1, X2, …, Xn be a random sample from a gamma distribution. The log of the likelihood function is: n r x ir 1 e xi ln L r , ln r i 1 n n i 1 i 1 nr ln r 1 ln x i n ln r x i ln L r , r ln x n n ln ' r i i 1 ln L r , n nr r x r 0 n x i 0 i 1 ' r ln x n r n and n ln i i 1 T here is no closed solution for r and . Sec 7-4.2 Method of Maximum Likelihood © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 48 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. Sec 7-4.2 Method of Maximum Likelihood © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 49 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 © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 50 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: f x1 , x 2 , ..., x n f x1 , x 2 , ..., x n , θ , for θ discrete θ f x1 , x 2 , ..., x n , θ d θ, for θ continuous • The desired posterior distribution is: f θ | x1 , x 2 , ..., x n f x1 , x 2 , ..., x n , θ f x1 , x 2 , ..., x n • And the Bayesian estimator of θ is the expected value of the posterior distribution 7-4.3 Bayesian Estimation of Parameters © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 51 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: f μ 1 2 0 e 0 2 2 0 2 1 2 e 2 0 0 2 2 2 0 2 2 0 The joint distribution of the sample is: f x1 , x 2 , ..., x n | 1 2 1 2 2 n 2 2 2 n 2 2 i 1 e ( x ) i e 1 2 1 n 2 2 n i 1 n 2 xi 2 i 1 7-4.3 Bayesian Estimation of Parameters © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 2 xi n 52 Example 7-16: Bayes Estimator for a Normal Mean-2 Now the joint distribution of the sample and μ is: f x1 , x 2 , ..., x n , f x1 , x 2 , ..., x n | f μ 1 2 2 n 2 1 2 w here u 2 h1 e u 2 1 n 2 2 2 0 0 xi 2 2 0 1 1 x 2 1 2 2 0 2 2 n 2 2 n 0 0 h2 e f | x1 , x 2 , ..., x n h3 e 2 0 e 2 2 xi 0 2 2 0 & com pleting the square 2 2 1 x 0 1 2 n 0 1 2 2 2 n 2 2 n 2 2 n 0 0 0 2 2 1 1 2 n 0 0 x 1 2 2 2 2 2 n 0 n 0 2 is the posterior distribution hi functi on to collect unneeded com ponents (not ) 7-4.3 Bayesian Estimation of Parameters © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 53 Example 7-16: Bayes Estimator for a Normal Mean-3 • After all that algebra, the bottom line is: E 2 n 0 0 x 2 0 2 1 1 V 2 2 0 n 2 n 1 0 2 0 2 n 2 2 n • 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. 7-4.3 Bayesian Estimation of Parameters © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 54 Example 7-16: Bayes Estimator for a Normal Mean-4 To illustrate: – The prior parameters: μ0 = 0, σ02= 1 – Sample: n = 10, x-bar = 0.75, σ2 = 4 2 n 0 0 x 2 0 2 4 2 n 10 0 1 0.75 1 4 10 0.536 7-4.3 Bayesian Estimation of Parameters © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 55 Important Terms & Concepts of Chapter 7 Bayes estimator Bias in parameter estimation Central limit theorem Estimator vs. estimate Likelihood function Maximum likelihood estimator Mean square error of an estimator Minimum variance unbiased estimator Moment estimator Normal distribution as the sampling distribution of the: – sample mean – difference in two sample means Parameter estimation Point estimator Population or distribution moments Posterior distribution Prior distribution Sample moments Sampling distribution An estimator has a: – Standard error – Estimated standard error Statistic Statistical inference Unbiased estimator Chapter 7 Summary 56 © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.