Method of Moments

More documents

Recommendations

Info

$Math 520a - Midterm take home exam Do 5 out of the 6 problems ...$

$page 1 of 2 Math 410 Matrix Analysis Fall 2007 Final Exam Answers ...$

Introduction to the Science of Statistics The Method of Moments 13.2 The Procedure More generally, for independent random variables X 1 ,X 2 ,...chosen according to the probability distribution derived from the parameter value ✓ and m a real valued function, if k(✓) =E ✓ m(X 1 ), then 1 n nX m(X i ) ! k(✓) as n !1. The method of moments results from the choices m(x) =x m . Write i=1 µ m = EX m = k m (✓). (13.1) for the m-th moment. Our estimation procedure follows from these 4 steps to link the sample moments to parameter estimates. • Step 1. If the model has d parameters, we compute the functions k m in equation (13.1) for the first d moments, µ 1 = k 1 (✓ 1 ,✓ 2 ...,✓ d ), µ 2 = k 2 (✓ 1 ,✓ 2 ...,✓ d ), ..., µ d = k d (✓ 1 ,✓ 2 ...,✓ d ), obtaining d equations in d unknowns. • Step 2. We then solve for the d parameters as a function of the moments. ✓ 1 = g 1 (µ 1 ,µ 2 , ··· ,µ d ), ✓ 2 = g 2 (µ 1 ,µ 2 , ··· ,µ d ), ..., ✓ d = g d (µ 1 ,µ 2 , ··· ,µ d ). (13.2) • Step 3. Now, based on the data x =(x 1 ,x 2 ,...,x n ) we compute the first d sample moments, x, x 2 ,...,x d . Using the law of large numbers, we have that µ m ⇡ x m . • Step 4. We replace the distributional moments µ m by the sample moments x m , then the solutions in (13.2) give us formulas for the method of moment estimators (ˆ✓ 1 , ˆ✓ 2 ,...,ˆ✓ d ). For the data x, these estimates are ˆ✓ 1 (x) =g 1 (¯x, x 2 , ··· , x d ), ˆ✓2 (x) =g 2 (¯x, x 2 , ··· , x d ), ..., ˆ✓d (x) =g d (¯x, x 2 , ··· , x d ). How this abstract description works in practice can be best seen through examples. 13.3 Examples Example 13.1. Let X 1 ,X 2 ,...,X n be a simple random sample of Pareto random variables with density f X (x| )= x +1 , x > 1. The cumulative distribution function is F X (x) =1 x , x > 1. The mean and the variance are, respectively, µ = 1 , 2 = ( 1)2 ( 2) . 196
Introduction to the Science of Statistics The Method of Moments In this situation, we have one parameter, namely . Thus, in step 1, we will only need to determine the first moment µ 1 = µ = k 1 ( )= 1 to find the method of moments estimator ˆ for . For step 2, we solve for as a function of the mean µ. = g 1 (µ) = µ µ 1 . Consequently, a method of moments estimate for mean ¯X. is obtained by replacing the distributional mean µ by the sample ˆ = ¯X ¯X 1 . A good estimator should have a small variance . To use the delta method to estimate the variance of ˆ, we compute g 0 1(µ) = ✓ 1 (µ 1) 2 , giving g0 1 2ˆ ⇡ g 0 1(µ) 2 2 n . ◆ = 1 1 ( 1 1) = ( 1) 2 2 ( ( 1)) 2 = ( 1)2 and find that ˆ has mean approximately equal to and variance 2ˆ ⇡ g 0 1(µ) 2 2 n =( 1)4 n( 1) 2 ( 2) = ( 1)2 n( 2) As a example, let’s consider the case with =3and n = 100. Then, 2ˆ ⇡ 3 · 22 100 · 1 = 12 100 = 3 p 3 25 , ˆ ⇡ 5 =0.346. To simulate this, we first need to simulate Pareto random variables. Recall that the probability transform states that if the X i are independent Pareto random variables, then U i = F X (X i ) are independent uniform random variables on the interval [0, 1]. Thus, we can simulate X i with F 1 X (U i). If u = F X (x) =1 x 3 , then x =(1 u) 1/3 = v 1/3 , where v =1 u. Note that if U i are uniform random variables on the interval [0, 1] then so are V i =1 U i . Consequently, 1/ p V 1 , 1/ p V 2 , ··· have the appropriate Pareto distribution. > paretobar for (i in 1:1000){v mean(betahat) [1] 3.053254 > sd(betahat) [1] 0.3200865 197
Page 1: Topic 13 Method of Moments 13.1 Int
Page 5 and 6: Introduction to the Science of Stat
Page 7 and 8: Introduction to the Science of Stat
Page 9: Introduction to the Science of Stat

Method of Moments

Create successful ePaper yourself

Delete template?

Save as template?