MIT Spring 2015

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1 Assessing Goodness Of Fit MIT Dr. Kempthorne Spring 205

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3 Poisson Distribution Counts of events that occur at constant rate Counts in disjoint intervals/regions are independent If intervals/regions are constant in size, then identical distributions of counts Consider data: X, X 2,..., X n independent Poisson(λ i ), and testing: Null Hypothesis H 0 : X i are i.i.d. Poisson(λ). Alternate Hypothesis H : X i are independent Poisson(λ i ) (rates vary over i) Apply Generalized Likelihood Ratio Test 3

4 Generalized Likelihood Ratio Test: MLE of λ under H 0 : X j,..., X n i.i.d. Poisson(λ) n λ x i Lik(λ) = e λ i= x i! = λˆ = n n i= x i = x MLEs of λ i under H : X i Poisson(λ j i ), i =,..., n xi i Lik(λ λ i,..., λ n ) = n λ i= x i! e = λ j = x j, j =,..., n Likelihood Ratio Λ = Lik(λˆ)/Lik(λ, j..., λ n) n λˆx i λˆ x ˆ i i= x i! e = j = n λ λˆ+λ i x e n λ i i i= x i! e λ i λ i= i x i n n x nx+ x i n x = e = i= x i i= x i x i 4

5 Generalized Likelihood Ratio Test (continued) GLR Test Statistic LRStat = 2 log(λ) ) x n x i = 2 log( i= x i ) n = 2 i= x j ln( x j x ) ) n ˆx (x i x) 2 σ = n 2 x i= x Note: Last line applies Taylor Series Approximation x f (x) = x ln( ) (x x x 0) (x x 0) 2. Approximate Distribution under H 0 : LRStat χ 2 q, where q = dim(θ) dim(θ 0 ) = n. σˆ2 x H 0 is rejected when LRStat is high >> x (For a Poisson Distribution Var(X ) = E (X ) = λ.) 5

6 Example 9.6.A. Asbestos Fibers Steel et al. 980: Counts of asbestos fibers on filters (from Example 8.4.A) Data: x = c(3, 29, 9, 8, 3, 28,..., 24) (23 values) Test Statistic: n LRStat = 2 x j ln(x j /x) = 27. ) n ˆx (x i x) 2 σ = n 2 = Approximate P-Value: x x Asymptotic Distribution: LRStat χ 2, with q = n = 22. q P Value =

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8 Hanging Histograms Histograms Random sample from distribution with cdf F (x θ). Sample data: x, x 2,..., x n m interval bins in histogram: bin j = (b j, b j+ ], for j =, 2,..., m. m bin counts in histogram n j = #(x i bin j ) = #({x i : b j < x i b j+ }), Evaluate Goodness-of-Fit of F (x θ) Expected Counts nˆj = np j where p j = F (b j+ θ) F (b j θ) Observed Counts n j Binomial(n, p j ) Hanging Histogram: Instead of plotting n j, use (n j nˆj ) in Histogram. Correct for non-constant Var(n j nˆj ) = np j ( p j ) 8

9 Hanging Histogram Hanging Chigram Hanging Histogram: Instead of plotting n j, use (n j nˆj ) in Histogram. Correct for non-constant Var(n j nˆj ) = np j ( p j ) (n j nˆj ) (n j nˆj ) use nˆj np j ( p j ) J 2 (n j nˆj ) (O j E j ) 2 Note: = nˆj E j 9

10 Hanging Histogram (continued) Hanging Rootogram Hanging Rootogram: : Instead of plotting n j, use n j nˆj in Histogram g(x) = x is a Variance Stabilizing Transformation For a r.v. X : E [X ] = µ and Var[X ] σ 2 (µ) Then Y = g(x ) is a random variable with Var[Y ] [g ' (µ)] 2 Var[X ] const (if g ' (µ) = /σ(µ)) 0

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12 Sample from a Uniform(0, ) Distribution X, X 2,..., X n i.i.d. Uniform(0, ). Def: Order Statistics ordered sample values X () < X (2) < < X (n) CDF and PDF of X (n) F (n) (x) = P(max X i x) = P( all X i x) = [P(X i x)] n = x n d = f (n) (x) = dx F (n) (x) = nx n. n Note: E [X (n) ] = 0 xf (n) (x)dx = n+ 2

13 Sample from a Uniform(0,) Distribution CDF and PDF of X () F () (x) = P(min X i > x) = P( all X i > x) = [P(X i > x)] n = ( x) n = F () (x) = ( x) n = f () (x) = d [ ( x) n ] = n( x) n dx 3

14 Order Statistics from a Uniform(0,) Distribution PDF of X (j), the jth order statistic (j =, 2,..., n) Use the cdf of the original distribution: ) F (x) = P(X x) n! f (j) (x) = (j )!!(n j)! [F (x)] (j ) f (x)[ F (x)] (n j) For F (x) = x, the cdf of the Uniform(0, ) ) distribution n! (j ) f (j) (x) = (j )!!(n j)! x [ x] (n j) x j ( x) n j+ = Beta(j, n j + ) I.e., X (j) Beta(j, (n j) + ) By properties of Beta integrals: Beta(j+,(n j)+) j E[X (j) ] = Beta(j,(n j)+) = Note: n+ j j Var[X (j) ] = ( n+ ) ( n+ ) ( n+2 ) 4

15 Order Statistics from Sampling a Continuous Distribution X,..., X n i.i.d. with cdf F X (x) (assumption: F X ( ) is strictly increasing over its range) X () < X (2) < < X (n), the order statistics Definition: Probability Integral Transform Y = F X (X ) Y Uniform(0, ) (See Rice Proposition C of Section 2.3) Y i = F X (X i ) are i.i.d. Uniform(0, ) The jth order statistic Y (j) is Beta(j, n j + ) random variable and j E [Y (j) ] = n + 5

16 Definition: Probability Plot Given a sample X,..., X n H 0 : F ( ) is the cdf of each X i k Plot y = F (X (k) ) vs x = n+ The points should fall close to the line y = x if H 0 is true. QQ (Quantile-Quantile) Plots k Plot y = X (k) vs x = F ( ) n+ The vertical axis is the observed quantile and the horizontal axis is the theoretical quantile of the distribution. 6

17 Normal QQ Plots k n+ Plot y = X (k) vs x = F ( ) using F ( ) for a Normal(µ, σ 2 ) distribution In terms of the N(0, ) distribution cdf Φ( ), F (x µ, σ) = Φ( x µ σ ) k ), k =,..., n Denote z k = Φ ( n+ the theoretical quantiles of a N(0, ) distribution, then k F ( n+ ) = µ + σz k If the {X i } are a Normal(µ, σ 2 ) the Normal QQ Plot can be graphed as X (k) versus z k (without using µ and σ) The plot will be close to linear with Intercept = µ and Slope = σ Note: F ( k n+ ) E [X (k) ] (exact if F is uniform). 7

18 Testing for Normality Goodness of Fit Tests for Normal Distributions Normal QQ Plots Filliben(975) : Accept Normal if R Squared of QQ Plot is close to. Skewness Coefficient n n i=(x i x) 3 SKEW = s3 n 2 n where x = i= x n i, and s = n i=(x i x) 2. Reject Normality if SKEW large. Kurtosis Coefficient n n i=(x i x) 4 KURT = s4 n 2 n where x = i= x n i, and s = n i=(x i x) 2. Reject Normality if KURT large. 8

19 MIT OpenCourseWare Statistics for Applications Spring 205 For information about citing these materials or our Terms of Use, visit: