What to do today (Nov 22, 2018)?

Transcription

1 What to do today (Nov 22, 2018)? Part 1. Introduction and Review (Chp 1-5) Part 2. Basic Statistical Inference (Chp 6-9) Part 3. Important Topics in Statistics (Chp 10-13) Part 4. Further Topics (Selected from Chp 14-16) 4.1 Distribution-Free Procedures (Chp15.1, 15.2) Basic Concepts Nonparametric Testing Procedures 4.2 Quality Control Methods (Chp16.1, 16.2, 16.3) Introduction Examples of Control Charts

2 4.1 Distribution-Free Procedures (Nonparametric Methods) (Chp15) Basic Concepts Nonparametric Testing Procedures 4.1.2A Sign test (Binomial Test) 4.1.2B Wilcoxon signed-rank test 4.1.2C Wilcoxon rank-sum test 4.1.2D Kolmogorov-Smirnov test Distribution-free Confidence Intervals (Chp15.3) Distribution-free ANOVA (Chp15.4)

3 4.1 Distribution-Free Procedures (Nonparametric Methods) Basic Concepts order statistics. Definition. Suppose X 1,..., X n are iid observations from a continuous r.v. X f ( ) with cdf F ( ). The order statistics of the random sample are X (1), X (2),..., X (n) : X (1) < X (2) <... < X (n). X (1) = the smallest value of X 1,..., X n, X (2) = the 2nd smallest value of X 1,..., X n,......, X (n) = the largest value of X 1,..., X n. Distribution. X (k) n! (k 1)!(n k)! F (x)k 1 (1 F (x)) n k f (x) for k = 1,..., n.

4 rank statistics. Definition. The rank of X k, the kth observation in a random sample of size n, is r k such that X k = X (rk ), for k = 1,..., n. Example 10.1 Realizations of 5 iid observations X 1,..., X 5 from a population are given in the table below. x 1 x 2 x 3 x 4 x What are the order statistics? What are the rank statistics?

5 percentiles/quantiles. Definition. Suppose r.v. X f ( ) with a random sample X 1,..., X n. Population percentitles: π p is the (100p)th percentile of the population if P(X π p ) = p. That is, π p f (x)dx = p. Sample percentiles: Let X (1),..., X (n) be the order statistics. Then X (r) is the (r/n)100th (or (r/n + 1)100th) sample percentile.

6 empirical distribution function. Definition. Suppose r.v. X F ( ) with a random sample X 1,..., X n. Its empirical distribution is defined as ˆF n (x) = 1 n #{X i : X i x; i = 1,..., n}, x (, ). That is, ˆF n (x) = 0 if x < X (1) ; k/n, if X (k) x < X (k+1) when 1 k n 1; 1, if x X (n). For a fixed x, E[ ˆF n (x)] = F (x) Var[ ˆF n (x)] = F (x)[1 F (x)]/n.

7 4.1.2 Nonparametric Testing Procedures 4.1.2A Binomial test: (the sign test) setting. r.v. X f ( ) with a random sample X 1,..., X n. To test H 0 : its population median m = m 0 vs H 1 : otherwise. test statistic. S i = 1 or 0 if X i m 0 0 or not. S = n S i = #{i : X i m 0 0; i = 1,..., n} i=1 (i) S B(n, 1/2) under H 0. (ii) if n 1, Z = S n/2 n/4 N(0, 1) approximately under H 0. making inference. (i) Obtain the two critical values c 1 and c 2 from the binomial table: P H0 (S > c 2 ) = α/2; P H0 (S < c 1 ) = α/2. Reject H 0 if S obs > c 2 or S obs < c 1. [the exact approach] (ii) If n 1, reject H 0 if Z obs > z α/2. [the approximate approach]

8 4.1.2B Wilcoxon s Signed-Rank Test setting. r.v. X f ( ) with a random sample X 1,..., X n. To test H 0 : its population median m = m 0 vs H 1 : otherwise. test statistic. For any i, S i = 1 or 0 if X i m 0 0 or not; ri is the rank statistic of X i m 0 in the sample of X 1 m 0,..., X n m 0. S + = n i=1 S i r i (i) critical values with the distribution of S + under H 0 are tabulated in Table A.13 of the appendix (ii) if n 1, Z = S + n(n + 1)/4 n(n + 1)(2n + 1)/24 N(0, 1) approximately under H 0.

9 making inference. (i) Obtain the two critical values c 1 and c 2 from Table A.13: P H0 (S + > c 2 ) = α/2; P H0 (S + < c 1 ) = α/2. Reject H 0 if S +obs > c 2 or S +obs < c 1. [the exact approach] (ii) If n 1, reject H 0 if Z obs > z α/2. [the approximate approach]

10 Example 10.2 (p658) to change the numerical results a bit to make it more interesting. Study. a type of steel beam with a compressive strength 50K lb/in 2? Data. n = 25 beams (observations). (Assume they re iid.) Hypotheses. H 0 : m = 50K vs H 1 : m < 50K by the Binomial Test. by the Wilcoxon Signed-Rank Test.

11 4.1.2C Wilcoxon s Rank-Sum Test setting. r.v. X F x ( ) with a random sample X 1,..., X g, r.v. Y F y ( ) with a random sample Y 1,..., Y n, and X and Y are independent. To test H 0 : F x ( ) = F y ( ) vs H 1 : otherwise. reformulating: To test H 0 : m X Y = 0 vs H 1 : otherwise? (m X Y is the median of X Y.) test statistic. For any X i, let r Xi the sample of X 1,..., X g, Y 1,..., Y n : g W RS = i=1 be its the rank statistic in (i) critical values with the distribution of W RS under H 0 are tabulated in Table A.14 of the appendix (ii) if n 1, approximately under H 0. making inference r Xi Z = W RS g(g + n + 1)/2 gn(g + n + 1)/12 N(0, 1)

12 What will we study next? Part 1. Introduction and Review (Chp 1-5) Part 2. Basic Statistical Inference (Chp 6-9) Part 3. Important Topics in Statistics (Chp 10-13) Part 4. Further Topics (Selected from Chp 14-16) 4.1 Distribution-Free Procedures (Chp15.1, 15.2) Basic Concepts Nonparametric Testing Procedures 4.2 Quality Control Methods (Chp16.1, 16.2, 16.3) Introduction Examples of Control Charts