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STA4502 Chapter 9

# STA4502 Chapter 9 - Chapter 9 Nonparametric Regression...

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STA4502 Chapter 9, page 1 Chapter 9 Nonparametric Regression Analysis Some type of regression is one of the most used statistical techniques. It involves a relationship (usually linear) between a predictor (also called independent variable) x (assumed to be non- random!) and a random variable, Y, called the response variable (or dependent variable because its value depends on the value of x). General Setup Let x 1 , x 2 , …, x n be known values of an independent variable Model the response variable corresponding to each x i as o Y i = α + βx i + e i , for i = 1, 2, …, n, where α and β are the intercept and the slope of the true regression line (all unknown) e 1 , e 2 , …, e n is a random sample from a population that has continuous distribution. Since the e’s come from the same population, they have the same distribution. Also, because we have a random sample, the e’s are independent of each other. Objective: Make inferences about the slope, β o Point estimation of β o Confidence interval for β o Testing hypotheses about β Ho: β = β 0 vs. Ha: β > β 0 Ho: β = β 0 vs. Ha: β < β 0 Ho: β = β 0 vs. Ha: β ≠ β 0 Hypothesis Testing – Theil’s Test Suppose we want to test Ho: β = β 0 vs. Ha: β > β 0 Remember the regression model, Y i = α + βx i + e i , for i = 1, 2, …, n, For each i, i = 1, 2, …, n, define D i = Y i – β 0 x i Look at   1 1 0 1 1 1 0 1 0 1 1 D Y x x e x x e D i values for each x i are given in the general case, as well as under Ho and under Ha in the following table: x D D under Ho D under Ha x 1 D 1 = Y 1 – β 0 x 1 D 1 = α + e 1 D 1 = α + (β – β 0 )x 1 + e 1 x 2 D 2 = Y 2 – β 0 x 2 D 2 = α + e 2 D 2 = α + (β – β 0 )x 2 + e 2 x n D n = Y n – β 0 x n D n = α + e n D n = α + (β – β 0 )x n + e n Under Ho, the D i ’s are independent and identically distributed random variables and D i and x i are independent. Under Ha, D i is positively correlated with x=i At this stage we have n pairs of (x i , D i ): x x 1 x 2 x n D D 1 D 2 D n

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STA4502 Chapter 9, page 2 We can test the hypothesis Ho: β = β 0 vs. various alternatives by performing Kendall’s test on the pairs (x 1 , D 1 ), …, (x n , D n ). First we compute C = (Number of concordant pairs) – (Number of discordant pairs). Then depending on the alternative hypothesis we use one of the following decision rules: For Ha: β > β 0 , the decision rule is ―Reject Ho if C ≥ k (α, n) For Ha: β < β 0 , the decision rule is ―Reject Ho if C ≤ – k (α, n) For Ha: β ≠ β 0 , the decision rule is ―Reject Ho if C ≥ k (α/2, n) For large n, compute 0 * ~ (0,1) ( 1)(2 5) 18 C CN n n n  Then use the following decision rules: o For Ha: β > β 0 , the decision rule is ―Reject Ho if C* ≥ z α o For Ha: β < β 0 , the decision rule is ―Reject Ho if C* ≤ – z α o For Ha: β ≠ β 0 , the decision rule is ―Reject Ho if | C* | ≥ z α/2 Case of Tied observations Find the average of the ties that would be assigned if the observations were slightly different.
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STA4502 Chapter 9 - Chapter 9 Nonparametric Regression...

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