Question

2. Suppose Y ~ Exp(a), which has pdf f(y)-1 exp(-y/a). (a) Use the following R code to generate data from the model Yi ~ Exp(0.05/Xi), and provide the scatterplot of Y against X set.seed(123) n <- 500 <-rnorm (n, x 3, 1) Y <- rexp(n, X) (b) Fit the model Yi-Ao + Ax, + ε¡ using the lm function in R and provide a plot of the best fit line on the scatterplot of Y vs X, and the residual vs fitted plot. What assumptions of the linear regression model appear to be violated by the data? (c) Derive the variance stabilizing transformation for Y Exp() (d) Apply the variance stabilizing transformation to the data and fit the linear regression model to the transformed data. Provide the new residuals by fitted plot and comment on whether the linear regression model assumptions are satisfied by the transformed data

Answer 1

scatterplot OO 5 4 3 2 1 X 0.05 0.10 0.15 0.20 00 0 LO

residuals vs predicted 0.02 0.00 0.01 0.03 predicted residuals 0.05 0.10 0.15 00 0 O O

residuals vs predicted after variance stabilization transformation 2 -5.2 -5.0 4.8 4.6 -4.4 -4.2 4.0 predicted residuals 17

Set.seed (123) n=500 X-rnorm (n, 3, 1) yrexp (n, x/.05) #to draw sample from the given Exponential distribution > plot (x, Y, main="scatterplot",col="blue") #scatterplot of Y vs X > fit-lm (y-x) #fitting linear regression between Y and X >summary (fit) Call: Im (formula = y x) Residuals: Min 10 Median 30 Маx -0.030900 -0.013288 -0.004852 0.006677 0.178047 Coefficients: Estimate Std. Error t value Pr (>| tI) (Intercept) 0.040005 0.003308 12.095 < 2e-16 -6.546 1.47e-10 *** -0.006795 0.001038 X Signif. codes: 0.001 0 0.01 0.05 O.1 Residual standard error: 0.02256 on 498 degrees of freedom Adjusted R-squared: Multiple R-squared: F-statistic: 42.85 on 1 and 498 DF, 0.07923, 0.07738 p-value: 1.473e-10 > abline (1m (y x), col="red") # adding best fitted line to the scatterplot 2predicted=predict (fit) #saving fitted values residuals=residuals (fit) #saving residuals plot (predicted, residuals,main-"residuals vs predicted", xlab="predicted", ylab-S

abline (lm(y-x),col="red") tadding best fitted 1ine to the scatterplot predicted predict (fit) tsaving fitted values residuals=residuals (fit) #saving residuals > plot (predicted, residuals,main-"residuals va predicted",xlab="predicted", ylab=$ > z=log (y) fapplying variance stabilizing transformation to Y fit new=lm (z-x) tfitting linear reqression between X and variance stabilized rş >summary (fit new) Call: lm (formula = z X) Residuals: Median Min 10 30 Max -5.7091 -0.7317 0.1628 0.8627 2.4783 Coefficients: Estimate Std. Error t value Pr(> Itl) (Intercept) -3.90561 0.17776 -21.971 < 2e-16 -0.21062 0.05579 -3.775 0.000179 X 0.001 Signif. codes: 1 C 0.01 0.05 0.1 Residual standard error: 1.212 on 498 degrees of freedom Adjusted R-squared:| Multiple R-squared: F-statistic: 14.25 on 1 and 4 98 DF, 0.02783, 0.02587 P-value: 0.000179 predicted new-predict (fit new) #saving fitted values residuals new-residuals (fit new) #saving residuals a plot (predicted new, residuals new,main="residuals vs predicted after variance stabilizatzon transformation", xlab="predicted", vlab="residualss

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