Testing and constructing the confidence intervals

Assignment Help Applied Statistics

Reference no: EM132389121

Problem 1: In linear regression we do not assume the error term to be normally distributed to estimate the parameters. But normality assumption is the key for testing and constructing the confidence intervals of regression parameters. Assuming different distribution for error, check the coverage probability of the regression coefficients and verify how important the normality assumption.

Problem 2: The following data are an iid sample from a Cauchy (θ, 1) distribution.
1.77, -0.23, 2.76, 3.80, 3.47, 56.75, -1.34, 4.24, -2.44, 3.29, 3.71, -2.40, 4.53, 0.007, -1.05, -13.87, -2.53, -1.75, 0.27, 43.21

a) Graph the log likelihood function. Find the MLE for theta using the Newton-Raphson method. Try all of the following starting points: -11, -1, 0, 1.5, 4, 4.7, 7.8, and 38. Discuss your results. Is the mean of the data a good starting point?

b) Apply the bisection method with starting points -1 and 1. Use the additional runs to illustrate manners in which the bisection method may fail to find the global maximum.

c) From the starting value of (θ⁽⁰⁾, θ⁽¹⁾)) = (-2, -1), apply the secant method to estimate θ.. What happens when (θ(0), θ(1))) = (-3, 3), and for other start- ing choices?

d) Compare the speed and stability of the three methods employed based to this example.

Problem 3: Recall the zero truncated poisson distribution with parameter λ discussed in class. Consider the following data: (Develop your own code and use same criteria for convergence)
(x, f ) = (1, 14), (2, 9), (3, 8), (4, 2), (5, 1),
a) Graph the likelihood function between θ = 0 and 5
b) Use the golden ratio method to find the MLE
c) Use the bisection method to find the MLE
d) Use the N-R method to find the MLE
e) Use Nelder- Mead (Simplex) method to find the MLE (you may use optim function in R)
f) Compare the performance of all the above three methods based on speed of convergence.

Problem 4: Consider the San Francisco International airport customer sat- isfaction data set 2018

Q7ALL is the overall satisfaction score. Make some summary measures of this Q7ALL and use graphical tools to understand the problem and find out the root causes thereby ways to find out ways to improve this score.

Reference no: EM132389121

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