Assist with the following:
7. Parametric bootstrap Suppose we have a sample of size 100 drawn from a geom(p) distribution with unknown p. The MLE estimate for p is given by by p = 1/2. Assume for our data r = 3.30, so p = 1/= = 0.30303. (a) Outline the steps needed to generate a parametric bootstrap 90% confidence interval. (b) Suppose the following sorted list consists of 200 bootstrap means computed from a sample of size 100 drawn from a geometric(0.30303) distribution. Use the list to construct a 90% CI for p. 2. 68 2.77 2.79 2.81 2.82 2.84 2.84 2.85 2.88 2.89 2.91 2.91 2.91 2.92 2.94 2.94 2.95 2.97 2.97 2.99 3.00 3.00 3.01 3.01 3.01 3.03 3.04 3.04 3.04 3.04 3.04 3.05 3.06 3.06 3.07 3.07 3.07 3.08 3.08 3.08 3. 08 3.09 3.09 3. 10 3.11 3. 11 3. 12 3. 13 3. 13 3. 13 3. 13 3.15 3. 15 3. 15 3.16 3. 16 3. 16 3. 16 3. 17 3.17 3. 17 3.18 3.20 3.20 3.20 3.21 3.21 3.22 3.23 3.23 3. 23 3.23 3.23 3.24 3.24 3.24 3.24 3.25 3.25 3.25 3.25 3.25 3.25 3.26 3.26 3.26 3.26 3.27 3.27 3.27 3.28 3.29 3.29 3.30 3.30 3.30 3.30 3.30 3.30 3.31 3.31 3.32 3.32 3.34 3.34 3.34 3.34 3.35 3.35 3.35 3.35 3.35 3.36 3.36 3.37 3.37 3.37 3.37 3.37 3.37 3.38 3.38 3.39 3.39 3.40 3.40 3. 40 3. 40 3.41 3.42 3. 42 3.42 3.43 3.43 3.43 3. 43 3. 44 3.44 3.44 3.44 3.44 3.45 3.45 3.45 3.45 3. 45 3.45 3.45 3.46 3.46 3.46 3.46 3.47 3.47 3.49 3. 49 3.49 3.49 3.49 3.50 3.50 3.50 3.52 3.52 3.52 3.52 3.53 3.54 3.54 3.54 3.55 3.56 3.57 3.58 3.59 3.59 3.60 3.61 3.61 3.61 3. 62 3.63 3.65 3.65 3.67 3.67 3.68 3.70 3.72 3.72 3.73 3.73 3.74 3.76 3.78 3.79 3.80 3.86 3.89 3.9112. Least Squares and MLE. In this problem we will see that the least squares fit of a line is just the MLE assuming the error terms are normally distributed. For bivariate data (21, y1), .... (In, yn), the simple linear regression model says that y; is a random value generated by a random variable Yi = ari +b+ Ei where a, b, r; are fixed (not random) values, and &; is a random variable with mean 0 and variance of. (a) Suppose that each &; ~ N(0, o?). Show that Y; ~ N(ar; + b, o?). (b) Give the formula for the pdf fy (yi) of Yi. (c) Write down the likelihood of the data as a function of a, b, and o.9. Fitting a line to data using the MLE. Suppose you have bivariate data (21, 91), ..., (In, Un). A common model is that there is a linear relationship between z and y, so in principle the data should lie exactly along a line. However since data has random noise and our model is probably not exact this will not be the case. What we can do is look for the line that best fits the data. To do this we will use a simple linear regression model. For bivariate data the simple linear regression model assumes that the z; are not random but that for some values of the parameters a and b the value y; is drawn from the random variable Yimaritbtei where s; is a normal random variable with mean 0 and variance of. We assume all of the random variables &; are independent. Notes. 1. The model assumes that o is a known constant, the same for each ci. 2. We think of &; as the measurement error, so the model says that 1: = ar; + 6 + random measurement error. 3. Remember that (r;, y;) are not variables. They are values of data. (a) The distribution of Y depends on a, b, o and r;. Of these only a and b are not known. Give the formula for the likelihood function f(y: | a, b, ri, o) corresponding to one random value yi. (Hint: yi - ar; - b ~ N(0, 5?).) (b) (i) Suppose we have data (1,8), (3,2), (5, 1). Based on our model write down the likelihood and log likelihood as functions of a, b, and o. (ii) For general data (21,91), .... (In, yn) give the likelihood and and log likelihood functions (again as functions of a, b, and o). (c) Assume o is a constant, known value. For the data in part b(i) find the maximum likelihood estimates for a and b
