# A parameter 9 has a Oamma(2. 1) posterior distribution. Find the 95% highest posterior density..

2.4. A parameter 9 has a Oamma(2. 1) posterior distribution. Find the 95% highest posterior density interval for B. that is. the interval containing 95% of the posterior probability for which the posterior density for every point contained in the interval is never lower than the density for every point outside the interval. Since the gamma density is unimodal.,t, the interval is also the narrowest possible interval containing 954E of the posterice ,1 probability. 2.5. There were 46 crude oil spills of at least 1000 bagels from tankers in U.S. waters during ‘ 1974-1999. The website for this book contains the following data the number of spills in the ith year. NI; the estimated amount of oil shipped through US waters as pan of. US import/expon operations in the ith year. adjusted for spillage in international or foreign waters. 6,1; and the amount of oil shipped through U.S. waters during dornestk shipments In the ith year. b,2. The data are adapted from II 11 Oil shipment amounts are measured in billions of barrels (BMA). The volume of oil shipped is a measure of exposure to spill risk. Suppose:. we use the Poisson process assumption given by N,16,1. 162 — Poissonail where = alb,’ +alba. The parameters of this model am at andal, which represent the rate.rit of spill occurrence per Bbbl oil shipped during impon/expott and domestic shipments, respectively. 4. a. Derive the Newton-Raphson update for finding the MLEs of at and alt. b. Derive the Fisher scoring update for finding the MLEs of a, and at. c. Implement the Newton-Raphson and Fisher scoring methods for this problem, provide the MLEs. and compare the implementation case and performance of the two methods. d. Estimate standard errors for the NILEs of at and a2. e. Apply the method of steepest ascent Use step•halving backtracking as necessary. f. Apply quasi•Newton optimization with the Hessian approximation update given in (2.49). Compare perfomuince with and without step halving. g. Constma a graph resembling Figure 2.8 that compares the paths taken by methods used in ta)-(1). Choose the plotting region and starting point to best illustrate the features of the algorithms’ performance.
2.6. Table 2.3 provides counts of a flour beetle (Tribalitun confusion) population at various points in time 11031. Beetles in all stages of development were counted. and the food supply was carefully controlled. An elementary model for population growth is the logistic model given by