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Mathemagical Statistics – MATH 2606

2022

Problem IX.1.  (Weighted bootstrap.)

(a) Suppose we observe (yi , xi ) for i = 1, . . . , N  to which we associate wi , the weight of each observation.  Assume the linear model : yi  = a . xi + b + ci .  but also assume that the squared loss function is weighted by each wi , i.e.

N

SS(a, b) =       wi ci(2) .

i=0

Derive an expression for the weighted line of best fit.

(b)  Using the country_data.txt  data set, plot the  life  expectancy log0＋(GDP)  together

with the weighted line of best fit with the following requirements: label your axes appro- priately, color the data as grey, closed cirlces, and color the regression line in red.

(c) Design a bootstrap procedure to construct a distribution for your weighted least squares estimates.  Use at least 1,000 iterations.

(d)  Construct a 99% confidence interval for the life expectancy when a country has a GDP

per capita of $3000.

Problem IX.2.  (Odds are.)

(a)  Consider the  data  set found in logistic.csv.   Plot  the  data  set with  closed,  grey

squares and appropriately labeled axes.

(b)  The logistic model has that

yi  ～ BERNOULLI(pi ),

where pi  = b( .  Write down the log-likelihood for this data set and construct a function that calculates for a specified a,  b,  x, and y .

(c)  Construct a function that performs a stochastic search for the MLE of a and b .

(d)  Construct a bootstrap procedure to estimate the distirbution of (MLE , MLE ).   Use at least 1,000 iterations.

(e)  The odds of an event are defined as 0p一p .  The odds ratio for a logistic regression is the

ratio  of odds measured when x = 1  and the odds measured when x = 0.   Conduct a hypothesis test at the 99% confidence level for whether the odds ratio is equal to one.