Reliability and survival analysis is the Weibull distribution

1. Widely used in reliability and survival analysis is the Weibull distribution with the density
   f(y; , ) = (
   x1
   e(x/)
   
   for y 0,
   0 for y < 0, where  > 0 and > 0. Suppose that the data y1, y2,…,yn are modeled as the outcomes of
   independent random variables Y1, Y2,…,Yn, each with the Weibull distribution as above.
   (a) (Theoretical.) Derive the equations that the maximum likelihood estimates of the parameters and  have to satisfy.
   (b) Solve as much as you can of those equations in closed form. (I would say that you will be
   not able to solve both.)
   (c) (Computational.) Design and implement a numerical procedure to finish the problem, to
   solve the equation(s) that cannot be solved in closed form.
   In order to learn something by doing that, do it from the first principles: hence, please, no
   packages or downloaded code, no R functions like optim(), optimize(), nls(), and uniroot()
   and others. Program your own method, and use something better than bisection or straightforward search.
   (d) Test your implementation on two examples: one with n = 100, another with n = 1000, both
   of them with  = 2 and = 3. Generate the data; you may use the R function rweibull() for
   that this time, no need to repeat the already known skills. No need to report the data either
   – but report the results, that is, the estimates of  and in each case, and comment on the
   them.
2. Suppose that the data y1, y2,…,yn can be modeled as the outcomes of independent
   random variables Y1, Y2,…,Yn, which have all the same distribution – a mixture of two normal
   distributions with density
   f(yi; p, µ1, µ2, 1, 2) = p
   1
   p2⇡
   e
   (yiµ1)2
   22
   1 +
   1 p
   2
   p2⇡
   e
   (yiµ2)2
   22
   2 .
   The distribution, for the data that show length of eruptions of the geyser Old Faithful in the Yellowstone park (R dataset faithful), can be interpreted as switching
   1 2 3 4 5 6
   0.0 0.1 0.2 0.3 0.4 0.5
   density.default(x = faithful$eruptions)
   N = 272 Bandwidth = 0.3348
   Density
   between two possible regimes: with probabilities p and
   1 p, the length of the eruption follows respectively the
   normal distribution with µ1 and 1, and that with µ2
   and 2. The plot of the kernel estimate of the probability
   density, invoked by the R command

plot(density(faithful$eruptions))
can be seen on the right.
We want to estimate all five unknown parameters involved: µ1, µ2, 1, 2, and p. To this end, we consider a hypothetic situation that apart from y!, y2,….,yn, we would