In all problems do as much as you can analytically, next numerically and last using simulation.
You can use any written or online source but you can not discuss the exam with ANYONE !
Say we have observations \(X_1,...,X_n\sim N(\mu_x, \sigma_x)\) and \(Y_1,...,Y_m\sim N(\mu_y, \sigma_y)\), all independent, and \(\sigma_x,\sigma_y\) known.
Find a \((1-\alpha)100\%\) confidence interval for \(\delta=\mu_x-\mu_y\).
Do a simulation that shows that your solution works for the case \(\mu_x=0, \mu_y=1, \sigma_x=2, \sigma_y=3, n=10, m=7\) and \(90\%\) confidence interval.
Say that we can carry out a total of k=n+m experiments, and we are free to do any combination of n and m. Which should we choose? What are n and m for the case of part b, that is \(\sigma_x=2, \sigma_y=3, k=17\)?
Below is data from a distribution with density
\[f(x\vert \alpha)=\alpha+(1-\alpha)\frac1{\sqrt{2\pi0.1^2}}\exp \left\{-\frac1{2\times0.1^2}(x-0.5)^2 \right\}\]
that is a mixture of a U[0,1] and a N(0.5,0.1) random variable. Find a 95% Bayesian credible interval for \(\alpha\) if the prior is \(\alpha\sim U[0,1]\).
[1] 0.00 0.03 0.05 0.06 0.09 0.10 0.15 0.15 0.18 0.19 0.20 0.20 0.21 0.27 0.28
[16] 0.29 0.30 0.30 0.30 0.31 0.33 0.34 0.34 0.34 0.35 0.35 0.35 0.36 0.36 0.36
[31] 0.37 0.38 0.38 0.39 0.39 0.41 0.41 0.41 0.43 0.43 0.43 0.44 0.44 0.45 0.46
[46] 0.46 0.46 0.46 0.47 0.47 0.47 0.47 0.48 0.49 0.50 0.50 0.50 0.50 0.50 0.51
[61] 0.52 0.52 0.53 0.53 0.53 0.53 0.55 0.55 0.56 0.57 0.57 0.57 0.57 0.59 0.60
[76] 0.61 0.61 0.63 0.64 0.65 0.68 0.68 0.68 0.68 0.70 0.71 0.72 0.73 0.77 0.78
[91] 0.78 0.80 0.81 0.82 0.87 0.93 0.94 0.98 0.99 0.99In an experiment we observe the counts \(X\) of some event. However, a certain percentage of events fall below some threshhold and can not be observed. An auxiliary experiment is done to yield information on this efficiency. Overall, we have a model of the form
\[X\sim Pois(\epsilon\lambda),Z\sim Bin(n,\epsilon)\] where n is known, X and Z are independent. Specifically, consider the case \(X=67;n=1000;Z=780\)
Find the mles of \(\lambda\) and \(\epsilon\).
Find the p-value of the test \(H_0:\lambda=60\) vs \(H_a:\lambda>60\).
Say we have a single observation \(X\) which takes values in \(\{0,1,2\}\) with \(P(X=k) = \frac{p+k}{3(p+1)}\).
Find the method of moments estimator of p
Find the maximum likelihood estimator of p
Find the Bayesian estimator for p if the prior is \(p\sim U[0,1]\) and using the posterior mean.
Do all the work analytically!
Below we have observations from a rv \(X\) such that \(\frac{X-\mu}\sigma\sim t(3)\).
7.04 18.91 19.13 19.68 20.01 21.63 22.44 22.96 23.14 23.3 23.48 24.14 24.69 25.08 25.86 25.88 26.23 26.43 26.54 26.81 27.26 27.55 27.65 27.72 27.73 27.97 28 28.03 28.1 28.31 28.32 28.35 28.41 28.46 28.47 28.57 28.74 28.77 28.78 29.21 29.23 29.38 29.66 29.69 29.71 29.76 30.08 30.24 30.29 30.42 30.49 30.49 30.52 30.61 30.63 30.71 30.74 30.74 30.75 30.75 30.8 30.83 31.19 31.22 31.22 31.4 31.51 31.53 31.56 31.75 31.78 32.06 32.46 32.66 32.76 32.85 32.85 32.96 32.99 33.33 33.34 33.35 33.38 33.53 33.53 33.62 33.69 34.52 34.52 35.12 35.48 36.21 36.33 36.53 36.58 37.03 38.08 38.7 40.58 57.05Find the mleโs of \(\mu,\sigma\). Compare them to the mles based on the normal distribution.
Give some argument why this is a better solution than the one based on the normal distribution.