HW 1

All questions are from Chapter 2 in Robert and Casella. Unless otherwise mentioned, you need to do all the subquestions:

2.1
2.10
- It is fine to use the R runif function to generate uniform [0,1] RVs
- For summaries, look at the following summary statistics:
  - First three moments
  - 1st, 5th, 25th, 50th, 75th, 95th, 99th quantiles
  - Max and min
2.19
2.30a, 2.30c
2.42
- Part a should read: “Show that the product of log-concave densities is log-concave”

Graduate students need to complete 2.4 too

Notes on 2.4:

For the arcsine transform run the following transformation \[ D(x) = 4 x (1 - x) \] and the following algorithm for $n = 1e4$ \[ X_0 \sim \text{Unif}(0,1), \,X_{n+1} = D(X_n) \]

Then let \[ Y_n = 0.5 + \frac{\arcsin (2 X_n - 1)}{\pi} \] Then make a bivariate plot of $(Y_{1}, Y_{101}), (Y_{2}, Y_{102}), \dots, (Y_{9{,}901},Y_{10{,}001})$, and show that the samples look approximately uniform on $[0,1] \times [0,1]$.

You can then look at every $100^\mathrm{th}$ sample and make a qqplot against a uniform distribution to see how well these samples actually approximate a uniform distribution. In order to make this plot, it’ll be helpful to look at the qqplot help page.

For part (d), it suffices to show numerically the tent transformation convergence, no need to do the problem analytically.

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