All the exercises are independent and may be done in the order you want. You may use R software at any time it may help you. You can use the result stated in a previous question even if you have not been able to proove it. Please include all the commands you want to use (even the codes given during the lecture) in your script. At the end of the exam, send your script by email at laurent.delsol@univ-orleans.fr.

Exercise 1: Archery competition

A. Olympic scores

At the last olympic game, the winner of the archery competition obtained the following scores.

score 10 9 8 7 6 5
number 12 33 29 17 6 3
  1. Give the type of the collected data and the sample size.
  2. Compute the mean and the corrected variance of the data.
  3. Compute the proportion corresponding to each score, represent the distribution and give the modal value.
  4. From the shape of the previous graph, a \(Bin(10,p)\) modelization seems possible, propose a relevant estimator \(\hat p\) of \(p\) from the method of your choice.
  5. Compute the bias and quadratic error of this estimator.
  6. Compute the Fisher Information of the sample.
  7. Is \(\hat p\) an efficient estimator of \(p\)?
  8. Plot on the same graph the empirical distribution (as in question 3) and the estimated binomial distribution. Comment the result.

B. Distance to the center

We now observe more precisely the results of the winner through the distance between the arrows and the center of the target.

The data are collected and available in the variable x (load Archery.RData workspace). If the position of the arrows follow a bidimensional gaussian distribution \(\mathcal{N}(0,\sqrt{\theta} I_2)\), then \(f\) should have the following expression : \[ f(x,\sigma^2)=\frac{x}{\theta}e^{-\frac{x^2}{2\theta}}1_{\{x>0\}}.\]

  1. Give a nonparametric estimation of the density \(f\) of \(x\) using the kernel density estimator \(\hat f\) using as smoothing parameter 0.1, 1, 10 and the value given by cross-validation criterion (hint: share the graphic window into four plots). Comment.
  2. Give a nonparametric estimation of the mode of \(x\).
  3. Proove that \(\hat f\) is a density if \(K\) is a density.
  4. Now use the parametric model to propose an unbiased estimator \(\hat\theta\) of \(\theta\) from maximum likelihood method.
  5. Compute its bias and quadratic error (hint: \(Y=\frac{X^2}{\theta}\sim\mathcal{E}(\frac{1}{2})\) )
  6. Compute the Fisher Information of the sample.
  7. Is \(\hat \theta\) an efficient estimator of \(\theta\)?
  8. Plot on the same graph the empirical distribution (as in question 1 with the best choice of h) and the estimated parametric distribution. Comment the result.
  9. Give a parametric estimation of the mode of \(x\) form the parametric model and \(\hat \theta\).

C. Competition

A competition is organized between two archers Robin and Legolas. The data are collected and available in the variable y (the distance between the arrows and the center of the target) and archer (the name of the corresponding archer, the class here) (load Archery.RData workspace).

  1. Use the nonparametric discrimination method ( classif_NP ) on this sample and comment the results.
  2. Three new values 2.1, 3.5, and 7.1 for y are obtained, could you predict who is the corresponding archer?

Exercise 2: Uniform distribution

Let \((X_1,\dots,X_n)\stackrel{i.i.d.}\sim \mathcal{U}([-\theta,2\theta])\), with \(\theta>0\). Hereafter, as usually, \(X\) denotes a variable following the common distribution \(\mathcal{U}([-\theta,2\theta])\).

  1. Give the density of X and compute \(\mathbb{E}[X]\) and \(Var(X)\).
  2. Give the estimator \(\hat \theta_1\) obtained by the method of moments applied to \(X\).
  3. Compute the bias and variance of \(\hat \theta_1\).
  4. Proove that for all \((x,t)\in\mathbb{R}\times\mathbb{R_+}\), one gets \[\{-t\leq x\leq 2t\}\Leftrightarrow \{max(-x,\frac{x}{2})\leq t\}.\]
  5. Proove from the previous question \(Z=max(-X,\frac{X}{2})\sim \mathcal{U}([0,\theta])\) (hint: compute the cumulative distribution function of Z)
  6. Give the estimator \(\hat \theta_2\) obtained by maximum likelihood (hint: use the \(Z_i\)'s).
  7. Compute the bias and variance of \(\hat \theta_2\).
  8. If \(\hat \theta_2\) is biased, propose an unbiased estimator \(\hat \theta_3\) and compute its variance.
  9. Compare \(\hat \theta_1\), \(\hat \theta_2\), and \(\hat \theta_3\) in erms of quadratic errors.