2.6 Practice Problems
Problem 2.6.1. Define each of the following precisely, and give one example of each:
- (i)
- equal in distribution;
- (ii)
- a statistic distribution-free over a class \(\mathscr {Z}\);
- (iii)
- a non-parametric distribution-free statistic.
Explain what distinguishes (iii) from (ii).
Show solution
Solution. The distinction in (iii) is the size of the class. A statistic is distribution-free over \(\mathscr {Z}\) if its null distribution is the same for every member of \(\mathscr {Z}\); it is non-parametric distribution-free when \(\mathscr {Z}\) is the class of all continuous distributions rather than a parametric family. Thus \(\sqrt {n}(\overline {V}-\mu _0)/S\) is distribution-free over the normal family, since its \(t_{n-1}\) distribution does not depend on \(\sigma ^{2}\), but it is not distribution-free outside that family. The rank vector, by Corollary 2.2.2, is distribution-free over every continuous distribution and is therefore non-parametric distribution-free.
Problem 2.6.2. Let \(X_1,\dots ,X_n\) be a random sample from a continuous distribution and let \(R=(R_1,\dots ,R_n)\) be the vector of ranks. Prove that \(R\) is uniformly distributed over the \(n!\) permutations of \((1,\dots ,n)\), stating carefully where continuity and where identical distribution are used. Where to start: this is Corollary 2.2.2. Reproduce the proof and identify the line at which continuity is used and the line at which identical distribution is used.
Problem 2.6.3. Let \(X_1,\dots ,X_n\) be a random sample from a continuous distribution symmetric about \(\theta _0\), and put \(\Psi _i = \Psi (X_i-\theta _0)\) where \(\Psi (u)=1\) if \(u>0\) and \(0\) otherwise. Show that \(\Psi _1,\dots ,\Psi _n\) are independent Bernoulli variables with parameter \(\tfrac 12\), and that they are independent of the ranks of \(\left |X_i-\theta _0\right |\).
Show solution
Solution. Symmetry about \(\theta _0\) gives \(P(X_i>\theta _0)=P(X_i<\theta _0)=\tfrac 12\), and continuity makes \(P(X_i=\theta _0)=0\), so each \(\Psi _i\) is Bernoulli\((\tfrac 12)\); independence is inherited from that of the \(X_i\). For the second part write \(X_i-\theta _0 = S_i\left |X_i-\theta _0\right |\) with \(S_i = 2\Psi _i - 1\). Under symmetry the joint density of \((S_i,|X_i-\theta _0|)\) factorises, so the sign vector is independent of the vector of absolute values, and hence of any function of them — in particular of their ranks. This independence is exactly what makes the signed-rank statistic distribution-free.
Problem 2.6.4. Show that the Kolmogorov–Smirnov statistic \(D_n=\sup _x\left |F_n(x)-F(x)\right |\) is distribution-free over all continuous \(F\). Hint: apply the probability integral transform to the whole function, not to a single value. Where to start: put \(U_i=F(X_i)\) and show the supremum is unchanged, so \(D_n\) is the same functional computed on a uniform sample; Theorem 2.2.1.
Problem 2.6.5. Give an example of random variables that are exchangeable but not independent, and verify the exchangeability directly from the definition. Where to start: draw two cards without replacement from a deck and consider the indicators of an ace.
Problem 2.6.6. Using the Dvoretzky–Kiefer–Wolfowitz inequality, find the sample size needed for a \(95\%\) confidence band of half-width \(0.05\) about the entire distribution function. Compare this with the sample sizes usual in applied work and comment. Where to start: invert the bound of Theorem 2.2.9 as in the note following it.
Problem 2.6.7. Show that for \(s\leq t\), \[\operatorname {cov}\left (F_n(s),F_n(t)\right ) = \frac {F(s)\left [1-F(t)\right ]}{n},\] and explain why this rules out treating the values of the empirical distribution function at different points as independent estimates. Where to start: write both as averages of indicators and use \(\mathbb {I}\{X\leq s\}\mathbb {I}\{X\leq t\} = \mathbb {I}\{X\leq s\}\) for \(s \leq t\).
Problem 2.6.8. A permutation test is carried out on \(n=8\) observations using all \(8!=40{,}320\) relabellings. Which significance levels are exactly attainable? Explain why \(\alpha =0.05\) is not among them and what is conventionally done in consequence. Where to start: Theorem 2.2.5 attains levels that are multiples of \(1/n!\) only.
Problem 2.6.9. State and prove the result that a statistic which treats \(n\) \(iid\) random variables symmetrically, and which has finite expectation, is uncorrelated with any statistic that is a function of the ranks alone under \(H_0\). Where to start: condition on the vector of order statistics, which is sufficient, and use the symmetry of the joint density.
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