Statistics 583, Problem Set 7

Wellner; 5/10/2000

Reading:	Lecture Notes, Chapter 7, pages 1-15;
	Efron and Tibshirani, Chapter 4 (pages 31-37), Chapter 21, pages (296 - 205).
Due:	Wednesday, May 24, 2000.

1.

Let $U_{m,n} \equiv T( \FF_m , \GG_n )$ where $T(F,G) = \int F dG =P(X \le Y)$ is the Mann-Whitney functional and $\FF_m$ and $\GG_n$ are the empirical df's of $X_1 , \ldots , X_m$ i.i.d. with df F, $Y_1 , \ldots ,Y_n$ i.i.d. with df G.
A. Show that

$$mn U_{m,n} + n(n+1)/2 =W_{m,n} \equiv \sum_{j=1}^n Q_j =\sum_{j=1}^n R_{m+j} \, .$$

B. Show that $E U_{m,n} = P(X \le Y) = \int Fd G$ and that

$$\begin{eqnarray*}Var( \sqrt {mn} U_{m,n} ) & = & (n-1) \int (1-G)^2 dF + (m-1)\i... ...] +(m-1)Var[F(Y)] \\ & & \qquad + \ \int FdG (1-\int FdG ) \, . \end{eqnarray*}$$

C. When F=G use the results of A and B to compute E_(F,F) W_m,n and Var_(F,F) (W_m,n). (This should agree with calculations for the Wilcoxon rank sum form of the statistic under the null hypothesis via finite sampling calculations.)

2.

Consider the Mann-Whitney-Wilcoxon functional T(F,G) as in problem 1.
(a) Show that T(F,G) is continuous at every pair of distributions (F,G) with respect to the Kolmogorov distance

$$d_K (F, \tilde{F}) \equiv \sup_x \vert F(x) - \tilde{F}(x) \vert \equiv \Vert F - \tilde{F} \Vert _{\infty} \, :$$

if $\Vert F_n - F\Vert _{\infty} \rightarrow 0$ and $\Vert G_n - G\Vert _{\infty} \rightarrow 0$, then $T(F_n , G_n ) \rightarrow T(F,G)$.
(b) Use the result of (a) to prove that $T( \FF_n , \GG_n ) \rightarrow_{a.s.} T(F,G)$.
(c) Give an example to show that T(F,G) is not weakly continuous at pairs of distribution functions (F,G) with common discontinuity points.
(d) Extend the definition of Gateaux differentiable functions in a natural way to include T(F,G), and then calculate the Gateaux derivative of T(F,G).
(e) Use your calculation in (d) to ``guess'' the asymptotic variance of $T(\FF_m, \GG_n)$.

3.

Consider the collection ${\cal F}_0$ of distribution functions F on R⁺ with $0 < E_F X < \infty$ and $E_F X^2 < \infty$. Let $T(F) \equiv \sigma (F) / \mu (F)$ for $F \in {\cal F}_0$where $\sigma^2 (F) = Var_F (X)$ and $\mu (F) = E_F (X)$. This is the coefficient of variation of F. Find the influence function of T(F).

4.

Suppose that ${\cal F}_0$ is the same class of distribution functions as in problem 1, but now consider the functional T(F) defined for a fixed $x_0 \in R^+$ by

$$T(F) \equiv e_F (x_0) \equiv E_F (X - x_0 \vert X > x_0 ) = \frac{\int_{x_0}^{\infty} (1 - F(t))dt}{1 - F(x_0)} \, .$$

This functional is the mean residual life functional. Find the influence function of T(F).

5.

Let F be a distribution function on R² with finite second moments, and let $\rho(F)$ be the correlation coefficient

$$\rho(F) = \frac{Cov_F(X,Y)}{\sqrt{Var_F (X) Var_F(Y)} } \, .$$

Find a collection ${\cal F}$ of distribution functions on R² so that $\rho$ is weakly continuous on ${\cal F}$.

6.

For distribution functions F on R⁺ and t₀>0, consider the functional

$$T(F) = \Lambda (t_0) \equiv \int_0^{t_0} \frac{1}{1-F_{\_}} dF \, ,$$

the cumulative hazard function corresponding to F at t₀. Find the influence function of T(F). What does this mean about asymptotic normality of the natural estimator $T( \FF_n)$ of T(F)? Can you prove asymptotic normality of $T( \FF_n)$ directly?