1993 Prelim in Probability and Statistics

Let \(X_0=1\) and define \(X_n\) inductively by declaring that \(X_{n+1}\) is uniformly distributed over \((0,X_n)\). Prove that \(n^{-1}\log X_n\to c\) a.s. and compute \(c\).

Let \(X_1,X_2,\ldots \) be independent Poisson random variables with \(\mathbb {E} X_n=\lambda _n\), and let \(S_n=X_1+\cdots +X_n\). Show that if \(\sum \lambda _n=\infty \), then \(S_n/\mathbb {E} S_n\to 1\) a.s.

Hint: Show that (*) \(Y_n/c_n\to 1\) a.s., provided that \(Y_n\ge 0\) is nondecreasing in \(n\), and (*) holds for a subsequence \(n(k)\) that has \(c_{n(k+1)}/c_{n(k)}\to 1\).

Let \(X_1,X_2,\ldots \) be i.i.d. mean 0, variance \(\sigma ^2\in (0,\infty )\).

(a) Use the central limit theorem as well as Kolmogorov’s 0-1 law to conclude that \(\limsup _{n\to \infty }S_n/\sqrt n=\infty \) a.s.

(b) Show that \(S_n/\sqrt n\) does not converge in probability.

Suppose that \(X\) and \(Y\) are independent. Let \(f\) be a Borel function on \(\mathbb R^2\) with \(\mathbb {E}[|f(X,Y)|]<\infty \) and let \(g(x)=\mathbb {E}[f(x,Y)]\). Show that \(\mathbb {E}[f(X,Y)\mid X]=g(X)\).

A thinker who owns \(r\) umbrellas travels back and forth between home and office, taking along an umbrella (if there is one at hand) in rain (probability \(p\)) but not otherwise (probability \(q=1-p\)). Let the state be the number of umbrellas at hand, irrespective of whether the thinker is at home or at work. Set up the transition matrix, and show that the Markov chain approaches equilibrium (i.e., the ergodic theorem is applicable). Find the steady-state probability of his getting wet, and show that five umbrellas will protect him at the \(5\%\) level against any climate (any \(p\)).

Let \(U_1,U_2,\ldots ,U_n\) be independent identically distributed random variables, uniform on \([0,1]\). Let \(U_{1,n}\le U_{2,n}\le \cdots \le U_{n,n}\) denote the order statistics. Let \(X_1,X_2,\ldots ,X_{n+1}\) be independent identically distributed exponential random variables with \(\mathbb {E} X_1=1\). Define \(S(i)=X_1+\cdots +X_i\).

(a) Prove that the random vectors \[ \{U_{1,n},\ldots ,U_{n,n}\} \quad \text {and}\quad \left \{\frac {S(1)}{S(n+1)},\ldots ,\frac {S(n)}{S(n+1)}\right \} \] have the same distribution.

(b) Compute the asymptotic distribution of \(n(U_{i+3,n}-U_{i,n})\), as \(n\to \infty \) when \(i\) is fixed.

Let \(Y_i=\alpha ^{i/n}+\epsilon _i\), \(1\le i\le n\). We assume that \(\epsilon _1,\epsilon _2,\ldots ,\epsilon _n\) are independent, identically distributed random variables with \(\E \epsilon _i=0\), \(0<\sigma ^2=\operatorname {Var}\epsilon _i<\infty \) and \(\E \epsilon _i^4<\infty \).

(a) Find the least-squares estimator for \(\alpha \).

(b) Show that the estimator is asymptotically normal.

(c) Find an estimator for \(\sigma ^2\).

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables, uniformly distributed on \([0,\theta ]\), \(\theta >\theta _0\). We want to test \(H_0:\theta =\theta _0\) against \(H_A:\theta >\theta _0\).

(a) Find the uniformly most powerful test. (You must prove your claim.)

(b) Show that the uniformly most powerful test and the likelihood ratio test are equivalent,

(c) Compute the power function of the most powerful test.

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables with density function \(f\). We assume that \(f'\) is bounded. Let \(K\) be a function satisfying \(\int _{-\infty }^{\infty }K(u)du=1\), \(K'(u)\) is bounded and \(K(u)=0\), if \(|u|\ge a\) where \(a\) is a constant. The density \(f\) is estimated by \[ \hat f_n(t)=\frac {1}{nh}\sum _{1\le i\le n} K\left (\frac {t-X_i}{h}\right ). \] Show that \(\hat f_n(t)\) is an almost surely uniformly consistent estimator for \(f\) on \([\alpha ,\beta ]\), \(-\infty <\alpha <\beta <\infty \).

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables with \(\mathbb {P}\{X_i=1\}=p\), \(\mathbb {P}\{X_i=0\}=1-p\).

(a) Compute the maximum likelihood estimator of \(\sigma ^2=p(1-p)\).

(b) Compute the bias, the variance and the mean-square error of the estimator.

(c) Is the estimator asymptotically efficient?

(d) Find the uniformly minimum variance unbiased estimator for \(\sigma ^2\).