Student members of IMS are invited to submit solutions to bulletin@imstat.org (subject “Student Puzzle Corner”). If correct, we’ll publish your name (and photo, if there’s space), and the answer, in the next issue. The Puzzle Editor is Anirban DasGupta. His decision is final.
 
Puzzle editor Anirban DasGupta says, “It is enough to send a correct answer to either 56.1 or 56.2. For each part in 56.2, a correct answer (True or False) receives +3 points, each incorrect answer receives -2 points, and each item left unanswered receives -1 point. It would be really nice if you send an answer to both problems, though it is not required.”

Puzzle 56.1 Suppose we keep observing i.i.d. Poisson random variables with mean one, until the sum exceeds a given positive integer $k$. Let $u_k$ denote the expected overshoot when we stop. Give an analytical expression for $u_k$ and discuss the convergence of $\sum _{k=1}^{\mathrm{\infty }}{u}_k$.

Puzzle 56.2, the contest problem. For each question, just say True or False, without the need to provide a proof. But answers with some explanations are especially welcome. Here are the items.

(a) A fair coin is tossed $n$ times. Let $H$ be the number of heads and $T$ the number of tails. Then, $E(|H-aT|)$ is minimized at $a=1$.

(b) Two i.i.d. observations are obtained from a Cauchy distribution with location $\mu$ and scale parameter 1. The first observation is $x_1=5$. Then the set of all values of $x_2$, the second observation, for which the likelihood function is unimodal is an interval in the real line.

(c) Suppose $X\sim \text{Poisson}(\lambda )$. Then, $E(|X-\lambda |)$ is differentiable for almost all $\lambda$.

(d) Suppose we obtain iid observations $X_1,X_2,X_3$ from a uniform distribution on $[0,\theta ],\theta >0$. Denote the median of $X_1,X_2,X_3$ by $Y$. For testing $H_0:\theta =1$ against $\theta =2$ at level $\alpha =0.05$, there exists a test based on $Y$ with power $>0.5$.

(e) Suppose $X\sim C(0,1)$, the standard Cauchy distribution. Then $\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n}P(X>n)$ diverges.

(f) Let $X_{n\times p}$ be the design matrix in a standard linear model. Then $R(X^{\prime }X)\ge 2R(X)–n$, where $R(A)$ denotes the rank of $A$.

(g) Suppose $X_1,X_2,\cdots ,X_n$ are iid $N(\mu ,1)$, where $\mu$ is known to be a rational number. Then $\overline{X}$ is a minimal sufficient statistic.

Solution to Puzzle 55

A reminder of the questions is here.

IMS student member Ruiting Tong (Purdue University) sent correct and rigorous answers to most of the problems. And a mention for Eshan De (ISI Delhi) who also attempted many of them. Puzzle Corner Editor Anirban DasGupta writes on the previous puzzle:

Problem 55.1
Let $X_1,\dots ,X_n$ be i.i.d. $C(\mu ,1)$, the Cauchy distribution with median $\mu$ and scale parameter $1$.

(a) Prove rigorously that the number of roots $T_n$ of the likelihood equation is an odd integer between 1 and $2n-1$.

The likelihood equation is a rational function of $\mu$, with the denominator being a positive polynomial, and the numerator a polynomial of degree $2n-1$. Since complex roots will come in pairs, there must be an odd number of real roots.

(b) Find with precise reasoning lim${}_{n\to \mathrm{\infty }},P(T_n>1)$.

If we write $T_n=2,R_n+1$, then without any further centering and norming, $R_n$ has a limiting distribution, namely, a Poisson with mean $1/pi$; you can see, e.g., Erich Lehmann’s book on theory of point estimation.
Therefore, lim${}_{n\to \mathrm{\infty }},P(T_n>1)=1-e^{-1/\pi }$.

Problem 55.2: True or false?

(a) A quadratic $a{x}^2+bx+c$ is called a Gaussian quadratic if $a,b,c$ are i.i.d. standard normal. If $X_1,X_2$ denote the two roots of a Gaussian quadratic, then the expectation of $\frac{1}{X_1}+\frac{1}{X_2}$ does not exist.

TRUE. With probability 1, the roots are both nonzero, and $\frac{1}{X_1}+\frac{1}{X_2}=-\frac{b}{c}$, which has a standard Cauchy distribution, and so does not have an expectation.

(b) In the standard linear model $E(\mathbf{Y}\mathbf{)}\mathbf{=}\mathbf{X}\beta$, no quadratic function ${\mathbf{Y}}^{\prime }\mathbf{A}\mathbf{Y}$ can be an unbiased estimate of a linear function ${\mathbf{c}}^{\prime }\beta$.

TRUE. We can write an exact formula for the expectation of a general quadratic form. Simple linear algebra then shows that this function cannot equal a linear function of $\beta$ for all $\beta$.

(c) There exists a location parameter density on the real line such that the average of the three sample quartiles is asymptotically the most efficient among all convex combinations of the three sample quartiles.

TRUE. This one is tricky, in the sense that if you did not know from somewhere what this distribution is, you will not be able to guess it. Using the formula for the covariance matrix of the asymptotic multivariate normal distribution of a fixed number of sample percentiles, we can specialize it to the case of the three quartiles. Therefore, we can get from here the asymptotic variance of any convex combination of the three quartiles. It then follows, remarkably, that the simple average of the three quartiles has the least asymptotic variance among all consistent comvex combinations for a $t$-distribution, whose degree of freedom can be computed from the asymptotic variance formula.

(d) Suppose $f:[0,1]\to \mathbb{R}$ is a strictly increasing continuous function. Then there is a minimum value of $D(r)={\int }_0^1,|f(x)-r|,dx$, and the minimum is attained at a unique real number $r_0$.

TRUE. Just interpret the integral as the mean absolute deviation from $r$ of $f(X)$ if $X$ is a uniform on $[0,1]$ and then it follows that the minima is the median of $f(X)$, which is unique for a function $f$ as given. The continuity of $f$ is just a simple sufficient condition for the existence of $D(r)$.

(e) Let $X_n$ denote an $n\times n$ matrix all of whose elements are $\pm 1$. If $D_n$ denotes the supremum of the determinant of all such matrices $X_n$, then $(D_n{)}^{1/n}$ has a finite limit superior.

FALSE. Jacques Hadamard proved in 1893 in a classic article that if $n$ is a power of $2,n=2^k$, then $D_n^{1/n}$ is $2^{k/2}$. Therefore the limit superior of $(D_n{)}^{1/n}$ is infinite.

(f) Sixteen equally good soccer teams are going to play in a tournament, in which teams are paired up in random, and a team to lose a match is eliminated from the tournament. The probability that teams $1$ and $2$ will meet each other at some point in the tournament is more than 10%.

TRUE. A direct calculation will show that the probability that any given pair of teams will meet during the tournament is $1/8$.