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.

Anirban DasGupta says, “We are continuing with our contest model as in the previous puzzles. Each correct answer receives 3 points, each incorrect answer receives -2 points, and each item left unanswered receives -1 point. The top three scorers will be recognized. You can answer just one of the two problems, 51.1 and 51.2, although answering both will be very welcome. The bonus problem is to excite your scientific curiosity, and it will be a joy to hear your opinion..”

Puzzle 51.1
Let r be a standard normal variable, and A a cube in d-dimensions with side lengths equal to |r|. Find the expected volume, as analytically as you can, of the largest ball that can be put inside the cube A, and compute it for d = 2, 3, 5.

Puzzle 51.2 For our contest problem, answer True or False, without the need to provide a proof. But reasoned answers are especially welcome. Here are the items.

(a) The total variation distance between a binomial distribution with parameters n = 50 and p = 0.01 and the Poisson distribution with mean 0.5 is no more than 0.01.

(b) If X is a Cauchy distributed variable with a location parameter μ, then there is no unbiased estimator δ(X) of μ.

(c) If T_m denotes a symmetric t distributed variable on m degrees of freedom, then one can write T₁ as T₂+W where W is independent of T₂.

(d) If $\{S_n\}_{0}^\infty $ with $S_0 = 0$ stands for the simple symmetric random walk on the line, then there exist functions f(S_n) such that f is not one–one, but $\{f(S_n)\}_{0}^\infty $ is a Markov chain.

(e) For n i.i.d. observations from a univariate normal distribution with an unknown mean and a known variance of 1, the $N(\bar{X}, 1)$ density is a minimax estimator of the true density among all normal density estimators under an ℓ₂ loss.

(f) Given i.i.d. samples X₁, …, X_n from a d-dimensional normal distribution for general n≥2, d≥2, with a general mean vector and a general covariance matrix, the sample variance-covariance matrix can be written as A₁+A₂ where A₁, A₂ are nonsingular matrices.

Puzzle 51.3: Bonus Problem for Your Independent Exploration.
A point in the plane is called a Gaussian position if its Cartesian coordinates (X, Y ) are i.i.d. standard normals. Consider a triangle T with vertices chosen as independent Gaussian positions. Argue for or against the motion that we should predict that the slope of the Euler line of T is zero.

Solution to Puzzle 50

Deborshi Das

Well done to Deborshi Das (ISI Delhi, pictured here) for his correct (and what Anirban described as “nifty”) solutions! Here’s a reminder of Puzzles 50.1 and Puzzle 50.2. Anirban explains:

Puzzle 50.1
Since the normal distribution with a known variance is in the one-parameter exponential family, the posterior mean of the mean $\mu $ is an increasing function of $\bar{X}$ for any prior $G$. Hence the negative of the posterior mean is a decreasing function of $c\,\bar{X}$ for any $c > 0$. Now apply the Brouwer fixed-point theorem.

Puzzle 50.2

(a) The sample median is an inadmissible estimator of a one-dimensional normal mean $\mu $ if the loss function is $(\mu – a)^4$.

True; use the Rao–Blackwell theorem.

(b) If $X_1, X_2, \cdots , X_n$ are i.i.d. standard exponential, then $\limsup\, [E(\max (X_1, X_2, \cdots , X_n)) – \log n] < .50.$

False; actually, a limit exists and the limit equals the Euler constant $\gamma $ which is larger than 0.5.

(c) If $F(x)$ is the CDF of an absolutely continuous distribution on the real line with a completely unknown density $f(x)$, then it is possible to construct an unbiased estimate of $f$ provided the sample size is 5 or more.

False; no unbiased estimates exist for any value of $n$. This is a famous result of Murray Rosenblatt.

(d) If $Z$ has a standard normal distribution, then one can find independent random variables $X$ and $Y$, each possessing a density, such that $Z = X\,Y$in law.

True; a simple example is the Box–Muller transformation.

(e) If $F(x)$ is the CDF of an absolutely continuous distribution on the real line with a continuously differentiable density $f(x)$, and $|f'(x)| \leq 100$ for all $x$, then $f$ is bounded above by 20.

True; this follows from the Kolmogorov–Landau inequality on intermediate derivatives.

(f) The Jeffrey prior on a multinomial probability vector $(p_1, \cdots , p_k)$, where $p_k = 1- \sum_{i = 1}^{k-1}\,p_i$, is improper if $k \geq 3$.

False; in fact, the Jeffreys prior is a Dirichlet with all parameters equal to $1/2$, and so proper by definition of a Dirichlet.

(g) If $X$ has a Poisson distribution with mean 1, then $E(X^X) < \infty $.

False; since $\sum_{x = 1}^\infty \, \frac{(\lambda e)^x}{\sqrt{x}} < \infty $ i.f.f. $\lambda \,e < 1$, one has that for $X$ distributed as Poisson with mean $\lambda , E(X^X) < \infty $ i.f.f. $\lambda < 1/e$.

(h) Suppose $X, Y$ each has a finite variance, that $E(X\,|Y) = Y$ and $E(Y\,|X) = 0$. Then $Y = 0$ with probability 1.

True; this follows by using the iterated expectation formula on $E(XY)$ by conditioning once on $Y$ and once on $X$.

(i) Pick a permutation of $\{1, 2, \cdots , n\}$ at random. Then the probability that the cycle containing 1 has length $k$ is the same for all $k$.

True; this follows from the cycle structure of random permutations.