Deadline: August 15, 2026. 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.

Student Puzzle editor Anirban DasGupta sets these puzzles, returning to a familiar pattern. He says, “We state here three problems of a diverse nature for you. We will consider responses that solve just one of the three problems, but please consider sending solutions to more than one problem. We will be thrilled if you do!” Here are the three problems:

Puzzle 61.1
Consider iid observations $X_1, X_2, \cdots $ from a distribution on the real line having a density. Call $X_i$ a record value if $X_i > X_j$ for all $j < i$; we assume $X_1$ to be a record value. For $n \geq 1$, let $N_n$ denote the number of record values observed among $\{X_1, \cdots , X_n\}$.

(a) Find the mean and variance of $N_n$ for $n = 10$.

(b) Calculate approximately the value of $P(N_n > 8)$ for $n = 100$.

Puzzle 61.2
Let $\bf{Z}$ have a $p$-dimensional multivariate standard normal distribution. Denote the $p$-dimensional unit Euclidean ball by $\mathcal{B}$. Prove or disprove that $\lim_{p \to \infty }\, E[\sup_{\bf{c} \in \mathcal{B}}\, \bf{c}’\,\bf{Z}]/{\sqrt{p}}$ exists. Can you say what is the value of this limit, if a limit exists?

Puzzle 61.3
Suppose $X_1, X_2, \cdots , X_n$ are iid observations from $N(\theta , \theta), \theta > 0$. Prove that for any convex loss function, the sample mean $\bar{X}$ and the sample variance $s^2$ are inadmissible estimators of $\theta $.