Contributing Editor Anirban DasGupta sets this problem. Student members of the IMS are invited to submit solutions (to with subject “Student Puzzle Corner”). The deadline is June 25, 2019.

Here’s Anirban’s latest puzzle. He says:

To encourage many students to send an answer, we’re posing a very simple problem on random walks this time!

Let $n \geq 1$ be a given positive integer. A point in the plane starts a random walk on the finite lattice $(i,j), 0 \leq i, j \leq n$ at the origin $(0, 0)$. It moves one step up if a fair coin toss results in a head, and one step to the right if the coin toss results in a tail. If there is no more room to move up, i.e., if it is already at a point $(i, n)$, then it stays at that point; likewise, if it has no more room to move further to the right. Calculate $\mu _n$, the expected number of steps after which the coin arrives at the point $(n, n)$,

a) if $n = 3$;

b) for a general $n$.

c) For a special mention, what can you say about the asymptotic rate of $\mu _n$?

Solution to Puzzle 23

Contributing Editor Anirban DasGupta writes on the previous puzzle:

There are many different estimator sequences $T_n$ that have the stated properties. The important fact to utilize is that for $n \geq 7$, the Pitman estimate of $\mu $ is unique minimax, a result first proved by Charles Stein, and later generalized by Larry Brown. The Pitman estimate, say $S_n(X_1, \cdots , X_n)$, is unbiased (for such $n$) and has a constant risk. One may thus take the Pitman estimate sequence and use $T_n = S_{n-1}(X_1, \cdots , X_{n-1})$; that is, drop one data value and use the Pitman estimate. One can also use the MLE of $\mu $ for $T_n$.