May 15, 2019

Student Puzzle Corner 24

Contributing Editor Anirban DasGupta sets this problem. Student members of the IMS are invited to submit solutions (to bulletin@imstat.org 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 $T_n$.

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