The distribution of the argmax of two-sided Brownian motion with quadratic drift

The distribution of the argmax of two-sided Brownian motion with quadratic drift has been identified as the asymptotic distribution of several suggested estimators. Because of the intractability of this distribution, Narayanan and Sager (1989) performed a large-scale simulation to table the corresponding CDF. Here, this distribution is numerically derived and compared to the earlier simulations. It is also shown that a very good normal approximation exists which works very well for finding desired probabilities and percentiles. The normal approximation also serves to make the distribution of probability easily understood and conveyed, makes approximate percentiles and probabilities easily obtained using normal tables, and, since the distribution usually appears as a limiting distribution, the normal density can be used in place of the true limiting density as a benchmark of the closeness of the finite sample distribution to its limiting distribution.