MLE for the chance of an up day

Question

A desk labels each trading day's move as up, flat, or down, treating days as independent with fixed probabilities $p_{\text{up}}, p_{\text{flat}}, p_{\text{down}}$ that sum to 1. Over 200 days it records 90 up, 50 flat, 60 down.

Derive the maximum likelihood estimator of the category probabilities, and state the estimated probability of an up day.

Accepted Answer

With counts n1, \dots, nK out of n and probabilities p1, \dots, pK, the multinomial log-likelihood (dropping the constant multinomial coefficient) is \ell(p) = \sum{j} nj \ln pj, \qquad 	ext{subject to } \sumj pj = 1. Maximize under the constraint with a Lagrange multiplier. Set \dfrac{\partial}{\partial pj}\Big[\sumj nj\ln pj - \lambda\big(	extstyle\sumj pj - 1\big)\Big] = \dfrac{nj}{pj} - \lambda = 0, giving pj = nj/\lambda. Summing over j and using \sumj pj = 1 forces \lambda = \sumj nj = n, so \hat pj = \frac{nj}{n}. The MLE is just the observed frequency of each category. For up days, …

MLE for the chance of an up day

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