MLE for the share of limit orders

Question

An exchange feed labels each incoming message as a market order, a limit order, or a cancel, treated as independent with fixed probabilities summing to 1. Over 400 messages the counts are 120 market, 240 limit, 40 cancel.

Estimate the probability that a message is a limit order by maximum likelihood.

Accepted Answer

Maximizing the multinomial log-likelihood \sumj nj \ln pj subject to \sumj pj = 1 yields the sample proportions \hat pj = nj / n. For limit orders, \hat p{	ext{limit}} = \frac{240}{400} = 0.6. Takeaway. The category MLE is the observed relative frequency, here 240 of 400.

MLE for the share of limit orders

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