Method of moments for a trade-gap rate

Question

On an illiquid stock, the time in seconds between consecutive trades looks like independent draws from an exponential distribution with unknown rate $\lambda$ (trades per second). The average gap you observe is $8$ seconds.

Use the method of moments to estimate $\lambda$ .

Accepted Answer

Match the mean. The exponential distribution's mean is one over its rate: \mathbb{E}[X] = \frac{1}{\lambda}. The mean gap is "seconds per trade," and the rate is "trades per second," so they are reciprocals. Solve. Set the model mean equal to the observed average gap of 8 seconds: \frac{1}{\lambda} = 8 \implies \hat{\lambda} = \frac{1}{8} = 0.125 	ext{ trades per second}. Reading the units. A rate of 0.125 per second is one trade every 8 seconds on average, which is exactly the gap we measured. Longer average gaps mean a smaller rate (a quieter name); shorter gaps mean a larger rate. Takeawa…

Method of moments for a trade-gap rate

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