Is one over the average a consistent rate estimator?

Question

Gaps between consecutive trades are i.i.d. Exponential with unknown rate $\lambda$ , so each gap has mean $1/\lambda$ . You estimate the rate by inverting the average gap:

$\hat\lambda = \frac{1}{\bar X}.$

Is $\hat\lambda$ a consistent estimator of $\lambda$ ? Address its bias as well.

Accepted Answer

Step 1: what the average converges to. By the law of large numbers, the sample mean converges in probability to the true mean of a gap: \bar X \xrightarrow{p} \mathbb{E}[X] = \frac{1}{\lambda}. Step 2: push it through the inversion. The map g(t) = 1/t is continuous at the point t = 1/\lambda (which is positive, so not the problematic t = 0). The continuous mapping theorem says that applying a continuous function preserves convergence in probability: \hat\lambda = g(\bar X) \xrightarrow{p} g\!\left(\frac{1}{\lambda}\right) = \lambda. So \hat\lambda is consistent: with enough data it homes in o…

Is one over the average a consistent rate estimator?

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