Inter-trade times are iid Exponential(lambda). From n = 25 observations with sample mean 0.4 seconds, give the maximum likelihood estimate of lambda, show that the estimator is biased, and give the multiple of it that is unbiased.

Inter-trade times are iid Exponential(lambda). From n = 25 observations with sample mean 0.4 seconds, give the maximum likelihood estimate of lambda, show that the estimator is biased, and give the multiple of it that is unbiased.

Approach: Maximise the exponential log-likelihood, then use the fact that a sum of iid exponentials is gamma to take the expectation of the reciprocal.

2.5. The log-likelihood is n*log(lambda) - lambda*sum(x_i), whose derivative n/lambda - sum(x_i) vanishes at lambda_hat = 1/xbar = 1/0.4 = 2.5. The sum S of n iid exponentials has a Gamma(n, lambda) distribution, and E[1/S] = lambda/(n - 1), so E[lambda_hat] = n*E[1/S] = n*lambda/(n - 1) = 25*lambda/24, an upward bias of about 4.2%. Multiplying by (n - 1)/n removes it exactly, giving (24/25)*2.5 = 2.4. The bias is Jensen's inequality applied to the strictly convex map 1/x: the estimator of the mean is unbiased and the reciprocal of an unbiased estimator is not. Maximum likelihood is invariant under reparameterisation and unbiasedness is not, so the two properties cannot both survive the transformation.

Follow-up: What is the exact variance of the bias-corrected estimator at n = 25, and how does it compare with the Cramer-Rao bound lambda^2/n?

Key concepts: maximum likelihood, gamma distribution, bias correction, Jensen's inequality.