Maximum Likelihood Estimation for Exponential Waiting Times

Why Parameterization Matters

An exponential model is often used for a nonnegative waiting time, such as the time until a hypothetical device fails or a service request is completed. The same model can be described by its rate or by its mean waiting time. These parameters are reciprocals, so confusing them can produce an estimate with the right numerical ingredients but the wrong interpretation or units.

This tutorial derives the maximum likelihood estimates (MLEs) from independent waiting times. The derivation also makes clear an important boundary case: if every recorded waiting time is zero, the likelihood has no finite maximizer.

Here \(\lambda\) is a rate per unit of time, while \(\theta\) is an average waiting time in units of time. For example, if waiting times are measured in hours, \(\lambda\) is measured per hour and \(\theta\) in hours. Write the observed values as \(x_1,\ldots,x_n\), and define their total and sample mean by

The likelihood uses the joint density of the independent observations, viewed as a function of the unknown parameter with the observed data held fixed. Since all observations share the same exponential distribution, multiplying their densities gives the rate likelihood

This expression applies to observations in the nonnegative support. If the recorded values are censored, rounded, or otherwise not exact waiting times, their likelihood contributions are different and this formula does not directly apply.

Deriving the Rate Estimate

For \(T>0\), maximize the log-likelihood rather than the likelihood itself. The logarithm is strictly increasing, so it preserves which parameter value maximizes the likelihood, while turning products into sums:

Differentiate with respect to the positive rate parameter and set the score equal to zero:

The score equation identifies a candidate, so it remains to check that it is a maximum. The second derivative is negative for every \(\lambda>0\), and hence the log-likelihood is strictly concave throughout its parameter space. In addition, when \(T>0\), the likelihood tends to zero as \(\lambda\) approaches either zero or infinity. Thus the candidate is the unique finite maximum.

The condition \(T>0\) is essential. Because every observation is nonnegative, \(T=0\) means all observations are zero. In that case \(L(\lambda)=\lambda^n\), which has no finite maximum over \(\lambda>0\). It is not valid to substitute \(T=0\) into \(n/T\).

The Mean or Scale Parameterization

The exponential density can instead be written using the positive mean/scale parameter \(\theta\). Its likelihood for the same observations is

For \(T>0\), its log-likelihood and derivative are

Setting the derivative to zero gives \(-n\theta+T=0\), and therefore \(\theta=T/n=\bar{x}\). At this candidate, the second derivative is \(-n/\theta^2<0\), confirming a local maximum. The log-likelihood tends to negative infinity at both ends of the positive parameter space when \(T>0\), so this is also the unique global maximum.

The rate and mean results are two expressions for the same maximization. To see why, substitute \(\theta=1/\lambda\) into the scale likelihood:

Thus the parameterizations describe the same family of distributions and the same fitted model. Their numerical MLEs differ because they estimate reciprocal parameters: the rate estimate is \(n/T\), and the mean/scale estimate is \(T/n\). This is a direct result of maximizing the likelihood under each parameterization; it does not require treating a rate as though it were a mean.

Zeros in the Data and the Boundary Case

A zero among otherwise positive observations does not by itself prevent a finite MLE. The condition is on the total \(T\), not on whether every observation is strictly positive. The exponential density is defined at zero, with \(f(0;\lambda)=\lambda\), so an exact zero contributes a factor \(\lambda\) to the likelihood.

Common Mistakes and When the Formula Does Not Apply

The formulas are short, but their conditions and units matter. Before calculating, confirm that the sample consists of independent observations from one exponential distribution, that the values are complete and measured in a common unit, and that \(T>0\) if a finite estimate is required.

• Dropping the sample size. The rate estimate is \(n/T\), not \(1/T\). The reciprocal of the total is not the rate MLE unless the sample size happens to be one.
• Switching rate and mean. The estimate \(T/n\) is a time, so it estimates the mean/scale \(\theta\); its reciprocal estimates the rate \(\lambda\).
• Ignoring time units. A rate per minute and a rate per hour have different numerical values. If the unit of time is multiplied by 60, the numerical rate is also multiplied by 60.
• Claiming a finite estimate when \(T=0\). In that case \(n/T\) is undefined, and the likelihood is unbounded rather than maximized at a finite parameter value.
• Using the complete-data likelihood for censored or rounded times. A censored observation does not contribute an ordinary density value at an observed completion time. Rounding also means the data describe an interval rather than an exact point. Those observations require likelihood contributions appropriate to how they were recorded.

An exponential model is also inappropriate if the assumed common distribution or independence is not justified by the problem. Maximum likelihood does not make those assumptions true; it identifies the parameter value that maximizes the likelihood under the stated model. This estimation method is aligned with the SOA VEE Mathematical Statistics syllabus.