Epstein Test

TL;DR

• A hypothesis test for whether a dataset follows an exponential distribution, commonly applied to inter-arrival or inter-event times.
• Procedure: compute mean and standard deviation, inspect a histogram (log-log scale), then compute the Epstein Test statistic.
• The Epstein Test statistic is the difference between the observed mean and the expected mean under an exponential model; a statistically significant difference indicates the data are not exponentially distributed.

Definition

The Epstein Test is a statistical test used to determine whether a given set of data follows an exponential distribution.

Explanation

• The researcher calculates the sample mean and standard deviation for the data.
• The data are plotted on a histogram with the x-axis representing data values and the y-axis representing frequency. According to the description, if the data are exponentially distributed, the histogram should have a straight line shape when plotted on a log-log scale.
• The Epstein Test statistic is computed as the difference between the observed mean and the expected mean of the data if it were exponentially distributed:

$$T = \bar{x} - \mu_{\text{exp}}$$

• If the difference is statistically significant, the researcher concludes that the data are not exponentially distributed.

Examples

Time between consecutive arrivals (intersection)

A researcher studying the time between cars arriving at a particular intersection may use the Epstein Test to determine whether those inter-arrival times are exponentially distributed. If they are, the researcher may conclude the times between arrivals are random and independent.

Time between consecutive purchases (store)

A researcher analyzing the time between purchases at a store may apply the Epstein Test to see whether those intervals follow an exponential distribution. If the test indicates an exponential distribution, the researcher may conclude that purchases are made randomly and independently.

Use cases

• Verifying whether inter-arrival or inter-event times can be modeled as exponential, which supports conclusions about randomness and independence.

Notes or pitfalls

• The source describes expecting a straight line shape for the histogram when plotted on a log-log scale if the data are exponential.
• The test decision hinges on whether the difference between observed and expected mean is statistically significant.

Related terms

• Exponential distribution