MEAN STREETS - THE MEDIAN OF A SIZE-BIASED SAMPLE AND THE POPULATION MEAN

In a size-biased or weighted sample, the probability that a unit is sampled depends on a property of the sampling unit. Cars sampled by an observer traveling at a fixed velocity, v0, can be modeled as a size-biased sample because a car with a large absolute difference between its velocity and v0 is more likely to be sampled. A consistent estimator of the population mean velocity is found that depends solely on counts of the number of cars the observer passes and the number of cars that pass the observer. This estimator has the unusual property that when counts of passed and passing cars are equal, the sample median velocity and the estimated population mean velocity are both the observer's velocity, v0.