The Malmquist Bias is a selection bias applicable to Astronomical Surveys stemming from missing the dimmest objects. If some objects are too dim for the instruments to detect, and are dimmer with distance, then surveys will include only brighter objects, and the greater the distance, the stronger the preference toward brighter objects, meaning evaluation of the randomness of sample-sets must take this into consideration.
This is in contrast to another type of observation bias, the Eddington Bias. I've seen references that call two terms synonymous, but to my understanding, the Eddington Bias stems from brightness measurement errors (or perhaps "errors" from Seeing, Reddening, Extinction and the like), whereas the Malmquist Bias stems from distance alone and applies even if there are no such brightness mistakes. However, the means of compensating for the two biases might be equivalent.