Traditionally, babies weighing less than 2,500 grams at birth have been classified as low birth weight. In a given population, low birth weight babies have a significantly higher mortality rate than others. Populations with a higher rate of low birth weights typically also have higher rates of child mortality than other populations.
The children of smoking mothers are more likely to be of low birth weight, and also have a higher child mortality. The surprising observation is that low birth weight babies of smoking mothers have a lower child mortality than low birth weight babies of non-smokers. (This is not specific to smoking; poverty for example has the same effect.)
Since these findings seemed to suggest that, at least for some babies, having a smoking mother could be beneficial, some epidemiologists questioned the general assumption that smoking causes an increased child mortality.
The paradox can be explained by the Wilcox-Russel hypothesis: low birth weight does not cause high child mortality, but rather the two variables are independently acted on by smoking. Smoking of the mother lowers the birth weight, and also increases the risk of death for any given baby. The low birth weight paradox is then a consequence of the arbitrary cut-off point of 2,500 grams. The birth weight distribution of smoking mothers as a whole is shifted to lower weights, moving more otherwise healthy babies into the low birth weight category, and thus lowering the child mortality of this category. If one corrects for this effect by artificially moving the birth weight distribution curve for smoking mothers to match that for non-smoking mothers, then one finds that babies of a given (corrected) weight are always more likely to die if their mother smoked.
Additional support for the hypothesis that birth weight and mortality can be acted on independently came from the analysis of birth data from Colorado: compared to the birth weight distribution in the US as a whole, the distribution curve in Colorado is also shifted to lower weights. The overall child mortality of Colorado children is the same as that for US children however, and if one corrects for the lower weights as above, one finds that babys of a given (corrected) weight are just as likely to die, whether they are from Colorado or not. The likely explanation here is that the higher altitude of Colorado affects birth weight, but not mortality.