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One meaning is involved in what is called a biased sample: If some elements are more likely to be chosen in the sample than others, and those that are have a higher or lower value of the quantity being estimated, the outcome will be higher or lower than the true value.
A famous case of what can go wrong when using a biased sample, is found in the 1936 US presidential election polls. The Literary Digest held a poll that forecast that Alfred E. Landon[?] would defeat Franklin Delano Roosevelt by 57% to 43%. George Gallup, using a much smaller sample (300,000 rather than 2,000,000), predicted Roosevelt would win, and he was right. What went wrong with the Literary Digest poll? They had used lists of telephone and automobile owners to select their sample. In those days, these were luxuries, so their sample consisted mainly of middle and upper class citizens. These voted in majority for Landon, but the lower classes voted Roosevelt. Because their sample was biased towards wealthier citizens, their result was incorrect.
This kind of bias is usually regarded as a worse problem than statistical noise[?]: Problems with statistical noise can be lessened by enlarging the sample, but a biased sample will not go away that easily. In particular, a meta-analysis will distill good data for studies that themselves suffer from statistical noise, but a meta-analysis of biased studies will be biased itself.
Another kind of bias in statistics does not involve biased samples, but does involve the use of a statistic whose average value differs from the value of the quantity being estimated. For example, suppose X1, ..., Xn are independent and identically distributed random variables, each with a normal distribution with expectation μ and variance σ2. Let
A far more extreme case of a biased estimator being better than any unbiased estimator is well-known: Suppose X has a Poisson distribution with expectation λ. It is desired to estimate