On a Statistic Similar to Student's t.
WASHINGTON UNIV SEATTLE DEPT OF MATHEMATICS
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Consider a random variable X which has median mu. Let X1 or X2 or ... X2 m 1 be an ordered sample of X and let U Xm 1-r, V Xmu 1, W Xmu 1 r. The statistic S V - muW - U is independent of mu and of any scale parameter, hence is distribution-free with regard to any family of probability distributions Fx - ab where F. is a specified distribution function and a any real and b any positive number. A partial answer is given to the problem of a studentized Chebyshev inequality. Properties of S are discussed which, even in the special case of X with normal distribution, make it useful in practical situations in which Students t is traditionally used, but in which t cannot be applied because of incomplete data, e.g. in case of one-sided or two-sided censoring.
- Statistics and Probability