Sankhy¯a : The Indian Journal of Statistics 2002, Volume 64, Series B, Pt. 3, pp 289-300 DO TAILS OF SYMMETRIC DISTRIBUTIONS CONTAIN MORE FISHER INFORMATION ABOUT THE SCALE PARAMETER? By G. ZHENG National Heart Lung and Blood Institute, Bethesda, MD, USA and J.L. GASTWIRTH George Washington University, DC, USA SUMMARY. Several studies have shown that the tails of an ordered sample from a symmetric distribution contain more Fisher information about the scale parameter than the middle portion. Whether the tails are the most informative symmetric fraction of the ordered data remains of interest. In this note, we provide conditions for two symmetric portions of order statistics to contain the most limiting Fisher information about the scale parameter of Cauchy, Laplace, logistic, and normal distributions. This indicates that highly efficient estimators for the scale parameter of these distributions may be obtained using the order statistics in this most informative region. 1. Introduction Suppose X 1 ,...,X n are i.i.d. random variables from a symmetric location- scale distribution F θ (x)= F ((x − μ)/θ), where the location parameter μ is known and θ is the unknown scale parameter, with an absolutely continu- ous density f θ (x)= f ((x − μ)/θ)/θ. Let X 1:n ≤···≤ X n:n be their order statistics. Comparing the approximate efficiency of the best linear unbiased estimator (BLUE) of θ of the normal distribution N (0,θ) using doubly cen- sored data, X r:n , ..., X s:n ,1 ≤ r ≤ s ≤ n, for n ≤ 15, Sarhan and Greenberg (1956, 1958) found that, if two extreme order statistics were censored, the efficiency of the BLUE fell below 80%. Nagaraja (1994) studied Tukey’s linear sensitivity and its role in the comparison of linear estimators, and concluded that, for the normal scale parameter, the extreme order statistics contained more information than the middle ones. Examining how much Fisher information about θ is contained in two extreme tails of symmetric Paper received September 2001. AMS (2000) subject classification. Primary 62F07; secondary 62J05, 94A15. Keywords and phrases. Best linear unbiased estimator, censored data, Fisher information, folded distribution, order statistics.