Gertrude M. Cox 'l'HE "INEFFICIENCY" OF THE SJUviPLE MEDIAN FOR F(;.R MANY FAMILIAR SYMMETRIC DISTRIBUTIONS by John T. Chu Department of Statistics University of North Carolina, Chapel Hill Special report to the Office of Naval Research of ~ork at Chapel Hill under Contract NR 042 031, Project N7-onr284(02), for research in probability and statistics. Institute of Statistics Mimeograph Series No. 125 February 23, 1955 1 THE IlINEFFICIENCyllOF THE SAlt1PLE MEDIAN FOR MANY FAMILIAR SYMMETRIC DISTRIBUTIONSJ. By J.T.Chu, University of North Carolina 1. A lower bound. If the reciprocal of the ( asymptotic ) variance of an estimate is taken as a measure of its ( asymptotic ) efficiency, the s~ple median is often ( asymptotically ) less efficient than the sample mean symmetric distributions familiar to statisticians. x ,for many In fact, for a symmetric distribution having its maximum frequency at the point of symmetry, i f x, asymptotically less efficient than as x X then quite often x x is is not so efficient at all, with the possible exceptions of very small samples. To show these facts, we derive a very simple, yet sharp, lower bound for the variance of the sample median. Suppose that F(x) and f(x) are the cdf ( cumulative distribution function ) and pdf ( probability density function ) of a certain continuous distribution, and f(x) is ~ f(x) for all x. then, where Cn = (1) ( var x Let x symmet~ic with respect to x = £, and f(t) be the s~mple median of a sample of size 2n + 1, 2n + 1 )1 / n1 n~, = x - S )2 C /- F(x) n - - 7 nr 1 - F(x) - - 7 nf(x) = ~ Equality holds for Ci r 1 f(s») -2 J (F - 1/2 )2 C o Fn ( 1 _ F )n dF n rectangular distribution, 1 Sponsored by the Office of Naval Research under Contract NR 042 031 dx -22. Examples. It is well known that .for a normal or rectangular distribution X x is more efficient than We shall show that this is also true .for many .familiar symmetric distributions. (1) Tri~ngular distribution: i is less efficient than =1 f(x) - lxi, Ixl $ 1. .for samples of sizes 2n + 1 where n computation shows that this 1s also true if n > 1. For a t-distribution with k degrees o.f freedom, less efficient than x 2 2 (kj2) / 4 r ( ( k + 1 )/2 ) = 2m + >5 -x = = r(2p)r- 2 (p) i- 2, p. 244_7 x p - 1 ( 1 _ x )p - 1, o < x < 1; is less efficient than 2 4p (2n + 3) / (2n + 1). > 25. and n = > Computation shows, e.g., that the (3) Symmetric ~-distribution. The pdf is given by f(x) is 1, the LHS ( left hand side ) of the above in- equality is an incredsing function of m. inequality holdS if k x i f ( not necessarily only if ) te ( k - 2 ) r k= 2m and k Direct = 1. (2) t-distribution. For both x Following (1), x > O. p if 2 - 4 ( 2p + 1 ) r\p) / r (2p) >( 2n + 3 )/( 2n + 1 ) • The LHS becomes smaller if p is replaced by p + 1, and tends to tej2 as p tends to every p (~) 00. So it has a lower bound te/2. > 0 and n ~ 2. Cauchy type distribution. f(x) = Cal ( 1 + Ixja ), It is defined to be one with a pdf of the type -00 <x < known Cauchy distribution for i. Hence the inequality holds for 00; ~hich a -x > 1. If a = 2, is infinitely more efficient than It would be interesting to examine whether or not efficient as a increases. Now i x is less efficient than x where x = teja. 2 sin 3x I sin 3 x i becomes more > 3. C a 3, p. 118_7. It has finite variance only if a and var i can be obtained by using contour integration follows that we obtain the well >( i i- if 2n + 3 ) j ( 2n + 1 ), The LHS is a decreasing function of x, so an increasing , -3f .:.::ti'::;'" :Jf The least ex's for \.hieh the LHS is equal to 0:.. maximum and minimum of the RHS, are found to be 5/3 and 1, the 4.65 and 3.75 approximately. 3. Remarks. (~) Not for all symmetric distributions is x more efficient than the parent population has a Laplace distribution, e.g., for all samples of odd sizes x - x When is more efficient L· 1.7. (2) If f(x) satisfies certain continuity conditions, normal distribution and the asymptotic variance is x has an asymptotically (4' L· f( s).7 2( 2n + 1 ) J .1. Therefore if the sample size is not too small, the asymptotic variance is for all practical purposes a lower bound for efficient than , then x -x var x. And if is less efficient than x X is asymptotically less for all samples whose sizes are not too small. References i· 1.7 Chu,J.T., and Hotelling, H., The moments of the sample median, to be published. L· 2.7 Cramer, H., Mathematical Methods of Statistics, Princeton University Press, 1946. I: 3.7 Whittaker,E.T., and Watson, G.N., Press, 1952. Modern Analysis, Cambridge University
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