It is well known that (see e.g. [1]) the median of a sample of an odd
size is just the middle order statistic, and that the distribution of it can be easily
formulated as long as the pdf (probability density function) and the cdf (cumulative
distribution function) of the underlying distribution are represented by simple func-
tions. On the other hand, the median of a sample of an even size is the
... [Show full abstract] average of the
two middle order statistics, and deriving its distribution requires some calculations.
The complexity of the problem is doubled especially for the jump type distributions
with bounded supports. In this paper we obtain general results for the pdf and the
moments of the median of a sample of even size from a two-piece distribution. We
use integral calculus, which was also employed by several authors e.g. [2], [3], [4]
for different purposes, to derive the exact pdf and the moments in the case of even
sample size. The general results are then used to obtain the required distribution in
the case of the two-piece uniform distribution and the two-sided power distribution.
Further, the variances of the sample median are calculated for different sample sizes
and parameter values.
Keywords: jump type distribution, sample median, special functions.
References
1. Arnold B.C, Balakrishnan N, Nagaraja H.N. (2008). A First Course in Order Statis-
tics, Classic Edition, SIAM, Philadelphia.
2. Glickman T.S., Xu F. (2008). The distribution of the product of two triangular
random variables. Statistics and Probability Letters, 78, 2821{2826.
3. Gupta P.L., Gupta R.C. (2009). Distribution of a linear function of correlated
ordered variables. Journal of Statistical Planning and Inference, 139, 2490{2497.
4. Gunduz S., Genc A.I. Distribution of the product of a pair of independent two-sided
power variates. Communications in Statistics Theory and Methods (in press)
DOI:10.1080/03610926.2014.957861.