On the Distribution of the Ratio of the ith Observation in an Ordered Sample from a Normal Population to an Independent Estimate of the Standard Deviation
Pillai, K. C. S. ; Ramachandran, K. V.
Ann. Math. Statist., Tome 25 (1954) no. 4, p. 565-572 / Harvested from Project Euclid
This paper deals with the distribution of any observation, $x_i$, in an ordered sample of size $n$ from a normal population with zero mean and unit standard deviation. The distribution has been developed as a series of Gamma functions, and has been used to obtain the distribution of $q_i = (x_i/s)$, where $s$ is an independent estimate of the standard deviation with $\nu$ degrees of freedom. In a similar manner the distribution of the Studentized maximum modulus $u_n = | x_n/s |$ has been obtained and upper 5 per cent points of $q_n$ and upper and lower 5 per cent points of $u_n$ have been given. The method of obtaining the different distributions essentially depends on appropriate expansions of the normal probability integral and its powers in the intervals $- \infty$ to $x$ and 0 to $x$.
Publié le : 1954-09-14
Classification: 
@article{1177728724,
     author = {Pillai, K. C. S. and Ramachandran, K. V.},
     title = {On the Distribution of the Ratio of the ith Observation in an Ordered Sample from a Normal Population to an Independent Estimate of the Standard Deviation},
     journal = {Ann. Math. Statist.},
     volume = {25},
     number = {4},
     year = {1954},
     pages = { 565-572},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1177728724}
}
Pillai, K. C. S.; Ramachandran, K. V. On the Distribution of the Ratio of the ith Observation in an Ordered Sample from a Normal Population to an Independent Estimate of the Standard Deviation. Ann. Math. Statist., Tome 25 (1954) no. 4, pp.  565-572. http://gdmltest.u-ga.fr/item/1177728724/