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

This paper deals with the distribution of any observation, xᵢ, 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 qᵢ = (xᵢ/s), where s is an independent estimate of the standard deviation with degrees of freedom. In a similar manner the distribution of the Studentized maximum modulus uₙ = | xₙ/s | has been obtained and upper 5 per cent points of qₙ and upper and lower 5 per cent points of uₙ 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 - to x and 0 to x.