Some Monotonicity Theorems for χ2, F and T Distributions with Applications

Summary The probability that a non-central χ2, F or t variable exceeds c is a function of c, the degrees of freedom and the non-centrality parameter. The behaviour of these probabilities as functions of the degrees of freedom is investigated when c and the non-centrality parameters are fixed. It is also shown that, when c refers to the 100α% point of the corresponding central distribution, the probabilities are monotone in the degrees of freedom. Some allied results and possible applications are discussed.