Printed in Great Britain A note on time-ordered classification

θ1 ξ∗+n∑ i=ξ∗+1 {log f1(Xi; θ1) − log f2(Xi; θ∗2)} → − ∞ (1) with probability 1 as n → ∞, which was described to be from the ‘consistency assumptions’. Here θ1, θ2, f1, f2, log f1(Xi; θ1) and log f2(Xi; θ∗2) are used to replace respectively θ, ψ, f, g, l fi (θ) and lgi (ψ∗) in Hinkley (1972). Condition (1) fails whenever the distribution corresponding to the density f2(·; θ∗2) is in the closure of the set of distributions corresponding to densities of the form f1(·; θ1), in a certain sense. One such case occurs if f1, f2 have the same parametric form with parameters θ∗1, θ 2, respectively, satisfying θ∗1 = θ∗2. Another such case occurs when the distribution with density f2(·; θ∗2) can be viewed as a limit of the distributions with densities f1(·; θ1). For instance, an exponential distribution can be viewed as a limit of a Weibull distribution. Thus, the ‘consistency conditions ’ in Hinkley (1972) may not be satisfied by otherwise well-behaved models. Here we give consistency assumptions to remedy this. Let us define v(θ j; θ∗i)