A note on time-ordered classification

摘要

Department of Mathematics, University of Kansas, Lawrence, Kansas 66045-7523, U.S.A.nhhe@math.ku.edunLemma 2 in §4 of Hinkley (1972) shows that ξn† − ξn∗ = Op(1) under the condition:nsupnu0003θ1nξnu0002∗+nni=ξ ∗+1n{log f1(Xi ; θ1) − log f2(Xi ; θn∗n2 )} → −∞ (1)nwith probability 1 as n→∞, which was described to be from the ‘consistency assumptions’. Herenθ1, θ2, f1, f2, log f1(Xi ; θ1) and log f2(Xi ; θn∗n2 ) are used to replace respectively θ,ψ, f, g, l fi (θ) andnlgi (ψn∗) in Hinkley (1972). Condition (1) fails whenever the distribution corresponding to the densitynf2(·; θn∗n2 ) is in the closure of the set of distributions corresponding to densities of the form f1(·; θ1), inna certain sense. One such case occurs if f1, f2 have the same parametric form with parameters θn∗n1 , θn∗n2 ,nrespectively, satisfying θn∗n1nu0006= θn∗n2 . Another such case occurs when the distribution with density f2(·; θn∗n2 ) cannbe viewed as a limit of the distributions with densities f1(·; θ1). For instance, an exponential distributionncan be viewed as a limit of a Weibull distribution. Thus, the ‘consistency conditions’ in Hinkley (1972)nmay not be satisfied by otherwise well-behaved models. Here we give consistency assumptions to remedynthis.nLet us define v(θ j ; θn∗ni ) ≡nu0003 +∞n−∞ {log f j (x; θ j ) − log fi (x; θn∗ni )} fi (x; θn∗ni ) dx for i, j = 1, 2.nAssumption 1. We have f1(x; θn∗n1 ) u0006= f2(x; θn∗n2 ) on a set of nonzero measure.nAssumption 2. We have θ j and θn∗nj are in u0003θ j , where u0003θ j is compact ( j = 1, 2).nAssumption 3.n1. For any j, s, t satisfying j = 1, 2, 0u0002s < t u0002T , there exist r < 2,C > 0 such thatnEn⎧⎨n⎩ maxnθ j∈u0003θ jnu0007 u0002tni=s+1n[log f j (Xi ; θ j ) − E{log f j (Xi ; θ j )}]nb2n⎫⎬n⎭ u0002C(t − s)r .n2. For any j, s1, s2, t1, t2 satisfying j = 1, 2, 0u0002s1 < t1 u0002ξn∗, ξn∗ u0002s2 < t2 u0002T , there exist r u00021, D > 0nsuch thatnEn⎧⎪⎨n⎪⎩nmaxnθ j∈u0003θ jn⎛n⎝nu0002t jni=s j+1n|v(θ j ; θn∗nj )|−12n[{log f j (Xi ; θ j ) − log f j (Xi ; θn∗nj )} − v(θ j ; θn∗nj )]n⎞n⎠n2n⎫⎪⎬n⎪⎭nu0002 D(t − s)r .nIt is easy to verify that an exponential family satisfies the relatively weak Assumption 3.nUnder Assumptions 1–3, it can be proved that ξn† − ξn∗ = Op(1). The author may be contacted for anproof.nREFERENCEnHINKLEY, D. V. (1972). Time-ordered classification. Biometrika 59, 509–23.n[Received November 2007. Revised July 2008