A necessary and sufficient condition for justifying non-parametric likelihood with censored data

The non-parametric likelihood L(F) for censored data, including univariate or multivariate right-censored, doubly-censored, interval-censored, or masked competing risks data, is proposed by Peto (Appl Stat 22:86–91, <CitationRef CitationID="CR14">1973</CitationRef>). It does not involve censoring distributions. In the literature, several noninformative conditions are proposed to justify L(F) so that the GMLE can be consistent (see, for examples, Self and Grossman in Biometrics 42:521–530 <CitationRef CitationID="CR15">1986</CitationRef>, or Oller et al. in Can J Stat 32:315–326, <CitationRef CitationID="CR13">2004</CitationRef>). We present the necessary and sufficient (N&S) condition so that <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$L(F)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> is equivalent to the full likelihood under the non-parametric set-up. The statement is false under the parametric set-up. Our condition is slightly different from the noninformative conditions in the literature. We present two applications to our cancer research data that satisfy the N&S condition but has dependent censoring. Copyright Springer-Verlag Berlin Heidelberg 2014