This paper provides the self-consistent estimator (SCE) and the nonparametric maximum likelihood estimator (NPMLE) for ''middle-censored'' data, in which a data value becomes unobservable if it falls within a random interval. We provide an algorithm to find the SCE and show that the NPMLE satisfies the self-consistency equation. We find a sufficient condition for the SCE to be concentrated on the uncensored observations. In addition, we find sufficient conditions for the consistency of the SCE and prove that consistency holds for the special case when one of the ends is a constant. Some simulation results and an illustrative example, using Danish melanoma data set, are provided.

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In this article we discuss how such a censoring scheme applies to circular data and in particular when the original data is assumed to come from a parametric model such as the von Mises. Maximum likelihood estimation of the parameters as well as their largesample properties are considered under this censoring scheme. We also consider nonparametric estimation of the circular probability distribution under such a general censoring scheme and use Monte Carlo methods to investigate its effects on the estimation of the mean direction and concentration.

A nonparametric test labelled ‘Rao Spacing-frequencies test’ is explored and developed for testing whether two circular samples come from the same population. Its exact distribution and performance relative to comparable tests such as the Wheeler-Watson test and the Dixon test in small samples, are discussed. Although this test statistic is shown to be asymptotically normal, as one would expect, this large sample distribution does not provide satisfactory approximations for small to moderate samples. Exact critical values for small samples are obtained and tables provided here, using combinatorial techniques, and asymptotic critical regions are assessed against these. For moderate sample sizes in-between i.e. when the samples are too large making combinatorial techniques computationally prohibitive but yet asymptotic regions do not provide a good approximation, we provide a simple Monte Carlo procedure that gives very accurate critical values. As is well-known, the large number of usual rank-based tests are not applicable in the context of circular data since the values of such ranks depend on the arbitrary choice of origin and the sense of rotation used (clockwise or anti-clockwise). Tests that are invariant under the group of rotations, depend on the data through the so-called ‘spacing frequencies’, the frequencies of one sample that fall in between the spacings (or gaps) made by the other. The Wheeler-Watson, Dixon, and the proposed Rao tests are of this form and are explicitly useful for circular data, but they also have the added advantage of being valid and useful for comparing any two samples on the real line. Our study and simulations establish the ‘Rao spacing-frequencies test’ as a desirable, and indeed preferable test in a wide variety of contexts for comparing two circular samples, and as a viable competitor even for data on the real line. Computational help for implementing any of these tests, is made available online “” R package and is part of this paper.

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