It is an honour and a pleasure to be able to offer this commentary on some of D. In the early s, we thought that we might write a book together on sufficiency.
We had many discussions and exchanged a fair amount of material. In particular, we prepared a bibliography on sufficiency which was reasonably comprehensive at the time Basu and Speed But the planned book never came to fruition. I have not worked on this topic since the late s. So the experience of writing this commentary has been a pleasant walk down the memory lane. However, it also means that I may be unaware of some relevant later developments.
Accordingly, I begin with an apology in advance to the readers for any such oversight or errors. The paper focusses on analyzing data obtained from an experiment, making it a direct companion piece to Basu and Basu which consider very similar issues in analyzing data obtained from sample surveys. It is a classic Basu paper highlighting the hallmarks of his style: it is provocative and challenging, based on simple examples pushed to extremes, and illustrated in an entertaining way by a conversation between three people.
Underlying all this of course is deep thinking on serious issues. The full story is included in the box for easy reference. The point of the story is summarised in Figure 1 which shows the log-sampling distributions i. We plot the log-sampling distributions to improve the visual impact. An example was given by Neyman and Scott  to show that there are situations where the method of maximum likelihood leads to inconsistent estimators.
Selected Works of Debabrata Basu
In their example considered, the observations were supposed to be drawn from an infinite sequence of distinct populations involving an infinite sequence of nuisance parameters. Certain well-known results of distribution theory follow immediately from the above considerations. For instance, if. It is also deduced that if. Partly of an expository nature this note brings out the fact that an estimator, though asymptotically much less efficient in the classical sense than another, may yet have much greater probability concentration as defined in this article than the latter.
In an earlier paper the author stated that any statistic independent of a sufficient statistic must have the same distribution for all values of the unknown parameter. An example is given here to show that the proposition is not true in the generality stated above. Conditions under which the proposition is true are discussed.
Certain aspects of sampling with or without replacement, with equal or unequal probabilities, are considered here in some details. Some comparisons have been made between the with and without replacement sampling schemes. When we are sampling with replacement the estimate should not depend on the number of times that any particular unit may appear in the sample. Thus, certain estimation procedures in current use are shown to be inefficient.
As a Complement to Sufficient Statistics
Though the marginal distributions of the ancillary statistics are independent of the parameter they are not useless or informationless. A set of ancillaries may sometimes summarise the whole of the information contained in the sample. A classification of the ancillaries in terms of the partial order of their information content is attempted here. In general there are many maximal ancillaries, Among the minimal ancillaries there exists a unique largest one.
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When there exists a complete sufficient statistic, the problem of tracking down the maximal and minimal ancillaries becomes greatly simplified. The main upsurge of late Professor R. It was during this period that Fisher came out with the brilliant and now famous notions of a likelihood, b fiducial probability, c information and intrinsic accuracy, d sufficiency and e ancillary statistics and recovery of information — concepts around which the superstructure of the theory is built. In statistical theory one comes across various families of statistics subfields.
The author proves here the existence of such elements in a number of cases and leaves the question unsolved in a number of other cases. A number of problems of an allied nature are also discussed. In this paper we discuss a number of problems which have their origin in statistics but whose main interest is measure-theoretical.
It is to the statistician interested in abstract harmonic analysis and to the harmonic analyst interested in statistics that the paper is addressed. The corresponding minimal sufficient statistic is derived. We examine the role of the sufficiency and likelihood principles in the analysis of survey data and arrive at the revolutionary but reasonable conclusion that, once the sample has been drawn, the inference should not depend in any way on the sampling design.
This poses the problem of designing a survey which will yield a good representative sample.
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The randomisation principle is examined from this view point and it is noticed that there is very little, if any, use for it in survey designs. It is shown that in many situations the principle of invariance is strong enough to lead us to the standard reductions. View access options below. You previously purchased this article through ReadCube. Institutional Login. Log in to Wiley Online Library.
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http://ilaquhebow.ml Email or Customer ID. Several appropriate goodness-offit tests are described and evaluated by simulation studies. Markov chain proportional hazard regression model provides a powerful tool for analysis of multiple event times. We discuss estimation in absorbing Markov chains with missing covariates. We consider a MAR model assuming that the missing data mechanism depends on the observed covariates, as well as the number of events observed in a given time period, their types and times of their occurrence. For estimation purposes we use a piecewise constant intensity regression model.
The paper discusses the contributions Student W. Gosset made to the three stages in which small-sample methodology was established in the period i the distributions of the test- statistics under the assumption of normality, ii the robustness of these distributions against nonnormality, iii the optimal choice of test statistics. The conclusions are based on a careful reading of the correspondence of Gosset with Fisher and E. Tree-structured methods for exploratory data analysis have previously been extended to right-censored survival data. We further extend these methods to allow for truncation and time-dependent covariates.
We apply the new methods to a data set on incubation times of acquired immunodeficiency syndrome AIDS , using calendar time as a time-dependent covariate. Contrary to expectation, we find that rates of progression to AIDS appear to be faster after August than before. Start a new search.
Author see all Lehmann, E. Publication see all Selected Works of E.