Showing papers on "Outlier published in 1971"

Estimation of Expected Life in the Presence of an Outlier Observation

Abstract: Consider a life testing experiment consisting of n items, (n – 1) of which have the same expected life while one of them could have a much longer expected life. The standard estimators based on a homogeneous model if used in such a case, would tend to overestimate and result in a large mean squared error (MSE). The present paper considers (n – 1) items having p.d.f. given by l/σ exp (–x/σ) while the remaining one has p.d.f. α/σ exp (–xα/σ), 0 < < 1. We consider estimators of c by linear combinations of the first few order statistics. An optimal estimator is suggested and its MSE is compared with that of the standard estimators. The cases of n = 3 and 4 are explicitly evaluated.

Outlier proneness of phenomena and of related distributions

Abstract: Publisher Summary This chapter discusses the outliner proneness of the phenomena and of related distributions. It discusses the distinction between cases where the tendency to suspect and to eliminate outlier observations may be justifiable and those in which it is not. The chapter discusses three concepts: (1) the concept of a k-outlier, (2) the concept of families of distributions that are outlier-resistant, and (3) the concept of families that are outlier-prone. If a substantial previous experience in a particular domain of study appears sufficient for the statistician to act on the assumption that the observable variables follow an outlier-resistant distribution, then the efforts to seek out and, possibly, to eliminate the outliers are justified. In statistical practice, when applying a test, it is important to see that the theory underlying the test is not in conflict with the phenomenon studied. The customary requirements on the test of a hypothesis H are two: (i) if H is true, the use of the test should ensure the maintenance of the desired level of significance and (ii) if H is false and some contemplated alternative hypothesis H1 is true, the power with regard to H1 should be high.

Significance tests based on residuals

Abstract: SUMMARY The known distribution of residuals from linear regression models may be used to construct exact tests of significance. New tests for the presence of one or more outliers are considered in detail. Applications of the theory to other tests are discussed. Exact results are worked out for the normal and exponential error distributions; formulae are given for other nonnormal cases. All statistical tests are based on some model specifying the form or structure of the response. Linear models are a large and important class of the models currently used. The statistical tests based on linear models fall generally into two categories: (i) tests within the model that are sensitive to departures from some hypothesis about the parameters of the model; and (ii) tests of the model that are sensitive to departures from the assumptions of the model regardless of the parameters within the model. Tests of the latter type are based on the normalized residual vector, or some function of it, which has a known marginal distribution independent of the parameters of the model. Tests within the model are made conditionally given this ancillary residual vector. However, the distinction between these tests, at least for normal models, exists more in theory thain in practice. Section 2 contains some preliminary definitions and results which lead, in ? 3, to the distribution of residuals in both normal and nonnormal cases. In ? 4 a class of significance tests based on the structure of the regression problem is proposed and in ? 5 it is shown that the common analysis of variance tests for normal models belong to this class. In ? 6 this theory is applied to the problem of testing for the presence of one or more outliers. Examples of the derived tests are given for normal and exponential cases. Finally, in ? 7 the relation to other tests for nonadditivity and nonnormality is discussed.

Tests for One or Two Outliers in Normal Samples with Unknown Variance

Abstract: The performances of three procedures for treatment of outliers in normal samples are evaluated. The first procedure is the sequential application of the usual maximum residual test. The largest observation is declared an outlier if the largest studentized residual exceeds a predetermined value. If one outlier is detected, the test is repeated on t.he remaining observations, the process continuing until no further outliers are detected. In the second procedure the two largest observations are declared outliers if the sum of the two largest studentized residuals exceeds a predetermined value. In the third procedure the two largest observations are considered outliers if the ratio of the corrected sum of squares omitting these values to the total corrected sum of squaresis less than a critical ratio. The performances of these procedures are evaluated for samples in which two of the observations have means different from the common mean of the remainder of the sample.

Detection of outliers

Abstract: Outlier detection can be regarded as a decision process within several of the theoretical frameworks of statistical inference. A semi-Bayesian approach is described which uses as inputs to the decision process separate Bayesian analyses for each contemplated number k of outliers. Significance tests can be used for assessing k but should be supplemented by Bayesian analysis for judging which observations are outliers. In §3, an asymptotic theorem is given to clarify the difference between the Bayesian and significance testing messages. Solutions to computational problems are outlined in §4.

Regime Testimation in a Time Sequence

Abstract: Two years of irregularly spaced measurements of seawater temperature at 200 feet at a fixed North Pacific Ocean location are considered. The data are heteroscedastic in time and the average path jumps about erratically. We seek points of significant change in the data-generating process and wish to study their nature. The set of all properties characterizing a data-generating process is its regime. Simultaneous testing and estimation is testimation. We must testimate abrupt changes in the regime as a whole instead of property by property. A moving mean square of error values is posed as a statistic and its relation to x 2 and F derived. Methods of testimation and decomposition of the statistic into proportions due to each possible cause are obtained. The pattern of behavior of the statistic which identifies an outlier is considered. It is possible for two properties in the regime to change jointly such that one change obscures the other. Methods to detect and solve this problem are derived. The temperatur...