SUMMARY Graphical methods of assessing the validity of a probability model, and of estimating location and scale parameters, have been widely applied over many years. A variety of different prescriptions have been advanced for the "plotting positions": that is, for the set of values at which the ordered observations in a sample should be plotted. The more formal proposals vary with the form of the probability model and reflect the function of the enquiry: whether it is predominantly one of model validation or parameter estimation. A closer tie-up of the estimation problem with known properties of order statistics estimators produces an economic recovery of earlier results on probability plotting estimators, some new results in this area, and some general observations on biased order statistics estimators with minimum mean square error.

Probability Plotting Methods and Order Statistics

A bibliography on life testing and related topics

Certain elementary properties of the theory of least squares are presented from the point of view of complex stochastic processes. The development parallels the real case. We consider the least squares estimate (LSE), the best linear unbiased estimate (BLUE), the Markov estimate (ME), and the relationships among them. In particular, we prove the Gauss-Markov theorem and give necessary and sufficient conditions that the LSE and ME be identical. The efficiency of the LSE is defined and a lower bound is obtained for the efficiency of a certain class of models. Estimation of the mean vector and variance for a complex normal population leads to the maximum likelihood estimates (MLE). We prove that the MLE of the mean vector is identical with the LSE, and deduce other analogous properties concerning the distribution of the MLE.

Complex Linear Least Squares

In this paper we present two methods of estimating a linear regression equation with Cauchy disturbances. The first method uses the maximum likelihood principle and therefore the estimators obtained are consistent. The asymptotic covariance is derived which provides with the necessary statistics for the purpose of making inference in large samples. The second method is the method of least lines which minimizes the sum of absolute errors (MSAE) from the fitted regression. Then these two methods are compared through a Monte Carlo study. The maximum likelihood method emerges superior over the MSAE method. However, the MSAE procedure which does not depend on the distribution of the error term appears to be a close competitor to the maximum likelihood estimator.

Estimation of regression equation with cauchy disturbances

Some upper bounds are derived for the variances of least squares estimators based on a subset of the ordered observations in a random sample (which may, for simplicity, be called best estimates or ordered least squares estimates) of (i) location, (ii) scale and (iii) both location and scale parameters of a distribution. The estimates achieving these upper bounds could, in some instances, be conservatively used in place of ordered least squares estimates. These results constitute generalizations of those due to Lloyd [6] and Downton [2], [3]. We also obtain Cramer-Rao type of lower bounds for the variance of a linear combination of censored observations (called a systematic statistic) which are valid even in the nonregular case. Finally, there is a discussion about an optimum choice of the function which can be used for generating lower bounds for variances of any estimators.

Certain General Properties of Unbiased Estimates of Location and Scale Parameters Based on Ordered Observations

A note on ordered least-squares estimation

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A note on ordered least-squares estimation