Dale Umbach

A few methods for fitting circles to data

Five methods are discussed to fit circles to data. Two of the methods are shown to be highly sensitive to measurement error. The other three are shown to be quite stable in this regard. Of the stable methods, two have the advantage of having closed form solutions. A positive aspect of all of these models is that they are coordinate free in the sense that the same estimating circles are produced no matter where the axes of the coordinate system are located nor how they are oriented. A natural extension to fitting spheres to points in 3-space is also given.

On inference for a mixture of a poisson and a degenerate distribution

Let be a random sample from Maximum likelihood estimates are calculated. UMPU tests for λ are shown to exist. Hypotheses testing about p is also discussed.

ESTIMATING THE QUANTILE FUNCTION OF A LOCATION-SCALE FAMILY OF DISTRIBUTIONS BASED ON FEW SELECTED ORDER STATISTICS

Some general asymptotic methods of estimating the quantile function, Q(ξ), 0<ξ<1, of location-scale families of distributions based on a few selected order statistics are considered, with applications to some nonregular distributions. Specific results are discussed for the ABLUE of Q(ξ) for the location-scale exponential and double exponential distributions. As a further application of the exponential results, we discuss a nonlinear estimator of Q(ξ) for the scale-shape Pareto distribution.

7 Optimal linear inference using selected order statistics in location-scale models

Publisher Summary This chapter discusses the optimal linear inference using selected order statistics in location-scale models. Mosteller advocated the estimation of location/scale parameters using a few optimally selected order statistics, particularly for large sample size. These procedures were developed as a compromise between the lack of efficiency and quickness and the ease of computation. In general, it has been observed that for most distributions efficiencies of 90% or more are achieved with seven or even fewer optimally chosen observations. These estimates are based on linear combinations of the selected order statistics, which are best linear unbiased estimates (BLUEs). Lloyd introduced BLUEs to construct linear estimates. As the coefficients of these linear combinations are functions of means and covariances of order statistics, the estimates can be numerically computed for small sample size. Ogawa considered the problem of estimating location/scale parameters for large samples and introduced the asymptotically best linear unbiased estimates (BLUEs). Sarhan and Greenberg gave a comprehensive account of the estimation problem using a few selected order statistics, which was addressed up until that point in time. Mostellers paper, along with Ogawas paper and Sarhan and Greenbergs book, laid the foundation for this area of research.

Large sample estimation of Pareto quantiles using selected order statistics

SummaryNonlinear estimates of the population quantile,49-1, of the shape-scale family of Pareto distributions are considered based on a few selected order statistics. Asymptotic relative efficiencies (A.R.E′.s) of the estimators are given relative to complete sample estimators and the usual nonparametric estimator of quantiles.

Estimating quantiles using optimally selected order statistics

This expository paper deals with the linear estimation of quantiles of location-scale families of distributions using a few selected order statistics.The general theory for the problem i s reviewed for the exact as well as the asymptotic cases.

ESTIMATING LIFE FUNCTIONS OF CHI DISTRIBUTION USING SELECTED ORDER STATISTICS

Abstract This paper deals with the large sample estimation of functions such as the quantile function, survival function, and the hazard function of the chi distribution using a few optimally selected order statistics. These functions arise in the study of life models and are functions of the location and scale parameters. The optimum ranks of the order statistics are obtained by maximizing the asymptotic relative efficiencies.

Tests of significance using selected sample quantiles

Large sample tests of significance for the location parameter, the scale parameter, and quantiles for a location-scale family of distributions based on a few optimally chosen sample quantiles are considered.

Estimating functions of location and scale of t-distribtion

The problem of optimally selecting a few, say k, order statistics from a sample of size n from a location-Scale family of t distributions for estimating sufficiently smooth functions, say g(λ, δ) of the location and scale parameters is considered. First, the asymptotically best linear estimators of λ and δ say, , are obtained for a fixed spacing of the order statistics. These are then used to estimate g(λ, δ) With . A lower bound for the efficiency of this estimator is introduced, which is independent of g. This lower bound is maximized to obtain the conservative spacings. The results of the maximization are presented in tables for k = 2, 3, 4, and 5, with degrees of freedom 1, 2, 3, 5, 10, 20, and ∞.

Conservative Spacings for the Estimation of Functions of Location and Scale

Abstract The problem of optimally selecting a few, say k, order statistics from a sample of size n from a location-scale family of distributions for estimating sufficiently smooth functions, say g(λ, δ) of the location and scale parameters is considered. First, the asymptotically best linear estimators of λ and δ, say and , are obtained for a fixed spacing of the order statistics. These are then used to estimate g(λ, δ) with . A lower bound for the efficiency of this estimator is introduced, which is independent of g. The lower bound is maximized to obtain the conservative spacings. The results of the maximization are presented in tables for the gamma distribution.