Unweighted Estimation Research Articles

A bias in estimating urban population density functions

This paper demonstrates that because of the rules used to delineate census tracts, unweighted estimation of an urban population density function using census tract observations leads to a severe upward bias in the estimated function. A weighted estimation procedure which leads to an unbiased estimate is proposed. The paper also points out that if one computes the integral of an unbiased estimate of a density function over the area of a city, that integral is not necessarily an unbiased estimate of total population. The paper thus explains the “disturbing” empirical results concerning density functions reported by McDonald and Bowman.

A two-stage model for growth curves which leads to Rao's covariance adjusted estimators

SUMMARY It is well known that the unweighted estimator for growth curve parameters may be justified by a random effects model. We show that the more complicated adjusted estimators suggested by Rao may also be justified in terms of a two-stage model whose interprQtation is simple. In studying growth it is common to make several observations over a period of time on each of the individuals in the study. If an average growth curve is then fitted to such data it is necessary to allow for the dependence between observations on the same individual. In the simplest cases this leads to a regression model with covariance: with the usual notation we have, independently,

Estimation of parameters in double uptake systems

A method is presented for estimating the kinetic parameters describing the cellular uptake of a single molecular species mediated by two independent transport processes. Two cases are considered: first, uptake by two carrier-mediated transport systems, and second, uptake by a single carrier-mediated system and diffusion. Unweighted and weighted least-squares estimations are described; the unweighted estimation is noniterative and provides excellent first approximations for the iterative weighted estimation procedure.

The Estimation of Blood Platelet Survival

SummaryThe characteristics of experimental error in measurement of platelet radioactivity have been explored by blind replicate determinations on specimens taken on several days on each of three Walker hounds.Analysis suggests that it is not unreasonable to suppose that error for each sample is normally distributed ; and while there is evidence that the variance is heterogeneous, no systematic relationship has been discovered between the mean and the standard deviation of the determinations on individual samples. Thus, since it would be impracticable for investigators to do replicate determinations as a routine, no improvement over simple unweighted least squares estimation on untransformed data suggests itself.

Least Squares Estimation of Constants in a Linear Recession Model

Least squares can be used for estimating constants in a linear recession model from published average daily streamflows. A model with two recession constants was derived and successfully tested on a number of Kentucky streams. It can be used to estimate recession constants from daily streamflows stored on magnetic tape or punched cards without resorting to time‐consuming graphical techniques. Equations and procedures are provided for recessions represented by either one or two recession constants using both a weighted and unweighted estimation procedure. In applying the two techniques, empirical evidence indicates the simpler unweighted procedure is preferable.

“Least-squares” Estimation of Distribution Mixtures

THE statistical estimation of the unknown fractions Π1 (i = 1, … k), and, when unknown, further parameters θj (j = 1, … m), in a mixed population which is denoted by its cumulative distribution (taken to be univariate) is, in general, cumbersome. It therefore seems worth noting the convenience and comparative efficiency, at least in relation to the estimation of Πi, of estimators of “least-squares” type, defined by the minimization of the integral where Fs(x) denotes the cumulative distribution of the empirical sample (from, for example, n independent observations), and G(x) is a suitable increasing function of x. Taking for simplicity the case m = 0, the estimators of the Πi are automatically unbiased with exact variances readily calculable, and with asymptotic normality. For example, if k= 2, we have an estimate p1 for Π1, for example, where H=(dF1−dF2)/dG, with the expected value E(p1)=Π1, and variance A considerable choice of H is possible. Thus, considering for definiteness the density case, we define the unweighted least-squares estimator of Π1 by the choice dG = dx, H=f1−f2. We should, however, expect weighted estimators to be better. In fact, because the variance of dFs is fdx/n, we should try to choose dG=fdx; the term ∫HdF in the expression for σ2(p1), arising from the covariance of dFs(x) and dFs(y), then vanishes, and σ2(p1) becomes identical with the reciprocal of the information function I(Π1). If we take dG =Π0dF1+(1−Π0)dF2, then this weighting will be most efficient when Π0 is close to Π1. Thus if we suspect Π1 to be near 1, ½ or 0, suitable choices for Π0 would be 1, ½ or 0, respectively. An alternative to ½(dF1+dF2) is the geometric mean of dF1 and dF2, and another is max(dF1, dF2). The geometric mean has some convenience over the arithmetic mean in theoretical investigations of efficiency (for example, for f1 and f2 normal), but the latter (or alternatively max(dF1, dF2)) has the advantage of approximating to the maximum likelihood estimator, for any value of Π1, in both the extreme cases of f1 and f2 well separated and of f1→f2. It has been shown by Hill1 that the information function I(Π1) for Π1 may be written n[1−S(Π1)]/(Π1Π2), where If f1 and f2 differ by some parameter μ (for example, the mean), and Δμ=μ2−μ1, then also as f1→f2 where I(μ) is the information function for μ. Thus for the maximum likelihood estimator Π1 the variance, which is Π1Π2/n for f1 and f2 well separated, is 1/[n(Δμ)2I(μ)] as f1→f2.