An extension of the validity of »Student»-Fisher's law of distribution

Abstract

One of the fundamental laws of the distribution of errors within modern theoretical statistics is the »Student»-Fisher law of distribution, which answers the question:     

Survey Estimates by Calibration on Complex Auxiliary Information

In the last decade, calibration estimation has developed into an important field of research in survey sampling. Calibration is now an important methodological instrument in the production of statistics. Several national statistical agencies have developed software designed to compute calibrated weights based on auxiliary information available in population registers and other sources. This paper reviews some recent progress and offers some new perspectives. Calibration estimation can be used to advantage in a range of different survey conditions. This paper examines several situations, including estimation for domains in one‐phase sampling, estimation for two‐phase sampling, and estimation for two‐stage sampling with integrated weighting. Typical of those situations is complex auxiliary information, a term that we use for information made up of several components. An example occurs when a two‐stage sample survey has information both for units and for clusters of units, or when estimation for domains relies on information from different parts of the population. Complex auxiliary information opens up more than one way of computing the final calibrated weights to be used in estimation. They may be computed in a single step or in two or more successive steps. Depending on the approach, the resulting estimates do differ to some degree. All significant parts of the total information should be reflected in the final weights. The effectiveness of the complex information is mirrored by the variance of the resulting calibration estimator. Its exact variance is not presentable in simple form. Close approximation is possible via the corresponding linearized statistic. We define and use automated linearization as a shortcut in finding the linearized statistic. Its variance is easy to state, to interpret and to estimate. The variance components are expressed in terms of residuals, similar to those of standard regression theory. Visual inspection of the residuals reveals how the different components of the complex auxiliary information interact and work together toward reducing the variance.     

On tests of independence and evaluation of the magnitude of the dependence between two categorized data

Abstract

A unit is characterized by two variables x and y, which need not be quantitatively defined but which make possible a grouping of the units into classes by increasing values. According to the value of the first property, the population can be divided into k classes with dividing points x ₁, x ₂, ... x _κ?1, and according to the value of the second property the population can be divided into l classes with dividing points Y ₁, Y ₂, ... Y _l?1. (Furthermore, we introduce the class limits x ₀ = Y ₀ = ? ∞, xκ = Yl = + ∞.)     

Changes in fertility following birth restriction

During the years following 1880 the population of Sweden has exhibited an almost uninterrupted downward trend in fertility. Even if the decline in the illegitimate fertility is considerable, that in the legitimate is of much greater importance. Whereas according to available records there was a roughly constant fertility up to about 1880, there has been a steady fall in fertility since that year. During the three decades 1880–1910 the legitimate fertility underwent a moderate fall, though not more than that the annual number of births kept fairly constant between 130 000 and 140 000. Since the year 1910, however, the decline has been so great that even the absolute number of births has fallen, this now being about 90 000 annually.     

Calculation of Death-Rates with Regard to Migrations

Abstract

As is known, the death-rates of a population are usually worked out by the aid of the formula where D_x denotes the number of deaths at the age x - x + 1 year and where M_x signifies the average population in the same age group or the total observed risktime (expressed in years).     

Trade policy and ecology

Trade sanctions on product exports are often used as measures for conservation of stocks of living resources. Two opposing approaches are investigated. The harvest approach argues that sanctions reduce the harvest, and thus protects the stock. It is shown that this does not consider the long run effects nor the effects of sanctions on the management system. The investment approach argues that increased price protects the stock, making the species a profitable investment. It is shown that this approach does not consider the asset effects of price changes, and that the sanctions usually increase the stock in an one species analysis. If the wildlife competes for land the conclusions may be different, but still sanctions usually works. If the manager has a joint management of several species, the stock effects of sanctions are ambiguous, depending on both the species interaction, and the profitability of the harvesting from each of them. In this case it is not possible to use intuitive reasoning, sanctions give distortions to all stocks simultaneously. The threat of extinction depends crucially on the unit cost in harvesting of depleted stocks. The paper concludes that trade policy is a too general measure for the management of living resources, and may implicate important economic distortions to the ecological system.This study is partially funded by the Research Council of Norway (Environment and Development). I thank Derek Clark, Tore Thonstad, Frode Steen and two anonymous referees for helpful comments.