The accuracy of population projections

A review of past population projection errors is presented as a means for constructing confidence intervals for future projections. The author first defines a statistic to measure projection errors independent of the size of population and the length of the projection period. A sample of U.S. and U.N. projections is used to show that the distributions of components of the error statistic are relatively stable. This information is then used to construct confidence intervals for the U.S. population up to the year 2000.     

An evaluation of population projection errors for census tracts

"In this article we evaluate the accuracy and bias of projections of total population and population by age group for census tracts in three counties in Florida. We use [U.S. census] data from 1970 and 1980 and several simple extrapolation techniques to produce projections for 1990; we then compare these projections with 1990 census counts and evaluate the differences. For the total sample, we find mean absolute errors of 17%-20% for the three most accurate techniques for projecting total population and find no indication of overall bias. For individual age groups, mean absolute errors range from 20%-29%." This is a revised version of a paper presented at the 1993 Annual Meeting of the Population Association of America.     

Improving the accuracy of intercensal estimates and postcensal projections of the civilian noninstitutional population: a parameterization of institutional prevalence rates

The authors first note that current official U.S. population estimates and projections are based on the assumption that certain characteristics of the institutionalized population remain constant between censuses. The article "examines the empirical validity of this assumption by using data from the decennial censuses for 1940-1980 and, in light of substantial decade to decade changes in the age patterns of the institutional proportions for sex- and race-specific populations, seeks to develop alternative methods." As part of these alternative methods, "parametric curves are fit to the age-specific institutional proportions for each population for each decade. A study of the observed historical variation in the parameters of these curves then leads to some suggestions about how their shapes can be estimated between censuses and projected beyond the latest available census to provide more accurate estimates and projections of the civilian noninstitutional population." This is a revised version of a paper originally presented at the 1984 Annual Meeting of the Population Association of America (see Population Index, Vol. 50, No. 3, Fall 1984, p. 439).     

Multivariate time series projections of parameterized age-specific fertility rates

1 solution to the dimensionality problem raised by projection of individual age-specific fertility rates is the use of parametric curves to approximate the annual age-specific rates and a multivariate time series model to forecast the curve parameters. Such a method reduces the number of time series to be modeled for women 14-45 years of age from 32 to 40 (the number of curve parameters). In addition, the curves force even longterm fertility projections to exhibit the same smooth distribution across age as historical data. The data base used to illustrate this approach was age-specific fertility rates for US white women in 1921-84. An important advantage of this model is that it permits investigation of the interactions among the total fertility rate, the mean age of childbearing, and the standard deviation of age at childbearing. In the analysis of this particular data base, the contemporaneous relationship between the mean and standard deviation of age at childbearing was the only significant relationship. The addition of bias forecasts to the forecast gamma curve improves forecast accuracy, especially 1-2 years ahead. The most recent US Census Bureau projections have combined a time series model with longterm projections based on demographic judgment. These official projections yielded a slightly higher ultimate mean age and slightly lower standard deviation than those resulting from the model described in this paper.     

Statistical materials

Official population data for the USSR are presented for 1985 and 1986. Part 1 (pp. 65-72) contains data on capitals of union republics and cities with over one million inhabitants, including population estimates for 1986 and vital statistics for 1985. Part 2 (p. 72) presents population estimates by sex and union republic, 1986. Part 3 (pp. 73-6) presents data on population growth, including birth, death, and natural increase rates, 1984-1985; seasonal distribution of births and deaths; birth order; age-specific birth rates in urban and rural areas and by union republic; marriages; age at marriage; and divorces.     

Orthogonal projections and the geometry of estimating functions

In this paper, a notion of generalized inner product spaces is introduced to study optimal estimating functions. The basic technique involves an idea of orthogonal projection first introduced by Small and McLeish (1988, 1989, 1991, 1992, 1994). A characterization of orthogonal projections in generalized inner product spaces is given. It is shown that the orthogonal projection of the score function into a linear subspace of estimating functions is optimal in that subspace, and a characterization of optimal estimating functions is given. As special cases of the main results of this paper, we derive the results of Godambe (1985) on the foundation of estimation in stochastic processes, the result of Godambe and Thompson (1989) on the extension of quasi-likelihood, and the generalized estimating equations for multivariate data due to Liang and Zeger (1986). Also we have derived optimal estimating functions in the Bayesian framework.     

Optimal sample size for record data and associated cost analysis for exponential distribution

Estimation of the mean of an exponential distribution based on record data has been treated by Samaniego and Whitaker [F.J. Samaniego, and L.R. Whitaker, On estimating popular characteristics from record breaking observations I. Parametric results, Naval Res. Logist. Quart. 33 (1986), pp. 531–543] and Doostparast [M. Doostparast, A note on estimation based on record data, Metrika 69 (2009), pp. 69–80]. When a random sample Y ₁, …, Y _n is examined sequentially and successive minimum values are recorded, Samaniego and Whitaker [F.J. Samaniego, and L.R. Whitaker, On estimating popular characteristics from record breaking observations I. Parametric results, Naval Res. Logist. Quart. 33 (1986), pp. 531–543] obtained a maximum likelihood estimator of the mean of the population and showed its convergence in probability. We establish here its convergence in mean square error, which is stronger than the convergence in probability. Next, we discuss the optimal sample size for estimating the mean based on a criterion involving a cost function as well as the Fisher information based on records arising from a random sample. Finally, a comparison between complete data and record is carried out and some special cases are discussed in detail.     

Population projections for the USSR

Population projections for the Soviet Union are presented up to the year 2015. Separate figures are provided for rural and urban population by republic, and for the economically active population by rural and urban area and republic.     

Total error in PES estimates of population

"We describe a methodology for estimating the accuracy of dual systems estimates (DSE's) of population, census estimates of population, and estimates of undercount in the census. The DSE's are based on the census and a post-enumeration survey (PES). We apply the methodology to the 1988 dress rehearsal census of St. Louis and east-central Missouri and we discuss its applicability to the 1990 [U.S.] census and PES. The methodology is based on decompositions of the total (or net) error into components, such as sampling error, matching error, and other nonsampling errors. Limited information about the accuracy of certain components of error, notably failure of assumptions in the 'capture-recapture' model, but others as well, lead us to offer tentative estimates of the errors of the census, DSE, and undercount estimates for 1988. Improved estimates are anticipated for 1990." Comments are included by Eugene P. Ericksen and Joseph B. Kadane (pp. 855-7) and Kenneth W. Wachter and Terence P. Speed (pp. 858-61), as well as a rejoinder by Mulry and Spencer (pp. 861-3).     

A projection pursuit index for large <Emphasis Type="Italic">p</Emphasis> small <Emphasis Type="Italic">n</Emphasis> data

In high-dimensional data, one often seeks a few interesting low-dimensional projections which reveal important aspects of the data. Projection pursuit for classification finds projections that reveal differences between classes. Even though projection pursuit is used to bypass the curse of dimensionality, most indexes will not work well when there are a small number of observations relative to the number of variables, known as a large p (dimension) small n (sample size) problem. This paper discusses the relationship between the sample size and dimensionality on classification and proposes a new projection pursuit index that overcomes the problem of small sample size for exploratory classification.     

Overview of current population trends in the Federal Republic of Germany and prospects for the future

Current population trends in the Federal Republic of Germany are first reviewed. Topics discussed include demographic aging, ethnic composition, declining fertility, and households. Population projections to the year 2030 are then presented by age group using several different models. Finally, the economic and social policy consequences of long-term population trends are examined, with particular reference to the economically active population and the dependency burden.     

The relationship between the length of the base period and population forecast errors

"The base period of a population forecast is the time period from which historical data are collected for the purpose of forecasting future population values. The length of the base period is one of the fundamental decisions made in preparing population forecasts, yet very few studies have investigated the effects of this decision on population forecast errors. In this article the relationship between the length of the base period and population forecast errors is analyzed, using three simple forecasting techniques and data from 1900 to 1980 for states in the United States. It is found that increasing the length of the base period up to 10 years improves forecast accuracy, but that further increases generally have little additional effect. The only exception to this finding is long-range forecasts of rapidly growing states, in which a longer base period substantially improves forecast accuracy for two of the forecasting techniques."     

Estimation with the lognormal distribution on the basis of records

Record scheme is a method to reduce the total time on test of an experiment. In this scheme, items are sequentially observed and only values smaller than all previous ones are recorded. In some situations, when the experiments are time-consuming and sometimes the items are lost during the experiment, the record scheme dominates the usual random sample scheme [M. Doostparast and N. Balakrishnan, Optimal sample size for record data and associated cost analysis for exponential distribution, J. Statist. Comput. Simul. 80(12) (2010), pp. 1389–1401]. Estimation of the mean of an exponential distribution based on record data has been treated by Samaniego and Whitaker [On estimating population characteristics from record breaking observations I. Parametric results, Naval Res. Logist. Q. 33 (1986), pp. 531–543] and Doostparast [A note on estimation based on record data, Metrika 69 (2009), pp. 69–80]. The lognormal distribution is used in a wide range of applications when the multiplicative scale is appropriate and the log-transformation removes the skew and brings about symmetry of the data distribution [N.T. Longford, Inference with the lognormal distribution, J. Statist. Plann. Inference 139 (2009), pp. 2329–2340]. In this paper, point estimates as well as confidence intervals for the unknown parameters are obtained. This will also be addressed by the Bayesian point of view. To carry out the performance of the estimators obtained, a simulation study is conducted. For illustration proposes, a real data set, due to Lawless [Statistical Models and Methods for Lifetime Data, 2nd ed., John Wiley & Sons, New York, 2003], is analysed using the procedures obtained.     

Effects of future population trends on the labor market--analyses based on projections prepared by the Institute for Research on the Labor Market and Occupations

The effects of future population trends, such as demographic aging, declining fertility, and changes in migration, on the labor market in the Federal Republic of Germany are analyzed up to the year 2000. The study is based on projections prepared by the Institute for Research on the Labor Market and Occupations. Topics discussed include demographic trends as a cause of current unemployment, labor market phases and demographic trends since 1950, the projection model used, age-specific projections of the potential labor force, and labor market projections.     

Constructing optimal projection designs

ABSTRACT

The early stages of many real-life experiments involve a large number of factors among which only a few factors are active. Unfortunately, the optimal full-dimensional designs of those early stages may have bad low-dimensional projections and the experimenters do not know which factors turn out to be important before conducting the experiment. Therefore, designs with good projections are desirable for factor screening. In this regard, significant questions are arising such as whether the optimal full-dimensional designs have good projections onto low dimensions? How experimenters can measure the goodness of a full-dimensional design by focusing on all of its projections?, and are there linkages between the optimality of a full-dimensional design and the optimality of its projections? Through theoretical justifications, this paper tries to provide answers to these interesting questions by investigating the construction of optimal (average) projection designs for screening either nominal or quantitative factors. The main results show that: based on the aberration and orthogonality criteria the full-dimensional design is optimal if and only if it is optimal projection design; the full-dimensional design is optimal via the aberration and orthogonality if and only if it is uniform projection design; there is no guarantee that a uniform full-dimensional design is optimal projection design via any criterion; the projection design is optimal via the aberration, orthogonality and uniformity criteria if it is optimal via any criterion of them; and the saturated orthogonal designs have the same average projection performance.     

Parameterized multistate population dynamics and projections

"This article reports progress on the development of a population projection process that emphasizes model selection over demographic accounting. Transparent multiregional/multistate population projections that rely on parameterized model schedules are illustrated [using data primarily from a number of developed countries, particularly Sweden], together with simple techniques that extrapolate the recent trends exhibited by the parameters of such schedules." The author notes that "the parameterized schedules condense the amount of demographic information, expressing it in a language and variables that are more readily understood by the users of the projections. In addition, they permit a concise specification of the expected temporal patterns of variation among these variables, and they allow a disaggregated focus on demographic change that otherwise would not be feasible."     

Some criteria for projection pursuit

The idea of searching for orthogonal projections, from a multidimensional space into a linear subspace, as an aid to detecting non-linear structure has been named exploratory projection pursuit.Most approaches are tied to the idea of searching for interesting projections. Typically, an interesting projection is one where the distribution of the projected data differs from the normal distribution. In this paper we define two projection indices which are aimed specifically at finding projections that best show grouped structure in the plane, if this exists in the multi-dimensional space. These involve a numerical optimization problem which is tackled in two stages, the projection and the pursuit; the first is based on a procedure to generate pseudo-random rotation matrices in the sense of the grand tour by D. Asimov (1985), and the second is a local numerical optimization procedure. One artificial and one real example illustrate the performance of the suggested indices.     

World population

A review of world population growth over the past 60 years is presented, and projections are made to the year 2000. Data are provided on the total population and rate of growth in 1983 of 202 countries, together with some regional data and estimates of the size of some major urban agglomerations.     

Hits-and-misses for the evaluation and combination of forecasts

Error measures for the evaluation of forecasts are usually based on the size of the forecast errors. Common measures are, e.g. the mean squared error (MSE), the mean absolute deviation (MAD) or the mean absolute percentage error (MAPE). Alternative measures for the comparison of forecasts are turning points or hits-and-misses, where an indicator loss function is used to decide if a forecast is of high quality or not. Here, we discuss the latter to obtain reliable combined forecasts. We apply several combination techniques to a set of German macroeconomic data. Furthermore, we perform a small simulation study for the combination of two biased forecasts.     

Projection Measures And Estimation Variances Of Intensities

The paper investigates random processes of geometrical objects in Euclidean spaces. General properties of the measure of total projections are derived by means of Palm distribution. Explicit formulas for variances of the projection measure are obtained for Poisson point processes of compact sets.

Intensity estimators of fibre (surface) processes are then studied by means of projection measures. Classification of direct and indirect probes is introduced. The indirect sampling design of vertical sections and projections is generalized and its statistical properties derived.