Stochastic population forecasts based on conditional expert opinions

The paper develops and applies an expert-based stochastic population forecasting method, which can also be used to obtain a probabilistic version of scenario-based official forecasts. The full probability distribution of population forecasts is specified by starting from expert opinions on the future development of demographic components. Expert opinions are elicited as conditional on the realization of scenarios, in a two-step (or multiple-step) fashion. The method is applied to develop a stochastic forecast for the Italian population, starting from official scenarios from the Italian National Statistical Office.     

Stochastic population forecasts for the United States: beyond high,medium, and low

"This article presents and implements a new method for making stochastic population forecasts that provide consistent probability intervals. We blend mathematical demography and statistical time series methods to estimate stochastic models of fertility and mortality based on U.S. data back to 1900 and then use the theory of random-matrix products to forecast various demographic measures and their associated probability intervals to the year 2065. Our expected total population sizes agree quite closely with the Census medium projections, and our 95 percent probability intervals are close to the Census high and low scenarios. But Census intervals in 2065 for ages 65+ are nearly three times as broad as ours, and for 85+ are nearly twice as broad. In contrast, our intervals for the total dependency and youth dependency ratios are more than twice as broad as theirs, and our ratio for the elderly dependency ratio is 12 times as great as theirs. These items have major implications for policy, and these contrasting indications of uncertainty clearly show the limitations of the conventional scenario-based methods."     

Scenarios, Uncertainty and Conditional Forecasts of the World Population

Current official population forecasts differ little from those that Whelpton made 50 years ago either in the cohort–component methodology used or in the arguments used to motivate the assumptions. However, Whelpton produced some of the most erroneous forecasts of this century. This suggests that current forecasters should ensure that they give users an assessment of the uncertainty of their forecasts. We show how simple statistical methods can be combined with expert judgment to arrive at an overall predictive distribution for the future population. We apply the methods to a world population forecast that was made in 1994. Accepting that point forecast, we find that the probability is only about 2% that the world population in the year 2030 will be less than the low scenario of 8317 million. The probability that the world population will exceed the high scenario of 10 736 million is about 13%. Similarly, the probability is only about 51% that the high–low interval of a recent United Nations (UN) forecast will contain the true population in the year 2025. Even if we consider the UN high–low intervals as conditional on the possible future policies of its member states, they appear to have a relatively small probability of encompassing the future population.     

Berry-Esseen-type bounds for the kernel estimator of conditional distribution and conditional quantiles

Berry-Esseen bounds of order O(n^−1/2) have been obtained for several classes of statistics. In this paper, the rates of convergence in central limit theorem for conditional empirical functions and conditional sample quantiles based on kernel estimators are studied for both conditional and unconditional distributions.     

The mean of the conditional,truncated multinormal distribution

An expression is derived for the mean of the conditional, truncated multinormal distribution in the general case, and for the situation where the conditioning variates have iden¬tical correlation. This equicorrelated case occurs in some models of duration and in panel data. An example is taken from economics, where a cross section of firms consider the adoption of several technological innovations. The results of the paper are used to estimate the firm-specific.     

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.     

Interval estimation of the population mean under model uncertainty: Robust versus empirical statistics

With reference to the problem of interval estimation of a population mean under model uncertainty, we compare approaches based on robust and empirical statistics via expected lengths of the associated confidence intervals. An explicit expression for confidence intervals arising from a general class of robust statistics is worked out and this is employed to obtain a higher order asymptotic formula for the expected lengths of such intervals. Comparative theoretical results, as well as a simulation study, are then presented.     

Interpreting forensic DNA mixtures: allowing for uncertainty in population substructure and dependence

We consider the assessment of deoxyribonucleic acid (DNA) profiles from biological samples containing a mixture of DNA from more than one person. The problem has been investigated in the context of likelihood ratios by Weir and co-workers under the assumption of independent alleles in DNA profiles. However, uncertainty about independence may arise from various factors such as population substructure and relatedness. This issue has received considerable attention in recent years. Ignoring this uncertainty may seriously overstate the strength of the evidence and therefore disadvantage innocent suspects. Taking this uncertainty into account, we develop a general formula for calculating the match probabilities of DNA profiles. Thus, we extend the result derived by Weir and co-workers to the dependence situation, which is often more to the benefit of the defendant in comparison with the simple product rule result based on an independence assumption. The effect of dependence of alleles on likelihood ratio estimates can be seen in the analysis of two real data sets.     

Sequential parameter structure,conditional inference,and likelihood drop

Successive tests of hypotheses, as exemplified with an analysis of variance table, impose a set theoretic structure on the parameter space and yet allow much arbitrariness in the definition of nuisance parameters. Two major types of statistical model, the exponential and transformation, are shown to have by basic theory well defined conditional testing procedures. The two types of testing procedure are then shown to have opposite forms of set theoretic structure on the sample space, and to differ sharply from the commonly used deviance or likelihood drop methods. The two types of model have the normal linear model as the intersection model and the two opposite forms of testing procedure manage to coincide by product space structure and independence. Details of the two types of testing procedure are discussed, related to the arbitrariness in nuisance parameter definition, and organized to provide a general-case pattern for the development of conditional procedures as an alternative to the default likelihood-drop methods.     

Sensitivity,uncertainty, and decision analyses in the prioritization of research

The evaluation of hazards from complex, large scale, technologically advanced systems often requires the construction of computer implemented mathematical models. These models are used to evaluate the safety of the systems and to evaluate the consequences of modifications to the systems. These evaluations, however, are normally surrounded by significant uncertainties related to the uncertainty inherent in natural phenomena such as the weather and those related to uncertainties in the parameters and models used in the evaluation.

Another use of these models is to evaluate strategies for improving information used in the modeling process itself. While sensitivity analysis is useful in defining variables in the model that are important, uncertainty analysis provides a tool for assessing the importance of uncertainty about these variables. A third complementary technique, is decision analysis. It provides a methodology for explicitly evaluating and ranking potential improvements to the model. Its use in the development of information gathering strategies for a nuclear waste repository are discussed in this paper.     

Estimation of conditional mode with truncated,censored, and dependent data

In this paper, we studied the uniform convergence with rates for the kernel estimator of the conditional mode function for a left truncated and right censored model. It is assumed that the lifetime observations with multivariate covariates form a stationary α-mixing sequence. Also, the asymptotic normality of the estimator is established.     

On uncertainty of a proportion from a stratified random sample of a small population

This article considers the uncertainty of a proportion based on a stratified random sample of a small population. Using the hypergeometric distribution, a Clopper–Pearson type upper confidence bound is presented. Another frequentist approach that uses the estimated variance of the proportion estimator is also considered as well as a Bayesian alternative. These methods are demonstrated with an illustrative example. Some aspects of planning, that is, the impact of specified strata sample sizes, on uncertainty are studied through a simulation study.     

Modelling the persistence of conditional variances

This paper will discuss the current research in building models of conditional variances using the Autoregressive Conditional Heteroskedastic (ARCH) and Generalized ARCH (GARCH) formulations. The discussion will be motivated by a simple asset pricing theory which is particularly appropriate for examining futures contracts with risk averse agents. A new class of models defined to be integrated in variance is then introduced. This new class of models includes the variance analogue of a unit root in the mean as a special case. The models are argued to be both theoretically important for the asset pricing models and empirically relevant. The conditional density is then generalized from a normal to a Student-t with unknown degrees of freedom. By estimating the degrees of freedom, implications about the conditional kurtosis of these models and time aggregated models can be drawn. A further generalization allows the conditional variance to be a non-linear function of the squared innovations. Throughout empirical e imates of the logarithm of the exchange rate between the U.S. dollar and the Swiss franc are presented to illustrate the models.     

Non-parametric estimation of the conditional mode

The problem of estimating the mode of a conditional probability density function is considered. It is shown that under some regularity conditions the estimate of the conditional mode obtained by maximizing a kernel estimate of the conditional probability density function is strongly consistent and asymptotically normally distributed.     

Approximate conditional inference and the linear functional model

Approximate conditional inference is developed for the slope parameter of the linear functional model with two variables. It is shown that the model can be transformed so that the slope parameter becomes an angle and nuisance parameters are radial distances. If the nuisance parameters are known an exact confidence interval based on a location-type conditional distribution is available for the angle. More gen¬erally, confidence distributions are used to average the conditional distribution over the nuisance parameters yielding an approximate conditional confidence interval that reflects the precision indicated by the data. An example is analyzed.     

Improved autoregressive forecasts in the presence of non-normal errors

This paper is concerned with obtaining more accurate point forecasts in the presence of non-normal errors. Specifically, we apply the residual augmented least-squares (RALS) estimator to autoregressive models to utilize the additional moment restrictions embodied in non-normal errors. Monte Carlo experiments are performed to compare our RALS forecasts to forecasts based on the ordinary least-squares estimator and the least absolute deviations (LAD) estimator. We find that the RALS approach provides superior forecasts when the data are skewed. Compared to the LAD forecast, the RALS forecast has smaller mean squared prediction errors in the baseline case with normal errors.     

Canonical expansions,correlation structure,and conditional distributions of bivariate distributions generated by mixtures

A simple result concerning the canonical expansions of mixed bivariate distributions is considered. This result is then applied to analyze the correlation structures of the Bates-Neyman accident proneness model and its generalization, to derive probability inequalities based on the concept of positive dependence, and to construct a bivariate beta distribution with positive correlation coefficient applicable in computer simulation experiments. The mixture formulation of the conditional distribution of this class of mixed bivariate distributions is used to define and generate first-order autoregressive gamma and negative binomial sequences.     

The demographic,social, and occupational distribution of the population

Some results from Poland's 1985 micro-census of population are presented. They concern population changes by age, sex, marital status, educational status, occupation, and income. The data are presented separately for rural and urban areas.