Multivariate L1 mean

The center of a univariate data set {x ₁,…,x _n} can be defined as the point μ that minimizes the norm of the vector of distances y′=(|x ₁−μ|,…,|x _n−μ|). As the median and the mean are the minimizers of respectively the L ₁- and the L ₂-norm of y, they are two alternatives to describe the center of a univariate data set. The center μ of a multivariate data set {x ₁,…,x _n} can also be defined as minimizer of the norm of a vector of distances. In multivariate situations however, there are several kinds of distances. In this note, we consider the vector of L ₁-distances y′₁=(∥x ₁- μ∥₁,…,∥x _n- μ∥₁) and the vector of L ₂-distances y′₂=(∥x ₁- μ∥₂,…,∥x _n-μ∥₂). We define the L ₁-median and the L ₁-mean as the minimizers of respectively the L ₁- and the L ₂-norm of y ₁; and then the L ₂-median and the L ₂-mean as the minimizers of respectively the L ₁- and the L ₂-norm of y ₂. In doing so, we obtain four alternatives to describe the center of a multivariate data set. While three of them have been already investigated in the statistical literature, the L ₁-mean appears to be a new concept. Received January 1999     

Approximations of the eigenvalues of the covariance matrix of a first-order autoregressive process

This paper provides explicit estimates of the eigenvalues of the covariance matrix of an autoregressive process of order one. Also explicit error bounds are established in closed form. Typically, such an error bound is given by $\text{ε}{}_{\mathrm{k}}\text{= (}\text{4}\text{(n+1)}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{ρ}{}^2\text{sin}\text{(}\text{kπ}\text{(n+1)}\text{)}$, so that the approximations improve as the size of the matrix increases. In other words, the accuracy of the approximations increases as direct computations become more costly.     

Botswana's Macroeconomic Management of Its Mineral-Based Growth: It Used Mining Revenues for Development and Services but Must Now Broaden the Beneficiaries

Abstract . The economic performance and the political history of Botswana during 1974-84 contrasted significantly with the experience of virtually every other African country. The South African republic, immediately north of the Republic of South Africa, achieved steady real economic growth with improved social services and provision for its people's basic needs. This was achieved in spite of world recession and drought because its mineral wealth Was reserved for the people and mining companies had to pay for the privilege of extracting that wealth through a tax program limited to appropriating its surplus, while assuring the investors and entrepreneurs an adequate long-term return on capital and enterprise. But sound macroeconomic policies failed to provide even reasonably equitable benefits for the majority of the people. Sound micropolicies are needed to widen access to employment and earned income and to asset accumulation.     

Convergence of a branching type recursion with non-stationary immigration

The asymptotic distribution of a branching type recursion with non-stationary immigration is investigated. The recursion is given by , where (X _l ) is a random sequence, (L _n ⁻¹⁽¹⁾ ) are iid copies ofL _n−1,K is a random number andK, (L _n ⁻¹⁽¹⁾ ), {(X _l ),Y _n } are independent. This recursion has been studied intensively in the literature in the case thatX _l ≥0,K is nonrandom andY _n =0 ∀n. Cramer, Rüschendorf (1996b) treat the above recursion without immigration with starting conditionL ₀=1, and easy to handle cases of the recursion with stationary immigration (i.e. the distribution ofY _n does not depend on the timen). In this paper a general limit theorem will be deduced under natural conditions including square-integrability of the involved random variables. The treatment of the recursion is based on a contraction method. The conditions of the limit theorem are built upon the knowledge of the first two moments ofL _n . In case of stationary immigration a detailed analysis of the first two moments ofL _n leads one to consider 15 different cases. These cases are illustrated graphically and provide a straight forward means to check the conditions and to determine the operator whose unique fixed point is the limit distribution of the normalizedL _n .     

Estimating process capability index C′′pmk for asymmetric tolerances: Distributional properties

Pearn et al. (1999) considered a capability index C ^′′ _pmk, a new generalization of C _pmk, for processes with asymmetric tolerances. In this paper, we provide a comparison between C ^′′ _pmk and other existing generalizations of C _pmk on the accuracy of measuring process performance for processes with asymmetric tolerances. We show that the new generalization C ^′′ _pmk is superior to other existing generalizations of C _pmk. Under the assumption of normality, we derive explicit forms of the cumulative distribution function and the probability density function of the estimated index . We show that the cumulative distribution function and the probability density function of the estimated index can be expressed in terms of a mixture of the chi-square distribution and the normal distribution. The explicit forms of the cumulative distribution function and the probability density function considerably simplify the complexity for analyzing the statistical properties of the estimated index . Received April 2000     

Preliminary test ridge regression estimators with student’s <Emphasis Type="Italic">t</Emphasis> errors and conflicting test-statistics

The preliminary test ridge regression estimators (PTRRE) based on the Wald (W), Likelihood Ratio (LR) and Lagrangian Multiplier (LM) tests for estimating the regression parameters has been considered in this paper. Here we consider the multiple regression model with student t error distribution. The bias and the mean square errors (MSE) of the proposed estimators are derived under both null and alternative hypothesis. By studying the MSE criterion, the regions of optimality of the estimators are determined. Under the null hypothesis, the PTRRE based on LM test has the smallest risk followed by the estimators based on LR and W tests. However, the PTRRE based on W test performs the best followed by the LR and LM based estimators when the parameter moves away from the subspace of the restrictions. The conditions of superiority of the proposed estimators for both shrinkage parameter, k and the departure parameter, are provided. Some tables for the maximum and minimum guaranteed efficiency of the proposed estimators have been given, which allows us to determine the optimum level of significance corresponding to the optimum estimator. Finally, we conclude that the estimator based on Wald test dominates the other two estimators in the sense of having highest minimum guaranteed efficiency.     

Asymptotic expansion of the minimax value in survey sampling

Summary Suppose that a real numbery _u is associated with each unitu of a populationU and that the functiony:u →y _u onU is known to be an element of the parameter space Θ. The statistician has to select a samples ⊂U ofn units and to employy _u;u ∈s to estimate the arithmetic mean of ally _u,u ∈U. The performance of such a strategy is assessed by its mean square error or, more simply, by the supremum of the mean square error. This supremum cannot be determined exactly for the parameter space of Scott/Smith (1975). We propose, therefore, an asymptotic approximation; this approximation is based on the assumption, that the sample sizen is fixed and that linear estimators have to be used.     

Two-sample rank tests with truncated populations

LetX ₁,…,X _m andY ₁,…,Y _n be two independent samples from continuous distributionsF andG respectively. Using a Hoeffding (1951) type theorem, we obtain the distributions of the vector S=(S ₍₁₎,…,S _(n)), whereS _(j)=# (X _i ’s≤Y _(j)) andY _(j) is thej-th order statistic ofY sample, under three truncation models: (a)G is a left truncation ofF orG is a right truncation ofF, (b)F is a right truncation ofH andG is a left truncation ofH, whereH is some continuous distribution function, (c)G is a two tail truncation ofF. Exploiting the relation between S and the vectorR of the ranks of the order statistics of theY-sample in the pooled sample, we can obtain exact distributions of many rank tests. We use these to compare powers of the Hajek test (Hajek 1967), the Sidak Vondracek test (1957) and the Mann-Whitney-Wilcoxon test. We derive some order relations between the values of the probagility-functions under each model. Hence find that the tests based onS ₍₁₎ andS _(n) are the UMP rank tests for the alternative (a). We also find LMP rank tests under the alternatives (b) and (c).     

Some Determinants of Property Crime: Economic Factors Influence Criminal Behavior But Cannot Completely Explain the Syndrome

Abstract . An empirical analysis of the property crimes, robbery, burglary and larceny, is presented for all 120 counties in Kentucky. While this analysis is based on an economic model of crime, certain sociological and legal variables are included as well in the system of equations. Overall, the empirical results support prior studies’findings with the exception that a quadratic relationship is found to exist between urbanization and each of the property crimes. Furthermore, neither the economic nor the non economic influences measured appear more important for affecting crime rates. Specifically, results indicate that the level of poverty, the degree of tourism, the presence of police, the unemployment rate and the apprehension rate all affect property crimes. In contrast, the length of sentence, the degree of industrialization, the level of public assistance payments and the proportion of youth in the county have no affect on property crime rates in these areas.     

Henry George's Theory of State's Agenda: The Origins of His Ideas on Economic Policy in Adam Smith's Moral Theory

Abstract . The tension between Henry George's reformism and his laissezfaire liberalism was resolved through a system of natural liberty George derived from the relation between Adam Smith's ethics and economics. Crucial for George's nonutilitarian philosophy of government was the interdependence between the moral sense (sympathy) and the prevailing socioeconomic order. In the appropriate institutional environment, the role of the government was diminished since the pervasive moral sense insured justice by monitoring the individual's pursuit of economic self-interest. In contrast, a defective socio economic order required government intervention. For example, land monopoly and the maldistribution of income undermined the role of sympathy, promoted excessive self interest and the breakdown of the system of natural liberty. Government action through the single tax eliminated the “fear of want,” restored an operative moral sense and guaranteed justice in society. Under these conditions, government can provide additional services for a growing society without being susceptible to “corrupt and tyrannous” behavior.     

Information in generalized method of moments estimation and entropy-based moment selection

In this paper, we make five contributions to the literature on information and entropy in generalized method of moments (GMM) estimation. First, we introduce the concept of the long run canonical correlations (LRCCs) between the true score vector and the moment function f(_vt,_θ0)$f(v_t,{\theta }_0)$ and show that they provide a metric for the information contained in the population moment condition E[f(_vt,_θ0)]=0$\mathrm{E}[f(v_t,{\theta }_0)]=0$. Second, we show that the entropy of the limiting distribution of the GMM estimator can be written in terms of these LRCCs. Third, motivated by the above results, we introduce an information criterion based on this entropy that can be used as a basis for moment selection. Fourth, we introduce the concept of nearly redundant moment conditions and use it to explore the connection between redundancy and weak identification. Fifth, we analyse the behaviour of the aforementioned entropy-based moment selection method in two scenarios of interest; these scenarios are: (i) nonlinear dynamic models where the parameter vector is identified by all the combinations of moment conditions considered; (ii) linear static models where the parameter vector may be weakly identified for some of the combinations considered. The first of these contributions rests on a generalized information equality that is proved in the paper, and may be of interest in its own right.     

Combining information for prediction in linear regression

This article deals with the prediction problem in linear regression where the measurements are obtained using k different devices or collected from k different independent sources. For the case of k=2, a Graybill-Deal type combined estimtor for the regression parameters is shown to dominate the individual least squares estimators under the covariance criterion. Two predictors ŷ _c and ŷ _p are proposed. ŷ _c is based on a combined estimator of the regression coefficient vector, and ŷ _p is obtained by combining the individual predictors from different models. Prediction mean square errors of both predictors are derived. It is shown that the predictor ŷ _p is better than the individual predictors for k≥2 and the predictor ŷ _c is better than the individual predictors for k=2. Numerical comparison between ŷ _c and ŷ _p shows that the former is superior to the latter for the case k=2.     

Bounds for the mean residual life function of a <Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis> system

In the reliability studies, k-out-of-n systems play an important role. In this paper, we consider sharp bounds for the mean residual life function of a k-out-of-n system consisting of n identical components with independent lifetimes having a common distribution function F, measured in location and scale units of the residual life random variable X _t  = (X−t|X > t). We characterize the probability distributions for which the bounds are attained. We also evaluate the so obtained bounds numerically for various choices of k and n.     

Performance Measures in Dose-Finding Experiments

In the first phase of pharmaceutical development, and assuming that the probability of positive response increases with dose, the main statistical goal is to estimate a percentile of the dose–response function for a given target value $\mathrm{\Gamma }$. We compare the Maximum Likelihood and centred isotonic regression estimators of the target dose and we discuss several performance criteria to assess inferential precision, the amount of toxicity exposure and the trade-off between them for a set of some exemplary adaptive designs. We compare these designs using graphical tools. Several scenarios are considered using simulation, including the use of several start-up rules, the change of slope of the dose-toxicity function at the target dose and also different theoretical models, as logistic, normal or skew-normal distribution functions.     

Work and Welfare in Economics:

Abstract . Today's conventional economics typically ignores the impact of alternative forms of work organization upon the welfare of the worker. In effect, its methodology is concerned with the welfare of the individual as a consumer, but not with the welfare of the individual as a worker. Hence, welfare conclusions of economics are subject to challenge on grounds of being incomplete. Whether the worker is alienated or achieves self-fulfillment, etc., stands in no necessary relationship to either the formal lines of enterprise ownership, political ideology or form of economic system. Questions of authority and power in work organizations and the workplace conditions affecting the worker on the job transcend them.     

An outlier test for linear processes — II. Large contamination

In Flak/Schmid (1993) an outlier test for linear processes was introduced. The test statistic bases on a comparison of each observation with a one-step predictor. It was assumed that an upper bound for the total number of outlierss _n is known, wheren denotes the sample size. The asymptotic distribution of the test statistic was derived under the assumption thats _n/n → 0 ands _n → ∞ asn → ∞. This note deals with the asymptotic behaviour of this quantity, ifs _n/n →p ₀ ∈ (0, 1).     

Redistributing the energy tax burden in the Philippines

This paper uses an aggregate modelling approach to assess the impacts of a redistribution of the taxes and duties that currently exist on crude oil and refined petroleum products on the Philippine economy. The approach used in the analysis consists of a general equilibrium model composed of fourteen producing sectors, fifteen consuming sectors, three household categories classified by income and a government. The effects of replacing the taxes and duties on crude oil and refined petroleum products with a more broad based tax on manufacturing and service sectors output on prices and quantities are examined. The results are revealing. For example, the consequences of redistributing the tax burden away from petroleum products to the manufacturing and service sectors of the Philippine economy will be an increase in output by all producing sectors of about 3.5 percent or about 2.4 hundred billion Philippine pesos, a rise in the consumption of goods and services by about 6.1 percent or 1.6 hundred billion Philippine pesos, a rise in total utility by 6.9 or 1.9 hundred billion Philippine pesos, and virtually no change in tax revenue for the government. When subjected to a sensitivity analysis, the results are reasonably robust with regard to the assumption of the values of the substitution elasticities. That is, while the model's equilibrium values do vary in response to different assumptions of the values of these elasticities, the fluctuations are not so enormous to suggest that the model is unrealistically sensitive to these parameters.Notation Y _j Total production in sectorj (j=1, 2, ..., 14) - CD _j Consumer demand for productj - GE _j Government endowment of productj - UM _j Imports of productj - _LRAS_jl RAS balanced input-output intermediate demands - GD _j Government demand for productj - INV _j Investment in sectorj - UX _j Exports of productj - SL _c Supply of labor by householdc (c=1, 2, 3) - SK _c Supply of capital by householdc - SD _c Supply of land by householdc - DL _j Demand for labor in the industryj - DK _j Demand for capital in the industryj - DD _j Demand for land in industryj - GDL Government demand for labor - GDD Government demand for land - TL _j Tax on labor in industryj - TK _j Tax on capital in industryj - TD _j Tax on land in industryj - GCE _i Consumer demand for consumer producti (i=1, 2, ..., 15) - Z _ji A 14×15 transformation matrix - RCS _ic RAS balanced matrix of each household's demand for each consumer good - TC _j Excise tax on consumer goodj - TRN _c Transfer payment to householdc - PIT _c Personal income tax payment for householdc - TAU _c Marginal income tax rate for householdc - SAV _c Savings in householdc - GC _c Gross consumption of householdc - ZTA Consumption plus leisure coefficient - TE Total government endowments - EM _j Demand elasticity of export demand - FE _j Endowment/demand sector of adjusted elasticity of export demand - GSK _j Government endowment of capital in industryj - GDK _j Government demand for capital in industryj - GTL Government wage taxes on its own employees - TXO _j Government output tax on industryj - TC _c Consumption taxes on householdc - CG _c Total government consumption by householdc - SAV _c Total savings by householdc - INV _j Total investment by industryj The views expressed are those of the authors and do not necessarily represent the policies of the organizations with which they are affiliated. They would like to thank Wildrido Cruz of the World Bank and Climenta Habido of the Philippine government for help in acquiring the requisite data to calibrate the model used in the analysis. They would also like to thank an anonymous referee for helpful suggestions.