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.     

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).     

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 .     

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.     

Using an approximate kiefer-weiss solution for testing insect population densities

Summary A sequential testing procedure called 2-SPRT for the meanμ of the negative binomial distribution with known exponentk is presented. For any fixedμ ₀(μ ₁<μ ₀<μ ₂), the 2-SPRT provides an asymptotic solution to the modified Kiefer-Weiss problem. Determination ofμ ₀ such that the test provides an approximate solution to the Kiefer-Weiss problem is described. The behavior of the 2-SPRT and the Sequential Probability Ratio Test is investigated using Monte Carlo methods. The error probabilities and the average sample numbers are compared. All the computations were carried out on the Alabama Supercomputer.     

Random mechanisms for house allocation with existing tenants

We study the house allocation problem with existing tenants: $n$ houses (stand for “indivisible objects”) are to be allocated to $n$ agents; each agent needs exactly one house and has strict preferences; $k$ houses are initially unowned; $k$ agents initially do not own houses; the remaining $n-k$ agents (the so-called “existing tenants”) initially own the remaining $n-k$ houses (each owns one). In this setting, we consider various randomized allocation rules under which voluntary participation of existing tenants is assured and the randomization procedure either treats agents equally or discriminates against some (or all) of the existing tenants. We obtain two equivalence results, which generalize the equivalence results in Abdulkadiroğlu and Sönmez (1999) and Sönmez and Ünver (2005).     

What Is the Economic Meaning of FDH?

The central feature of the FDH model is the lack of convexity for its production possibility set, TF. Starting with n observed (distinct) decision making units DMU_k , each defined by an input-output vector p _k = [y _k -x _k], domination is defined by ordinary vector inequalities. DMU_k is said to dominate DMU_j if p _k ≥ p _j , p _k ≠ p _j . The FDH production possibility set TF consists of the observed DMU_j together with all input-output vectors p=[y_k,?x_k] with y ≥ 0, x ≥ 0, y ≠ 0, x ≠ 0 which are dominated by at least one of the observed DMU_j. DMU_k is defined as “FDH efficient” if no DMU_j dominates it. In the BCC (or variable return to scale) DEA model the production possibility set TB consists of the observed DMU_k together with all input-output vectors dominated by any convex combination of them and DMU_k is DEA efficient if it is not dominated by any p in TB. In the DEA model, economic meaning is established by the introduction of (non negative) multiplier (price) vectors w = [u,v]. If DMU_k is undominated (in TB) then there exists a positive multiplier vector w for which (a) w ^T p _k = u ^T y _k ? v ^T x _k ≥ w ^T p for every p ∈ TB. In everyday language, the net return (or profit) for DMU_k relative to the given multiplier vector w is at least as great as that for any production possibility p. On the other hand, if DMU_k is FDH but not DEA efficient then it is proved that there exists no positive multiplier vector >w for which (a) holds, i.e. for any positive w there exists at least one DMU_j for which w ^T p _j > w^T p _k . Since, therefore, FDH efficiency does not guarantee price efficiency what is its economic significance? Without economic significance, how can FDH be considered as being more than a mathematical system however logically soundly it may be conceived?

     

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.     

Characterization of the multivariate normal distribution by conditional normal distributions

Summary LetX andY be two random vectors with values in ℝ ^k and ℝ∝, respectively. IfZ=(X ^T,Y ^T) ^T is multivariate normal thenX givenY=y andY givenX=x are (multivariate) normal; the converse is wrong. In this paper simple additional conditions are stated such that the converse is true, too. Furthermore, the case is treated that the random vectorZ=(X ₁ ^T , …,X _t ^T ) ^T is splitted intot≥3 partsX ₁, …,X _t.     

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.     

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).     

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.     

A complete asymptotic series for the autocovariance function of a long memory process

An infinite-order asymptotic expansion is given for the autocovariance function of a general stationary long-memory process with memory parameter d∈(−1/2,1/2)$d\in (-1/2,1/2)$. The class of spectral densities considered includes as a special case the stationary and invertible ARFIMA(p,d,q$p,d,q$) model. The leading term of the expansion is of the order O(1/k^1−2d)$O(1/k^{1-2d})$, where k$k$ is the autocovariance order, consistent with the well known power law decay for such processes, and is shown to be accurate to an error of O(1/k^3−2d)$O(1/k^{3-2d})$. The derivation uses Erdélyi’s [Erdélyi, A., 1956. Asymptotic Expansions. Dover Publications, Inc, New York] expansion for Fourier-type integrals when there are critical points at the boundaries of the range of integration - here the frequencies {0,2π}$\{0,2\pi \}$. Numerical evaluations show that the expansion is accurate even for small k$k$ in cases where the autocovariance sequence decays monotonically, and in other cases for moderate to large k$k$. The approximations are easy to compute across a variety of parameter values and models.     

Criterion-robust optimal product designs for discrimination between nested response surface models

Consider the problem of discriminating between two rival response surface models and estimating parameters in the identified model. To construct designs serving for both model discrimination and parameter estimation, the M _γ-optimality criterion, which puts weight γ (0≤γ≤1) for model discrimination and 1 − γ for parameter estimation, is adopted. The corresponding M _γ-optimal product design is explicitly derived in terms of canonical moments. With the application of the maximin principle on the M _γ-efficiency of any M _γ'-optimal product design, a criterion-robust optimal product design is proposed.     

Biases,variances and covariances of raking ratio estimators for marginal and cell totals and averages of observed characteristics

Summary On the basis of a simple random sample from a population, on which a cross-classification is defined with known marginal frequenciesN _i. andN _.j , one wishes to estimate the cell frequenciesN _ij , as well as cell totalsY _ij , marginal totalsY _i. andY _.j , and the grand totalY for characteristics measured on the units. Various authors have discussed so-called raking ratio estimators, which are built up from the estimated cell values by addition. They have given the bias and variance of this estimator ofY. This paper derives biases, variances and covariances for the corresponding estimators of the cell and marginal totals and of the corresponding marginal averages.