Restricted Least-squares Research Articles

Un teorema di rappresentazione per lo stimatore vincolato

A recent result in partitioned inversion (Faliva and Zoia, 2000) paves the way to shed some more light on constrained estimation, as the paper shows for the linear model. Indeed, a novel closed-form expression for the restricted least-squares(RLS) estimator is derived, which reads as a mirror image of the well-known Theil's(1961) solution and actually turns out to cover the latter as a special case. This leads eventually to an elegant representation theorem for the RLS estimator which combines classical and new results as well.

Restricted ridge estimation

In this paper, we introduce a ridge estimator for the vector of parameters in a linear regression model when additional linear restrictions on the parameter vector are assumed to hold. The estimator is a generalization of the well-known restricted least-squares estimator and is confined to the (affine) subspace which is generated by the restrictions. Necessary and sufficient conditions for the superiority of the new estimator over the restricted least-squares estimator are derived. Our new estimator is not to be confounded with the restricted ridge regression estimator introduced by Sarkar (Comm. Statist. Theory Methods 21 (1992) 1987).

Restricted least squares revisited

In this paper new light is shed on restricted least-squares thanks to a recent result on partitioned inversion [2] which paves the way to finding a novel closed-form expression for the constrained estimator that is more general than the classical Theil’s solution.

A sampling scheme for estimating the mean extractable phosphorus concentration of fields for environmental regulation

A soil sampling scheme for estimating the mean extractable P concentration of fields is designed to be used as a tool for environmental regulation of the application rates of manure. The field to be sampled, is split up into geographically compact blocks of equal area that are used as strata. From each stratum one sampling point is selected by Simple Random Sampling. These samples are bulked into one composite for the field. The geographical stratification is performed by restricted least-squares clustering of raster cells using the coordinates of the midpoints as classification variables and the within-group sum of squares as the minimisation criterion. Using a variance model and a cost model, the numbers of sample points and laboratory analyses are optimised simultaneously, given a maximum allowed variance of the total error (sampling error plus measurement error). To predict the sampling variance, variograms have been estimated for 16 fields differing in land-use, soil parent material and phosphate level. A pooled relative variogram was used to predict the sampling variance for various sample sizes (5 to 50), field-areas (1 to 10 ha) and phosphate levels (for grassland 20 to 80 mg P 2O 5 extracted in ammonimum lactate per 100 g soil, for arable land 20 to 80 mg P 2O 5 extracted in water per 1 dm 3 soil). The cost model consists of three components: (i) fieldwork cost; (ii) field equipment cost and, (iii) laboratory cost. For the 16 fields, the predicted sampling variance of the Stratified Sampling design is 0.8 to 0.4 times the predicted variance of Simple Random Sampling if 40 points were sampled. To estimate the mean extractable P concentration with a total variance ≤9, replicate measurement of the composite only pays if the mean extractable P concentration of the field exceeds 40 to 50. This critical phosphate level increases with the maximum allowed variance of the total error.

On preliminary test and shrinkage estimation in linear models with long-memory errors

This paper discusses the problem of estimating a subset of parameters when the complementary subset is possibly redundant, in a linear regression model when the errors are generated from a long-memory process. Such a model arises due to the overmodelling of a situation involving long-memory data. Along with the classical least-squares estimator and restricted least-squares estimator, preliminary test least-squares estimator and shrinkage least-squares estimator are investigated in an asymptotic set-up and their relative performances are studied under contiguous alternatives. The contiguous alternatives under such dependence are fundamentally different from those under the independent errors case.

Large sample inference for a multivariate linear model with autocorrelated errors

We consider a multivariate linear model with autocorrelated errors. The mean vector of the process is assumed to be linear in the time-trend parameter β and the within-group variation parameter γ. The least-squares estimators of β and γ, and the related estimators of the autoregressive parameter θ and the error covariance matrix Σ are derived and their asymptotic distributions are obtained. Large sample tests of H 1: γ=0 and H 2: β=0are derived and the limit distributions of the restricted least-squares estimators β̂ H 1 and γ̂ H 2 are obtained under H 1 and H 2, respectively.

A comparison of micro versus macro point estimators for markov-process models

We compare, by means of factorially designed Monte Carlo simulation experiments, the performance of (macro-data) restricted least-squares point estimators with that of (micro-data) maximum likelihood estimators for Markov-process models. We find, by various measures of estimator accuracy, that micro data are approximately ten times more valuable than macro data. We also find evidence that a small state space, a long time series, and a large number of entities observed enhance performance.

A note on comparing the unrestricted and restricted least-squares estimators

The problem of comparing the ordinary least-squares estimator β̂ and the restricted least-squares estimator β∗ with respect to a weighted quadratic risk under the normal linear regression model in which restrictions are not known to be true is approached in the literature either (i) by choosing the weight matrix in a special form such that the corresponding necessary and sufficient conditions become operational or (ii) by replacing these necessary and sufficient conditions with weaker sufficient conditions, which may lead, however, to a region of uncertainty where neither dominance of β∗ over β̂ nor dominance of β̂ over β∗ is demonstrable. This note shows that the two approaches are partially reconciliable, in the sense that the class of weight matrices which are appropriate for approach (i) coincides with the class of weight matrices for which the region of uncertainty disappears in approach (ii). Moreover, a number of alternative characterizations of this class are given, and a useful invariance property is established.

On Linear Restricted and Interval Least-Squares Problems

On etudie deux classes d'algorithmes pour la resolution du probleme des moindres carres lineaire sous contraintes

A further result on the sign of restricted least-squares estimates

Within the context of the linear regression model, the variable of interest may be termed the focus variable while those that can be deleted from the regression may be termed doubtful variables. In this note we derive the necessary condition for a sign reversal in the coefficient of the focus variable when the doubtful variables appear in any arbitrary linear combination.

Estimation of the voter transition matrix

This paper is concerned with an estimation of the voter transition matrix using programming methods. Thus, applying the Minimum Absolute Deviations (MAD) and the Restricted Least Squares (RLS) principles, the corresponding estimator has been determined, where the first one seems to be more efficient than the second.

A note on the intracyclical balance-sheet behavior of large commercial banks 1972–1978

Using a Parkin-type model, this paper tests for the responsiveness of large banks' portfolio decisions to exogenous factors over the period 1972 to 1978. Based upon aggregate weekly data (adjusted for first-order autocorrelation) and a restricted least-squares estimation procedure, the empirical results indicate significant differences in behavior (i.e., structural changes) over the business cycle. Previous studies have implicitly assumed that such behavior is constant over the cycle or that differences over the cycle are not significant. This research shows that the assumption is untenable and that the differences over the cycle are statistically significant. Since these patterns vary over the business cycle, the differences should be incorporated into models of bank balance-sheet behavior to avoid misleading interpretations and implications.

On the admissibility of restricted least squares estimators

This article examines the question of admissibility of the restricted least squares (RLS) estimator applied to the linear model , where . Straightforward proofs based on generalizations of the results of Cohen (1966) demonstrate that RLS estimators are admissible under predictive squared error risk iff the dimension of the β parameter space minus the number of linearly independent restrictions on β is less than three. A Stein-type estimator that dominates the inadmissible RLS estimator is presented. Some extensions of the results are indicated.

Restricted Least-Squares Estimates of Age Composition from Length Composition

An exploited stock's length distribution over a set of fixed length intervals can be represented as the product of its age distribution vector and a matrix whose columns are the length distributions of the various age-groups. This matrix should be reasonably stable over time, so it can be estimated from age–length sampling done at any time. Then, so long as the number of length intervals exceeds the number of age-groups, a restricted least-squares estimate of the age distribution underlying any sample length distribution can be found with certainty by quadratic programming. For practical work, the advantages of this method over numerical and closed-form alternatives are that the estimates are reproducible and never contain any negative values. A general statistical treatment is impossible owing to the exclusion of negative values and the presence of sampling error in the matrix of estimated age-specific length distributions, but simulation trials show that in commercial fishery assessments least-squares estimates can be obtained that are unbiased and nearly as precise as maximum-likelihood estimates.Key words: age composition, age–length key, linear regression, restricted least-squares, quadratic programming, simulation

Comment on “Splines and Restricted Least Squares”

Abstract The flexibility of the restricted least squares (RLS) method described by Buse and Lim (1977) for cubic spline regression models is used to consider other polynomial splines. Some of the conclusions of the analysis of a sample data set in the Buse and Lim article are criticized for failing to consider less restrictive models. The data are reanalyzed, the end result being a simpler quadratic spline regression model.

Cubic Splines as a Special Case of Restricted Least Squares

Abstract The standard cubic spline regression method is shown to be a special case of the restricted least-squares estimator. The equivalence of the two procedures under a common set of restrictions is proved. The greater flexibility of the restricted least-squares estimator in terms of the number of restrictions and tests of hypotheses that can be utilized is illustrated by an application to a set of data that has been previously analyzed by the spline method.

A result on the sign of restricted least-squares estimates

When a variable is dropped from a least-squares regression equation, there can be no change in sign of any coefficient that is more significant than the coefficient of the omitted variable. More generally, a constrained least squares estimate of a parameter β j must lie in the interval ( β ̂ j−V 1 2 jj|t|, β ̂ j + V 1 2 jj|t|) where β ̂ j is the unconstrained estimate of β j, V 1 2 jj is the standard error of β ̂ j and t is the t-value for testing the (univariate) restriction.