Model‐Assisted Regression Estimators for Longitudinal Data with Nonignorable Dropout

Estimation with longitudinal Y having nonignorable dropout is considered when the joint distribution of Y and covariate X is nonparametric and the dropout propensity conditional on (Y,X) is parametric. We apply the generalised method of moments to estimate the parameters in the nonignorable dropout propensity based on estimating equations constructed using an instrument Z, which is part of X related to Y but unrelated to the dropout propensity conditioned on Y and other covariates. Population means and other parameters in the nonparametric distribution of Y can be estimated based on inverse propensity weighting with estimated propensity. To improve efficiency, we derive a model‐assisted regression estimator making use of information provided by the covariates and previously observed Y‐values in the longitudinal setting. The model‐assisted regression estimator is protected from model misspecification and is asymptotically normal and more efficient when the working models are correct and some other conditions are satisfied. The finite‐sample performance of the estimators is studied through simulation, and an application to the HIV‐CD4 data set is also presented as illustration.     

Monotone evolution of resource allocations

We prove several existence theorems for solutions of multivalued differential equations whose trajectories (a) remain in a subset defined by scarcity constraints, and (b) are monotone with respect to a preordering; we give applications to the problem of allocation of scarce resources.     

Some notes on discount factor restrictions for dynamic optimization problems

We consider dynamic optimization problems on one-dimensional state spaces. Under standard smoothness and convexity assumptions, the optimal solutions are characterized by an optimal policy function h mapping the state space into itself. There exists an extensive literature on the relation between the size of the discount factor of the dynamic optimization problem on the one hand and the properties of the dynamical system x_t+1=h(x_t) on the other hand. The purpose of this paper is to survey some of the most important contributions of this literature and to modify or improve them in various directions. We deal in particular with the topological entropy of the dynamical system, with its Lyapunov exponents, and with its periodic orbits.     

Semiparametric Estimator for Binary‐outcome Sample Selection: Prejudice Matters in Election

a semiparametric estimator for binary‐outcome sample‐selection models is proposed that imposes only single index assumptions on the selection and outcome equations without specifying the error term distribution. I adopt the idea in Lewbel (2000) using a ‘special regressor’ to transform the binary response Y so that the transformed Y becomes linear in the latent index, which then makes it possible to remove the selection correction term by differencing the transformed Y equation. There are various versions of the estimator, which perform differently trading off bias and variance. A simulation study is conducted, and then I apply the estimators to US presidential election data in 2008 and 2012 to assess the impact of racial prejudice on the elections, as a black candidate was involved for the first time ever in the US history.     

Global dynamics in models of fluctuating growth Part I: Two dimensional systems

We consider a modified version of Goodwin's celebrated non-linear model of fluctuating growth, where the incomeY is a quadratic function of the capital stockk, in order to take into account the possibility of increasing or decreasing returns. The dynamics of the model is then defined by a non-autonomous system of two differential equations. Assuming the labour supply,N, and the productivity,a, to be constant in the time, the model becomes autonomous and can be embedded in a stable family of planar dynamical systems whose flows and bifurcations are globally described

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Riassunto Si considera una versione modificata del modello di crescita non-lineare di Goodwin, in cui il reddito,Y, è una funzione quadratica dello stock di capitale,k, così da permettere la possibilità di rendimenti crescenti o decrescenti. La dinamica del modello viene allora rappresentata da un sistema non autonomo di due equazioni differenziali. Con l'ipotesi che l'offerta di lavoro,N, e la produttività,a, siano costanti nel tempo, il modello diventa autonomo e può essere immerso in una famiglia stabile di sistemi dinamici piani, di cui vengono descritti i flussi globali e le biforcazioni.

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This paper relates to the activities of the M.P.I. Group Dinamiche Non-Lineari nelle Scienze Economiche e Sociali.     

A measure of mutual complete dependence

Two random variables X and Y on a common probability space are mutually completely dependent (m.c.d.) if each one is a function of the other with probability one. For continuous X and Y, a natural approach to constructing a measure of dependence is via the distance between the copula of X and Y and the independence copula. We show that this approach depends crucially on the choice of the distance function. For example, the L ^p -distances, suggested by Schweizer and Wolff, cannot generate a measure of (mutual complete) dependence, since every copula is the uniform limit of copulas linking m.c.d. variables. Instead, we propose to use a modified Sobolev norm, with respect to which mutual complete dependence cannot approximate any other kind of dependence. This Sobolev norm yields the first nonparametric measure of dependence which, among other things, captures precisely the two extremes of dependence, i.e., it equals 0 if and only if X and Y are independent, and 1 if and only if X and Y are m.c.d. Examples are given to illustrate the difference to the Schweizer–Wolff measure.     

The association between two random elements: A complete characterization and odds ratio models

For random elements X and Y (e.g. vectors) a complete characterization of their association is given in terms of an odds ratio function. The main result establishes for any odds ratio function and any pre-specified marginals the unique existence of a corresponding joint distribution (the joint density is obtained as a limit of an iterative procedure of marginal fittings). Restricting only the odds ratio function but not the marginals leads to semi-parmetric association models for which statistical inference is available for samples drawn conditionally on either X or Y. Log-bilinear association models for random vectors X and Y are introduced which generalize standard (regression) models by removing restrictions on the marginals. In particular, the logistic regression model is recognized as a log-bilinear association model. And the joint distribution of X and Y is shown to be multivariate normal if and only if both marginals are normal and the association is log-bilinear.Acknowledgements The author thanks both referees for their helpful comments which improved the first draft of the paper.     

A note on reliability properties of k-record statistics

LetY _k,n denote the nth (upper) k-record value of an infinite sequence of independent, identically distributed random variables with common continuous distribution function F. We show that if the nth k-record valueY _k,n has an increasing failure rate (IFR), thenY _l,n (l<k) andY _k+1,n+1(n≤k+1) also have IFR distributions. On the other hand, ifY _k,n has a decreasing failure rate (DFR), thenY _l,n (1>k) has also a DFR distribution. We also present some results concerning log convexity and log concavity ofY _k,n .     

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).     

Evaluation of P(Xt>Yt ) when both Xt and Yt are residual lifetimes of two systems

Let the random variables X and Y denote the lifetimes of two systems. In reliability theory to compare between the lifetimes of X and Y there are several approaches. Among the most popular methods of comparing the lifetimes are to compare the survival functions, the failure rates and the mean residual lifetime functions of X and Y. Assume that both systems are operating at time t > 0. Then the residual lifetimes of them are X_t=X?t | X>t and Y_t=Y?t | Y>t, respectively. In this paper, we introduce, by taking into account the age of systems, a time‐dependent criterion to compare the residual lifetimes of them. In other words, we concentrate on function R(t ):=P(X_t>Y_t) which enables one to obtain, at time t, the probability that the residual lifetime X_t is greater than the residual lifetime Y_t. It is mentioned, in Brown and Rutemiller (IEEE Transactions on Reliability, 22 , 1973) that the probability of type R(t) is important for designing as long‐lived a product as possible. Several properties of R(t) and its connection with well‐known reliability measures are investigated. The estimation of R(t) based on samples from X and Y is also discussed.     

Your Life Chances Affect Where You Live: A Critique of the ‘Cottage Industry’ of Neighbourhood Effects Research

This article advances a critique of the ‘neighbourhood effects’ genre in urban studies, by arguing that an acceptance of the ‘where you live affects your life chances’ thesis, however well‐intentioned, misses the key structural question of why people live where they do in cities. By examining the structural factors that give rise to differential life chances and the inequalities they produce, and by inverting the neighbourhood effects thesis to: your life chances affect where you live, the problem becomes one of understanding life chances via a theory of capital accumulation and class struggle in cities. Such a theory provides an understanding of the injustices inherent in letting the market (buttressed by the state) be the force that determines the cost of housing and therefore being the major determinant of where people live. The article draws on Marxist urban theory to contend that the residential mobility programs advocated by neighbourhood effects proponents stand on shaky ground, for if it is true that ‘neighbourhood effects’ exceed what would be predicted by poverty alone, moving the poor to a richer place would only eliminate that incremental difference, without addressing the capitalist institutional arrangements that create poverty.     

Nonlinear dynamical systems and inventory management

Surprisingly, deterministic time series can generate highly irregular, random-appearing trajectories. These deterministic time series result from nonlinear dynamical systems of differential or difference equations. The random appearance displayed by these systems is called nonlinear dynamical complexity. Properties of nonlinear complex systems include aperiodic random appearance, sensitive dependence on initial conditions and model parameters, and nonstationarity. Experiments involving the operation of simulated business environments and theoretical nonlinear dynamical models for inventory are reviewed to explore motivating factors that can give rise to demand with nonlinear complexities. The experimental and theoretical evidence reviewed indicates that nonlinear complexities in demand have significant implications for inventory management. Thus, researchers and practitioners in inventory management need to consider these properties when choosing inventory management methods. Characteristics of nonlinear dynamical systems and their implications for inventory management are presented in this paper. The use of the Brock, Dechert, and Scheinkman (1987) (BDS) test for nonlinear dependence is demonstrated on actual demand data.     

Alternative functional forms for production,cost and returns to scale functions

We consider generalized production functions, introduced in Zellner and Revankar (1969), for output y=g(f) where g is a monotonic function and f is a homogeneous production function. For various choices of the scale elasticity or returns to scale as a function of output, differential equations are solved to determine the associated forms of the monotonic transformation, g(f). Then by choice of the form of f, the elasticity of substitution, constant or variable, is determined. In this way, we have produced and generalized a number of homothetic production functions, some already in the literature. Also, we have derived and studied their associated cost functions to determine how their shapes are affected by various choices of the scale elasticity and substitution elasticity functions. In general, we require that the returns to scale function be a monotonically decreasing function of output and that associated average cost functions be U- or L-shaped with a unique minimum. We also represent production functions in polar coordinates and show how this representation simplifies study of production functions' properties. Using data for the US transportation equipment industry, maximum likelihood and Bayesian methods are employed to estimate many different generalized production functions and their associated average cost functions. In accord with results in the literature, it is found that the scale elasticities decline with output and that average cost curves are U- or L-shaped with unique minima. © 1998 John Wiley & Sons, Ltd.     

On testing for independence against right tail increasing in bivariate models

A random variableY is right tail increasing (RTI) inX if the failure rate of the conditional distribution ofX givenY>y is uniformly smaller than that of the marginal distribution ofX for everyy0. This concept of positive dependence is not symmetric inX andY and is stronger than the notion of positive quadrant dependence. In this paper we consider the problem of testing for independence against the alternative thatY is RTI inX. We propose two distribution-free tests and obtain their limiting null distributions. The proposed tests are compared to Kendall's and Spearman's tests in terms of Pitman asymptotic relative efficiency. We have also conducted a Monte Carlo study to compare the powers of these tests.Research supported by an NSERC Canada operating grant at the University of Alberta.Part of this research was done while visiting the University of Alberta supported by the NSERC Canada grant of the first author.     

Three dual regression schemes for the Lukacs theorem

In the paper we study regressional versions of Lukacs' characterization of the gamma law. We consider constancy of regression instead of Lukacs' independence condition in three new schemes. Up to now the constancy of regressions of U=X/(X + Y) given V=X + Y for independent X and Y has been considered in the literature. Here we are concerned with constancy of regressions for X and Y while independence of U and V is assumed instead.     

Learning curves—an axiomatic approach

A systematic mathematical analysis of learning curves is presented. It is shown that while a learning curve with learning factor k does necessarily satisfy the functional equation f(2x) = kf(x), this equation admits numerous other analytic, convex solutions as well so that it cannot be used to uniquely characterize learning curves. Rather, it is shown that a pair of functional equations f(rx) = kf(x) and f(sx) = jf(x) where rⁿ ≠ s^m for any positive integers n and m suffice to uniquely determine the learning curve; compatibility of the two equations requires that log_rk = log_sj or there will be no learning curve satisfying the pair of equations. Two classes of almost learning curves are generated and studied by means of suitable perturbation terms in the differential equation y = by/x of the true or standard learning curve and these curves are applied to describe data not exhibiting exact learning-curve behavior. Finally the concept of average marginal hours or cost is introduced and its behavior is found to also exhibit the learning-curve phenomenon except for an initial deviation.     

Estimating the conditional mode of a stationary stochastic process from noisy observations

Let {(X _i, Y _i,)}, i≥1, be a strictly stationary process from noisy observations. We examine the effect of the noise in the response Y and the covariates X on the nonparametric estimation of the conditional mode function. To estimate this function we are using deconvoluting kernel estimators. The asymptotic behavior of these estimators depends on the smoothness of the noise distribution, which is classified as either ordinary smooth or super smooth. Uniform convergence with almost sure convergence rates is established for strongly mixing stochastic processes, when the noise distribution is ordinary smooth. Received: April 1998     

Dynamic Modeling with Structural Equations and Stochastic Differential Equations: Applications with the German Socio-economic Panel

The paper discusses an application of linear dynamic models to multi-wave longitudinal data. Starting from three-wave and four-wave simplex models using standard structural equations, linear dynamic state space models with stochastic differential equations are presented. The main differences between longitudinal structural equations (static view) and stochastic differential equations (dynamic view) are emphasized. Substantively, the models prove the relation, stability and change of two concepts in a period of 10 years: National Identity and Intention to stay in Germany. Data from a sample of migrant workers in Germany included in the German Socio-economic Panel (GSOEP) are used for the analyses. Results and further developments of dynamic models are discussed in the final section.The authors thank Hermann Singer for his comments and discussions on applications of dynamic models.     

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.