Nonlinear regression models with single‐index heteroscedasticity

We consider nonlinear heteroscedastic single‐index models where the mean function is a parametric nonlinear model and the variance function depends on a single‐index structure. We develop an efficient estimation method for the parameters in the mean function by using the weighted least squares estimation, and we propose a “delete‐one‐component” estimator for the single‐index in the variance function based on absolute residuals. Asymptotic results of estimators are also investigated. The estimation methods for the error distribution based on the classical empirical distribution function and an empirical likelihood method are discussed. The empirical likelihood method allows for incorporation of the assumptions on the error distribution into the estimation. Simulations illustrate the results, and a real chemical data set is analyzed to demonstrate the performance of the proposed estimators.     

On efficiency of the ratio estimator

Summary The exact mean square error for the ratio estimator of a finite population total based on simple random sampling without replacement is shown to have an expected value less than that of the variance of the ratio estimator based on Midzuno’s scheme, under a usual super-population model.     

On two properties of an unequal probability sampling scheme

Summary For an inclusion probability proportional to size (IPPS) sampling scheme recently proposed by Saxena, Singh and Srivastava (1986), it is shown that under certain simple verifiable conditions (1) the Horvitz-Thompson (1952) estimator based on it has a smaller variance than the variance of the Hansen-Hurwitz (1943) estimator based on probability proportional to size (PPS) sampling with replacement (WR) both involving the same size-measures and the expected sample size in the former being equal to the number of draws in the latter and (2) the Yates-Grundy (1953) estimator for the variance of the Horvitz-Thompson estimator based on this IPPS scheme is uniformly non-negative.     

Residual‐based IV estimation of dynamic panel data models with fixed effects

In dynamic panel regression, when the variance ratio of individual effects to disturbance is large, the system‐GMM estimator will have large asymptotic variance and poor finite sample performance. To deal with this variance ratio problem, we propose a residual‐based instrumental variables (RIV) estimator, which uses the residual from regressing Δy_i,t?1 on as the instrument for the level equation. The RIV estimator proposed is consistent and asymptotically normal under general assumptions. More importantly, its asymptotic variance is almost unaffected by the variance ratio of individual effects to disturbance. Monte Carlo simulations show that the RIV estimator has better finite sample performance compared to alternative estimators. The RIV estimator generates less finite sample bias than difference‐GMM, system‐GMM, collapsing‐GMM and Level‐IV estimators in most cases. Under RIV estimation, the variance ratio problem is well controlled, and the empirical distribution of its t‐statistic is similar to the standard normal distribution for moderate sample sizes.     

Model assisted survey sampling strategy in two phases

Postulating a super-population regression model connecting a size variable, a cheaply measurable variable and an expensively observable variable of interest, an asymptotically optimal double sampling strategy to estimate the survey population total of the third variable is specified. To render it practicable, unknown model-parameters in the optimal estimator are replaced by appropriate statistics. The resulting generalized regression estimator is then shown to have a model-cum-asymptotic design based expected square error equal to that of the asymptotically optimum estimator itself. An estimator for design variance of the estimator is also proposed.     

MSE-improvement of the least squares estimator by dropping variables

It is well known that dropping variables in regression analysis decreases the variance of the least squares (LS) estimator of the remaining parameters. However, after elimination estimates of these parameters are biased, if the full model is correct. In his recent paper, Boscher (1991) showed that the LS-estimator in the special case of a mean shift model (cf. Cook and Weisberg, 1982) which assumes no “outliers” can be considered in the framework of a linear regression model where some variables are deleted. He derived conditions under which this estimator outperforms the LS-estimator of the full model in terms of the mean squared error (MSE)-matrix criterion. We demonstrate that this approach can be extended to the general set-up of dropping variables. Necessary and sufficient conditions for the MSE-matrix superiority of the LS-estimator in the reduced model over that in the full model are derived. We also provide a uniformly most powerful F-statistic for testing the MSE-improvement.     

Analytically pricing variance and volatility swaps under a Markov-modulated model with liquidity risks

This paper determines strike prices of discretely sampled variance/volatility swaps taking into account stochastic liquidity risks and the switching of economic conditions. We adopt nonlinear regime switching volatility to reflect how asset prices are affected by economic cycles, and market prices of assets are discounted according to the level of market liquidity. We then establish a risk-neutral measure under regime switching Esscher transform, so that analytical valuation of variance/volatility swaps can be completed based on the closed-form forward characteristic function. The limiting behavior of discretely sampled variance/volatility swaps is also considered through the investigation of pricing continuously sampled variance/volatility swaps. Finally, based on the results from numerical implementation, we confirm that the new model is very flexible in reflecting different influence associated with common real market observations.     

Simplified Jackknife Variance Estimates for Fuzzy Measures of Multidimensional Poverty

In this paper, we present a practical methodology for variance estimation for multi‐dimensional measures of poverty and deprivation of households and individuals, derived from sample surveys with complex designs and fairly large sample sizes. The measures considered are based on fuzzy representation of individuals' propensity to deprivation in monetary and diverse non‐monetary dimensions. We believe this to be the first original contribution for estimating standard errors for such fuzzy poverty measures. The second objective is to describe and numerically illustrate computational procedures and difficulties in producing reliable and robust estimates of sampling error for such complex statistics. We attempt to identify some of these problems and provide solutions in the context of actual situations. A detailed application based on European Union Statistics on Income and Living Conditions data for 19 NUTS2 regions in Spain is provided.     

Sampling with probabilities proportional to the variable of interest

To estimate the mean sojourn time, a sample of Tilburg fair visitors was asked for the duration of their stay on the fair grounds. The longer a visitor's sojourn, the larger his/her probability of being interviewed will be; therefore, longer sojourn times will be overrepresented in the sample. As a consequence, the arithmetic sample mean is not a good estimator.
The paper places this problem against a theoretical background. Sampling with unequal probabilities is considered in a general context. The special case that the sampling probabilities are a function of the variable under investigation, is discussed in detail. As a better estimator the harmonic mean of the observations is presented. Most properties of this estimator are difficult to derive analytically, but a suitable variance estimator is derived. The behavior of estimator and variance estimator is studied in a number of quite different examples.     

Estimating a restricted normal mean

Let X ₁, X ₂, ..., X _n be a random sample from a normal distribution with unknown mean μ and known variance σ ². In many practical situations, μ is known a priori to be restricted to a bounded interval, say [−m, m] for some m > 0. The sample mean , then, becomes an inadmissible estimator for μ. It is also not minimax with respect to the squared error loss function. Minimax and other estimators for this problem have been studied by Casella and Strawderman (Ann Stat 9:870–878, 1981), Bickel (Ann Stat 9:1301–1309, 1981) and Gatsonis et al. (Stat Prob Lett 6:21–30, 1987) etc. In this paper, we obtain some new estimators for μ. The case when the variance σ ² is unknown is also studied and various estimators for μ are proposed. Risk performance of all estimators is numerically compared for both the cases when σ ² may be known and unknown.     

Modelling response error in school effectiveness research

Statistical modelling of school effectiveness in educational research is considered. Variance component models are generally accepted for the analysis of such studies. A shortcoming is that outcome variables are still treated as measured without an error. Unreliable variables produce biases in the estimates of the other model parameters. The variability of the relationships across schools and the effects of schools on students' outcomes differ substantially when taking the measurement error in the dependent variables of the variance component models into account. The random effects model can be extended to handle measurement error using a response model, leading to a random effects item response theory model. This extended random effects model is in particular suitable when subjects are measured repeatedly on the same outcome at several points in time.     

The effects of sampling on measures of association between variables based on sample means

Many studies that involve people's perceptions or behaviors focus on aggregate rather than individual responses. For example, variables describing public perceptions for some set of events may be represented as mean scores for each event. Event mean scores then become the unit of analysis for each variable. The variance of these mean scores for a variable is not only a function of the variation among the events themselves, but is also due to the variation among respondents and their possible responses. This is also the case for the covariances between variables based on event mean scores. In many contexts the variance and covariance components attributable to the sampling of respondents and their responses may be large; these components can be described as measurement error. In this paper we show how to estimate variances and covariances of aggregate variables that are free of these sources of measurement error. We also present a measure of reliability for the event means and examine the effect of the number of respondents on these spurious components. To illustrate how these estimates are computed, forty-two respondents were asked to rate forty events on seven risk perception variables. Computing the variances and covariances for these variables based on event means resulted in relatively large components attributable to measurement error. A demonstration is given of how this error is removed and the resulting effect on our estimates.     

Locally minimum variance unbiased estimator in a discontinuous density function

The exact forms of the locally minimum variance unbiased estimators and their variances are given in the case of a discontinuous density function.     

Asymptotic distributions of impulse response functions in short panel vector autoregressions

This paper establishes the asymptotic distributions of the impulse response functions in panel vector autoregressions with a fixed time dimension. It also proves the asymptotic validity of a bootstrap approximation to their sampling distributions. The autoregressive parameters are estimated using the GMM estimators based on the first differenced equations and the error variance is estimated using an extended analysis-of-variance type estimator. Contrary to the time series setting, we find that the GMM estimator of the autoregressive coefficients is not asymptotically independent of the error variance estimator. The asymptotic dependence calls for variance correction for the orthogonalized impulse response functions. Simulation results show that the variance correction improves the coverage accuracy of both the asymptotic confidence band and the studentized bootstrap confidence band for the orthogonalized impulse response functions.     

Kostenoptimale Prüfpläne für ein quantitatives Merkmal

Summary A sample inspection plan is said to be optimal in the sense of the minimax regret principle, if it minimizes the difference between the expected total costs and the unavoidable costs. The results of this article can be used to calculate such sample inspection plans for a quantitative quality control with one-sided tolerance limits and known or unknown variance of the test variate. As an example of practical importance the case of a normal variate with unknown variance is considered. Formulae are given to estimate the error that arises if the assumed distribution of the test variate differs from the actual distribution.     

Reparameterized inverse gamma regression models with varying precision

In this article, we propose a mean linear regression model where the response variable is inverse gamma distributed using a new parameterization of this distribution that is indexed by mean and precision parameters. The main advantage of our new parametrization is the straightforward interpretation of the regression coefficients in terms of the expectation of the positive response variable, as usual in the context of generalized linear models. The variance function of the proposed model has a quadratic form. The inverse gamma distribution is a member of the exponential family of distributions and has some distributions commonly used for parametric models in survival analysis as special cases. We compare the proposed model to several alternatives and illustrate its advantages and usefulness. With a generalized linear model approach that takes advantage of exponential family properties, we discuss model estimation (by maximum likelihood), black further inferential quantities and diagnostic tools. A Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples with a discussion of the obtained results. A real application using minerals data set collected by Department of Mines of the University of Atacama, Chile, is considered to demonstrate the practical potential of the proposed model.     

On the Generalized Bootstrap for Sample Surveys with Special Attention to Poisson Sampling

We study the generalized bootstrap technique under general sampling designs. We focus mainly on bootstrap variance estimation but we also investigate the empirical properties of bootstrap confidence intervals obtained using the percentile method. Generalized bootstrap consists of randomly generating bootstrap weights so that the first two (or more) design moments of the sampling error are tracked by the corresponding bootstrap moments. Most bootstrap methods in the literature can be viewed as special cases. We discuss issues such as the choice of the distribution used to generate bootstrap weights, the choice of the number of bootstrap replicates, and the potential occurrence of negative bootstrap weights. We first describe the generalized bootstrap for the linear Horvitz‐Thompson estimator and then consider non‐linear estimators such as those defined through estimating equations. We also develop two ways of bootstrapping the generalized regression estimator of a population total. We study in greater depth the case of Poisson sampling, which is often used to select samples in Price Index surveys conducted by national statistical agencies around the world. For Poisson sampling, we consider a pseudo‐population approach and show that the resulting bootstrap weights capture the first three design moments of the sampling error. A simulation study and an example with real survey data are used to illustrate the theory.     

Estimation procedures for a family of density functions representing various life-testing models

A family of density functions is considered which contains several life-testing models as specific cases. Uniformly minimum variance unbiased estimators are obtained for the positive and negative powers of the parameter, moments and reliability function. These general results provide the estimators for the specific models.     

Randomized response: Estimating mean square errors of linear estimators and finding optimal unbiased strategies

Summary General procedures are described to generate quantitative randomized response (RR) required to estimate the finite population total of a sensitive variable. Permitting sample selection with arbitrary probabilities a formula for the mean square error (MSE) of a linear estimator of total based on RR is noted indicating the simple modification over one that might be based on direct response (DR) if the latter were available. A general formula for an unbiased estimator of the MSE is presented. A simple approximation is proposed in case the RR ratio estimator is employed based on a simple random sample (SRS) taken without replacement (WOR). Among sampling strategies employing unbiased but not necessarily linear estimators based on RR, certain optimal ones are identified under two alternative models analogously to well-known counterparts based on DR, if available. Unlike Warner’s (1965) treatment of categorical RR we consider quantitative RR here.     

On estimating nonnegative definite quadratic forms

It is often required to estimate a quadratic form in survey sampling, especially when one has to estimate the mean squared error of a linear estimator of the population total. In this note we consider the problem of obtaining uniformly nonnegative quadratic unbiased estimators for nonnegative definite quadratic forms. The estimators considered here are necessarily quadratic. Received January 1997