Model‐Based Non‐parametric Variance Estimation for Systematic Sampling

Abstract. Systematic sampling is frequently used in surveys, because of its ease of implementation and its design efficiency. An important drawback of systematic sampling, however, is that no direct estimator of the design variance is available. We describe a new estimator of the model‐based expectation of the design variance, under a non‐parametric model for the population. The non‐parametric model is sufficiently flexible that it can be expected to hold at least approximately in many situations with continuous auxiliary variables observed at the population level. We prove the model consistency of the estimator for both the anticipated variance and the design variance under a non‐parametric model with a univariate covariate. The broad applicability of the approach is demonstrated on a dataset from a forestry survey.     

Functional convergence of quantile-type frontiers with application to parametric approximations

Nonparametric estimators of the upper boundary of the support of a multivariate distribution are very appealing because they rely on very few assumptions. But in productivity and efficiency analysis, this upper boundary is a production (or a cost) frontier and a parametric form for it allows for a richer economic interpretation of the production process under analysis. On the other hand, most of the parametric approaches rely on often too restrictive assumptions on the stochastic part of the model and are based on standard regression techniques fitting the shape of the center of the cloud of points rather than its boundary. To overcome these limitations, Florens and Simar [2005. Parametric approximations of nonparametric frontiers. J. Econometrics 124 (1), 91–116] propose a two-stage approach which tries to capture the shape of the cloud of points near its frontier by providing parametric approximations of a nonparametric frontier. In this paper we propose an alternative method using the nonparametric quantile-type frontiers introduced in Aragon, Daouia and Thomas-Agnan [2005. Nonparametric frontier estimation: a conditional quantile-based approach. Econometric Theory 21, 358–389] for the nonparametric part of our model. These quantile-type frontiers have the superiority of being more robust to extremes. Our main result concerns the functional convergence of the quantile-type frontier process. Then we provide convergence and asymptotic normality of the resulting estimators of the parametric approximation. The approach is illustrated through simulated and real data sets.     

Generalised quasi‐likelihood inference in a semi‐parametric binary dynamic mixed logit model

There exists a recent study where dynamic mixed‐effects regression models for count data have been extended to a semi‐parametric context. However, when one deals with other discrete data such as binary responses, the results based on count data models are not directly applicable. In this paper, we therefore begin with existing binary dynamic mixed models and generalise them to the semi‐parametric context. For inference, we use a new semi‐parametric conditional quasi‐likelihood (SCQL) approach for the estimation of the non‐parametric function involved in the semi‐parametric model, and a semi‐parametric generalised quasi‐likelihood (SGQL) approach for the estimation of the main regression, dynamic dependence and random effects variance parameters. A semi‐parametric maximum likelihood (SML) approach is also used as a comparison to the SGQL approach. The properties of the estimators are examined both asymptotically and empirically. More specifically, the consistency of the estimators is established and finite sample performances of the estimators are examined through an intensive simulation study.     

A sign based loss approach to model selection in nonparametric regression

In parametric regression models the sign of a coefficient often plays an important role in its interpretation. One possible approach to model selection in these situations is to consider a loss function that formulates prediction of the sign of a coefficient as a decision problem. Taking a Bayesian approach, we extend this idea of a sign based loss for selection to more complex situations. In generalized additive models we consider prediction of the sign of the derivative of an additive term at a set of predictors. Being able to predict the sign of the derivative at some point (that is, whether a term is increasing or decreasing) is one approach to selection of terms in additive modelling when interpretation is the main goal. For models with interactions, prediction of the sign of a higher order derivative can be used similarly. There are many advantages to our sign-based strategy for selection: one can work in a full or encompassing model without the need to specify priors on a model space and without needing to specify priors on parameters in submodels. Also, avoiding a search over a large model space can simplify computation. We consider shrinkage prior specifications on smoothing parameters that allow for good predictive performance in models with large numbers of terms without the need for selection, and a frequentist calibration of the parameter in our sign-based loss function when it is desired to control a false selection rate for interpretation.     

ASYMPTOTIC PROPERTIES OF AN OPTIMAL ESTIMATING FUNCTION APPROACH TO THE ANALYSIS OF MARK RECAPTURE DATA

ABSTRACT

A drawback of non parametric estimators of the size of a closed population in the presence of heterogeneous capture probabilities has been their lack of analytic tractability. Here we show that the martingale estimating function/sample coverage approach to estimating the size of a closed population with heterogeneous capture probabilities is mathematically tractable and develop its large sample properties.     

A new behavioural model for fertility schedules

Modelling age-specific fertility rates is of great importance in demography because of their influence on population growth. Although we have a variety of fertility models in the demographic literature, most of them do not have any demographic interpretation for their parameters. It is generally expected that models with behavioural interpretation are more universal than those without any interpretation. Even though the famous Gompertz model has some behavioural interpretation it suffers from other drawbacks. In the present work, we propose a new fertility model, which has its genesis in the generalization of logistic law. The proposed model has good behavioural interpretation, alongside having nice parameter interpretations.     

A taxonomy of mixing and outcome distributions based on conjugacy and bridging

Abstract

The generalized linear mixed model (GLMM) is commonly used for the analysis of hierarchical non Gaussian data. It combines an exponential family model formulation with normally distributed random effects. A drawback is the difficulty of deriving convenient marginal mean functions with straightforward parametric interpretations. Several solutions have been proposed, including the marginalized multilevel model (directly formulating the marginal mean, together with a hierarchical association structure) and the bridging approach (choosing the random-effects distribution such that marginal and hierarchical mean functions share functional forms). Another approach, useful in both a Bayesian and a maximum-likelihood setting, is to choose a random-effects distribution that is conjugate to the outcome distribution. In this paper, we contrast the bridging and conjugate approaches. For binary outcomes, using characteristic functions and cumulant generating functions, it is shown that the bridge distribution is unique. Self-bridging is introduced as the situation in which the outcome and random-effects distributions are the same. It is shown that only the Gaussian and degenerate distributions have well-defined cumulant generating functions for which self-bridging holds.     

Estimation of the Moment Generating Function

A number of statistical problems use the moment generating function (mgf) for purposes other than determining the moments of a distribution. If the distribution is not completely specified, then the mgf must be estimated from available data. The empirical mgf makes no assumptions concerning the underlying distribution except for the existence of the mgf. In contrast to the nonparametric approach provided by the empirical mgf, alternative estimators can be formed based on an assumed parametric model. Comparison of these approaches is considered for two parametric models; the normal and a one parameter gamma. Comparison criteria are efficiency and empirical confidence interval coverage. In general the parametric estimators outperform the empirical mgf when the model is correct. The comparisons are extended to underlying models which are two component mixtures from the distributional family assumed by the parametric estimators. Under the mixture models the superiority of the parametric estimator depends upon the model, value of the argument of the mgf, and the comparison criterion. The empirical mgf is the better estimator in some cases.     

Sequential Designs for Binary Data with the Purpose to Maximize the Probability of Response

Two kinds of sequential designs are proposed for finding the point that maximizes the probability of response assuming a binary response variable and a quadratic logistic regression model. One is a parametric optimal design approach, and the other one is a nonparametric stochastic approximation approach. The suggested sequential designs are evaluated and compared in a simulation study. In summary, the parametric approach performed very well whereas its competitor failed in some cases.     

An empirical likelihood goodness-of-fit test for time series

Summary. Standard goodness-of-fit tests for a parametric regression model against a series of nonparametric alternatives are based on residuals arising from a fitted model. When a parametric regression model is compared with a nonparametric model, goodness-of-fit testing can be naturally approached by evaluating the likelihood of the parametric model within a nonparametric framework. We employ the empirical likelihood for an α -mixing process to formulate a test statistic that measures the goodness of fit of a parametric regression model. The technique is based on a comparison with kernel smoothing estimators. The empirical likelihood formulation of the test has two attractive features. One is its automatic consideration of the variation that is associated with the nonparametric fit due to empirical likelihood's ability to Studentize internally. The other is that the asymptotic distribution of the test statistic is free of unknown parameters, avoiding plug-in estimation. We apply the test to a discretized diffusion model which has recently been considered in financial market analysis.     

The glm log-rank test:general linear modeling of log-rank scores as a method of analysis for survival data

The use of general linear modeling (GLM) procedures based on log-rank scores is proposed for the analysis of survival data and compared to standard survival analysis procedures. For the comparison of two groups, this approach performed similarly to the traditional log-rank test. In the case of more complicated designs - without ties in the survival times - the approach was only marginally less powerful than tests from proportional hazards models, and clearly less powerful than a likelihood ratio test for a fully parametric model; however, with ties in the survival time, the approach proved more powerful than tests from Cox's semi-parametric proportional hazards procedure. The method appears to provide a reasonably powerful alternative for the analysis of survival data, is easily used in complicated study designs, avoids (semi-)parametric assumptions, and is quite computationally easy and inexpensive to employ.     

Direct parametric inference for the cumulative incidence function

Summary.  In survival data that are collected from phase III clinical trials on breast cancer, a patient may experience more than one event, including recurrence of the original cancer, new primary cancer and death. Radiation oncologists are often interested in comparing patterns of local or regional recurrences alone as first events to identify a subgroup of patients who need to be treated by radiation therapy after surgery. The cumulative incidence function provides estimates of the cumulative probability of locoregional recurrences in the presence of other competing events. A simple version of the Gompertz distribution is proposed to parameterize the cumulative incidence function directly. The model interpretation for the cumulative incidence function is more natural than it is with the usual cause-specific hazard parameterization. Maximum likelihood analysis is used to estimate simultaneously parametric models for cumulative incidence functions of all causes. The parametric cumulative incidence approach is applied to a data set from the National Surgical Adjuvant Breast and Bowel Project and compared with analyses that are based on parametric cause-specific hazard models and nonparametric cumulative incidence estimation.     

Finding local departures from a parametric model using nonparametric regression

Goodness-of-fit evaluation of a parametric regression model is often done through hypothesis testing, where the fit of the model of interest is compared statistically to that obtained under a broader class of models. Nonparametric regression models are frequently used as the latter type of model, because of their flexibility and wide applicability. To date, this type of tests has generally been performed globally, by comparing the parametric and nonparametric fits over the whole range of the data. However, in some instances it might be of interest to test for deviations from the parametric model that are localized to a subset of the data. In this case, a global test will have low power and hence can miss important local deviations. Alternatively, a naive testing approach that discards all observations outside the local interval will suffer from reduced sample size and potential overfitting. We therefore propose a new local goodness-of-fit test for parametric regression models that can be applied to a subset of the data but relies on global model fits, and propose a bootstrap-based approach for obtaining the distribution of the test statistic. We compare the new approach with the global and the naive tests, both theoretically and through simulations, and illustrate its practical behavior in an application. We find that the local test has a better ability to detect local deviations than the other two tests.     

Consistent fractional Bayes factor for nested normal linear models

In the Bayesian approach to parametric model comparison, the use of improper priors is problematic due to the indeterminacy of the resulting Bayes factor (BF). The need for developing automatic and robust methods for model comparison has led to the introduction of alternative BFs. Intrinsic Bayes factors (Berger and Pericchi, 1996a) and fractional Bayes factors (FBF) (O'Hagan, 1995) are two alternative strategies for default model selection. We show in this paper that the FBF can be inconsistent. To overcome this problem, we propose a generalization of the FBF approach that leads to the usual FBF or to some variants of it in some special cases. As an important problem, we consider and discuss this generalization for model selection in nested linear models.     

A new variability order based on tail-heaviness

In this paper, we introduce a new variability order that can be interpreted in terms of tail-heaviness which we will call the tail dispersive order. We provide the new definition, its interpretation and properties and the main characterization. We also study the relationship with other classical variability orders. Finally, we study the tail dispersive order in some classical parametric families and provide some applications in insurance and finance. We conclude with a numerical example applied to log returns distributions.     

On the grouped selection and model complexity of the adaptive elastic net

Lasso proved to be an extremely successful technique for simultaneous estimation and variable selection. However lasso has two major drawbacks. First, it does not enforce any grouping effect and secondly in some situation lasso solutions are inconsistent for variable selection. To overcome this inconsistency adaptive lasso is proposed where adaptive weights are used for penalizing different coefficients. Recently a doubly regularized technique namely elastic net is proposed which encourages grouping effect i.e. either selection or omission of the correlated variables together. However elastic net is also inconsistent. In this paper we study adaptive elastic net which does not have this drawback. In this article we specially focus on the grouped selection property of adaptive elastic net along with its model selection complexity. We also shed some light on the bias-variance tradeoff of different regularization methods including adaptive elastic net. An efficient algorithm was proposed in the line of LARS-EN, which is then illustrated with simulated as well as real life data examples.     

Comparing Two Population Means and Variances: A Parametric Robust Way

Abstract

This article introduces a parametric robust way of comparing two population means and two population variances. With large samples the comparison of two means, under model misspecification, is lesser a problem, for, the validity of inference is protected by the central limit theorem. However, the assumption of normality is generally required, so that the inference for the ratio of two variances can be carried out by the familiar F statistic. A parametric robust approach that is insensitive to the distributional assumption will be proposed here. More specifically, it will be demonstrated that the normal likelihood function can be adjusted for asymptotically valid inferences for all underlying distributions with finite fourth moments. The normal likelihood function, on the other hand, is itself robust for the comparison of two means so that no adjustment is needed.     

Discrete Nonparametric Kernel and Parametric Methods for the Modeling of Pavement Deterioration

This article is concerned with one discrete nonparametric kernel and two parametric regression approaches for providing the evolution law of pavement deterioration. The first parametric approach is a survival data analysis method; and the second is a nonlinear mixed-effects model. The nonparametric approach consists of a regression estimator using the discrete associated kernels. Some asymptotic properties of the discrete nonparametric kernel estimator are shown as, in particular, its almost sure consistency. Moreover, two data-driven bandwidth selection methods are also given, with a new theoretical explicit expression of optimal bandwidth provided for this nonparametric estimator. A comparative simulation study is realized with an application of bootstrap methods to a measure of statistical accuracy.     

Specification of household engel curves by nonparametric regression

This paper demonstrates the usefulness of nonparametric regression analysis for functional specfication of houshold Engel curves.

After a brief review in section 2 of the literature on demand functions and equivalence scales and the functional specifications used, we first discuss in section 3 the issues of using income versus total expenditure, the origin and nature of the error terms in the light of utility theroy, and the interpretation of empirical demand functions. we shall reach the unorthodox view that household demand functions should be interpreted as conditional expectations relative to prices, household composition and either income or the conditional expectation of total expenditure (rather that total expenditure itself), where the latter conditional expectation is taken relative to income, prices and household composition. these two forms appear to be equivalent. this result also solves the simultaneity problem: the error variance matrix is no longer singular. Moreover, the errors are in general heteroskedastic.

In section 4 we discuss the model and the data, and in section 5 we review the nonparametric kernal regression approach.

In section 6 we derive the functional form of our household engel curves from nonparametric regression results, using the 1980 budget survey for the netherlands, in order to avoid model misspecification. thus the modl is derived directly from the data, without restricting its functional form. the nonparametric regression results are then translated to suitable parametric functional specifications, i.e., we choose parametric functional forms in accordance with the nanparametric regression results. these parametric specification are estimated by least squares, and various parameter restrictions are tested in order to simplify the models. this yields very simple final specifications of the household engel curves involved, namely linear functions of income and the number of children in two age groups.     

Inferences in dynamic logit models in semi-parametric setup for repeated binary data

Binary dynamic fixed and mixed logit models are extensively studied in the literature. These models are developed to examine the effects of certain fixed covariates through a parametric regression function as a part of the models. However, there are situations where one may like to consider more covariates in the model but their direct effect is not of interest. In this paper we propose a generalization of the existing binary dynamic logit (BDL) models to the semi-parametric longitudinal setup to address this issue of additional covariates. The regression function involved in such a semi-parametric BDL model contains (i) a parametric linear regression function in some primary covariates, and (ii) a non-parametric function in certain secondary covariates. We use a simple semi-parametric conditional quasi-likelihood approach for consistent estimation of the non-parametric function, and a semi-parametric likelihood approach for the joint estimation of the main regression and dynamic dependence parameters of the model. The finite sample performance of the estimation approaches is examined through a simulation study. The asymptotic properties of the estimators are also discussed. The proposed model and the estimation approaches are illustrated by reanalysing a longitudinal infectious disease data.