Using response probability and ratio imputation in the estimation under callbacks

Although the response model has been frequently applied to nonresponse weighting adjustment or imputation, the estimation under callbacks has been relatively underdeveloped in the response model. We propose an estimator under callbacks using both the response probability and the ratio imputation and a replication variance estimator of the estimator. We also study the estimation of the response probability. A simulation study illustrates our technique.     

Likelihood methods in randomized trials with noncompliance and subsequent nonresponse

This article considers likelihood methods for estimating the causal effect of treatment assignment for a two-armed randomized trial assuming all-or-none treatment noncompliance and allowing for subsequent nonresponse. We first derive the observed data likelihood function as a closed form expression of the parameter given the observed data where both response and compliance state are treated as variables with missing values. Then we describe an iterative procedure which maximizes the observed data likelihood function directly to compute a maximum likelihood estimator (MLE) of the causal effect of treatment assignment. Closed form expressions at each iterative step are provided. Finally we compare the MLE with an alternative estimator where the probability distribution of the compliance state is estimated independent of the response and its missingness mechanism. Our work indicates that direct maximum likelihood inference is straightforward for this problem. Extensive simulation studies are provided to examine the finite sample performance of the proposed methods.     

Imputation using response probability

The authors propose a new ratio imputation method using response probability. Their estimator can be justified either under the response model or under the imputation model; it is thus doubly protected against the failure of either of these models. The authors also propose a variance estimator that can be justified under the two models. Their methodology is applicable whether the response probabilities are estimated or known. A small simulation study illustrates their technique.     

New generalized regression estimator in the presence of non response under unequal probability sampling

In this paper, we propose a new generalized regression estimator for the problem of estimating the population total using unequal probability sampling without replacement. A modified automated linearization approach is applied in order to transform the proposed estimator to estimate variance of population total. The variance and estimated value of the variance of the proposed estimator is investigated under a reverse framework assuming that the sampling fraction is negligible and there are equal response probabilities for all units. We prove that the proposed estimator is an asymptotically unbiased estimator and that it does not require a known or estimated response probability to function.     

Estimation of finite population kurtosis under two-phase sampling for nonresponse

In this paper an estimator of finite population kurtosis computed under the two-phase sampling for nonresponse is proposed. The formulas characterizing its asymptotic properties are derived using Taylor linearization technique for the general situation of arbitrary sampling designs in both phases and stochastic nonresponse represented by arbitrary response distribution. An important special case of simple random sampling without replacement and deterministic nonresponse is also considered.     

Semiparametric efficient estimation for the auxiliary outcome problem with the conditional mean model

The authors consider semiparametric efficient estimation of parameters in the conditional mean model for a simple incomplete data structure in which the outcome of interest is observed only for a random subset of subjects but covariates and surrogate (auxiliary) outcomes are observed for all. They use optimal estimating function theory to derive the semiparametric efficient score in closed form. They show that when covariates and auxiliary outcomes are discrete, a Horvitz‐Thompson type estimator with empirically estimated weights is semiparametric efficient. The authors give simulation studies validating the finite‐sample behaviour of the semiparametric efficient estimator and its asymptotic variance; they demonstrate the efficiency of the estimator in realistic settings.     

A kernel smoothing method of adjusting for unit non‐response in sample surveys

Non‐response is a common problem in survey sampling and this phenomenon can only be ignored at the risk of invalidating inferences from a survey. In order to adjust for unit non‐response, the authors propose a weighting method in which kernel regression is used to estimate the response probabilities. They show that the adjusted estimator is consistent and they derive its asymptotic distribution. They also suggest a means of estimating its variance through a replication‐based technique. Furthermore, a Monte Carlo study allows them to illustrate the properties of the non‐response adjustment and its variance estimator.     

Robust quasi-randomization-based estimation with ensemble learning for missing data

Missing data analysis requires assumptions about an outcome model or a response probability model to adjust for potential bias due to nonresponse. Doubly robust (DR) estimators are consistent if at least one of the models is correctly specified. Multiply robust (MR) estimators extend DR estimators by allowing for multiple models for both the outcome and/or response probability models and are consistent if at least one of the multiple models is correctly specified. We propose a robust quasi-randomization-based model approach to bring more protection against model misspecification than the existing DR and MR estimators, where any multiple semiparametric, nonparametric or machine learning models can be used for the outcome variable. The proposed estimator achieves unbiasedness by using a subsampling Rao–Blackwell method, given cell-homogenous response, regardless of any working models for the outcome. An unbiased variance estimation formula is proposed, which does not use any replicate jackknife or bootstrap methods. A simulation study shows that our proposed method outperforms the existing multiply robust estimators.     

Nonresponse Assessment of a Consumer Price Index

A household budget survey often suffers from a high nonresponse rate and a selective response. The bias that may be introduced in the estimation of budget shares because of this nonresponse can affect the estimate of a consumer price index, which is a weighted sum of partial price index numbers (weighted with the estimated budget shares). The bias is especially important when related to the standard error of the estimate. Because of the impossibility of subsampling nonrespondents to the budget survey, no exact information on the bias can be obtained. To evaluate the nonresponse bias, bounds for this bias are calculated using linear programming methods for several assumptions. The impact on a price index of a high nonresponse rate among people with a high income can also be assessed by using the elasticity with respect to total expenditure. Attention is also given to the possible nonresponse bias in a time series of price index numbers. The possible nonresponse bias is much larger than the standard error of the estimate.     

Box‐Cox transformations in linear models: Large sample theory and tests of normality

The authors provide a rigorous large sample theory for linear models whose response variable has been subjected to the Box‐Cox transformation. They provide a continuous asymptotic approximation to the distribution of estimators of natural parameters of the model. They show, in particular, that the maximum likelihood estimator of the ratio of slope to residual standard deviation is consistent and relatively stable. The authors further show the importance for inference of normality of the errors and give tests for normality based on the estimated residuals. For non‐normal errors, they give adjustments to the log‐likelihood and to asymptotic standard errors.     

Response probability estimation

This paper extends the ideas in Giommi (Proc. 45th Session of the Internat. Statistical Institute, Vol. 2 (1985) 577–578; Techniques d'enquête 13(2) (1987) 137–144) and, in Särndal and Swenson (Bull. Int. Statist. Inst. 15(2) (1985) 1–16; Int. Statist. Rev. 55(1987) 279–294). Given the parallel between a ‘three-phase sampling’ and a ‘sampling with subsequent unit and item nonresponse’, we apply results from three-phase sampling theory to nonresponse situation. To handle the practical problem of unknown distributions at the second and the third phases of selection (the response mechanisms) in the nonresponse case, we use two approaches of response probability estimation: response homogeneity groups (RHG) model (Särndal and Swenson, 1985, 1987) and the nonparametric estimation (Giommi, 1985, 1987). To motivate the three-phase selection, imputation procedures for item nonresponse are used with the RHG model for unit nonresponse. By means of a Monte Carlo study, we find that the regression-type estimators are the most precise of those studied under the two approaches of response probability estimation in terms of lower bias, mean square error and variance; variance estimator close to the true variance and achieved coverage rates closer to the nominal levels. The simulation study shows how poor the variance estimators are under the single imputation approach currently used to handle the problem of missing values.     

Probability in a Contested Election

Elementary probability models can sometimes be used to analyze the results of an election where irregularities have occurred. In this article we give an illustrative example by considering the 1975 mayoral election in the city of Flint, Michigan. True vote is estimated with an estimator which is used in randomized response models.     

On the bootstrap in cube root asymptotics

The authors study the application of the bootstrap to a class of estimators which converge at a nonstandard rate to a nonstandard distribution. They provide a theoretical framework to study its asymptotic behaviour. A simulation study shows that in the case of an estimator such as Chernoff's estimator of the mode, usually the basic bootstrap confidence intervals drastically undercover while the percentile bootstrap intervals overcover. This is a rare instance where basic and percentile confidence intervals, which have exactly the same length, behave in a very different way. In the case of Chernoff's estimator, if the distribution is symmetric, it is possible to bootstrap from a smooth symmetric estimator of the distribution for which the basic bootstrap confidence intervals will have the claimed coverage probability while the percentile bootstrap interval will have an asymptotic coverage of 1!     

Estimation of an indicator of the representativeness of survey response

Nonresponse is a major source of estimation error in sample surveys. The response rate is widely used to measure survey quality associated with nonresponse, but is inadequate as an indicator because of its limited relation with nonresponse bias. Schouten et al. (2009) proposed an alternative indicator, which they refer to as an indicator of representativeness or R-indicator. This indicator measures the variability of the probabilities of response for units in the population. This paper develops methods for the estimation of this R-indicator assuming that values of a set of auxiliary variables are observed for both respondents and nonrespondents. We propose bias adjustments to the point estimator proposed by Schouten et al. (2009) and demonstrate the effectiveness of this adjustment in a simulation study where it is shown that the method is valid, especially for smaller sample sizes. We also propose linearization variance estimators which avoid the need for computer-intensive replication methods and show good coverage in the simulation study even when models are not fully specified. The use of the proposed procedures is also illustrated in an application to two business surveys at Statistics Netherlands.     

Projection estimators of Pickands dependence functions

The authors consider the construction of intrinsic estimators for the Pickands dependence function of an extreme‐value copula. They show how an arbitrary initial estimator can be modified to satisfy the required shape constraints. Their solution consists in projecting this estimator in the space of Pickands functions, which forms a closed and convex subset of a Hilbert space. As the solution is not explicit, they replace this functional parameter space by a sieve of finite‐dimensional subsets. They establish the asymptotic distribution of the projection estimator and its finite‐dimensional approximations, from which they conclude that the projected estimator is at least as efficient as the initial one.     

Inference for domains under imputation for missing survey data

The authors study the estimation of domain totals and means under survey‐weighted regression imputation for missing items. They use two different approaches to inference: (i) design‐based with uniform response within classes; (ii) model‐assisted with ignorable response and an imputation model. They show that the imputed domain estimators are biased under (i) but approximately unbiased under (ii). They obtain a bias‐adjusted estimator that is approximately unbiased under (i) or (ii). They also derive linearization variance estimators. They report the results of a simulation study on the bias ratio and efficiency of alternative estimators, including a complete case estimator that requires the knowledge of response indicators.     

Estimation of a residual distribution with small numbers of repeated measurements

The authors consider the estimation of a residual distribution for different measurement problems with a common measurement error process. The problem is motivated by issues arising in the analysis of gene expression data but should have application in other similar settings. It is implicitly assumed throughout that there are large numbers of measurements but small numbers of repeated measurements. As a consequence, the distribution of the estimated residuals is a biased estimate of the residual distribution. The authors present two methods for the estimation of the residual distribution with some restriction on the form of the distribution. They give an upper bound for the rate of convergence for an estimator based on the characteristic function and compare its performance with that of another estimator with simulations.     

Consistent maximum likelihood estimation of a unimodal density using shape restrictions

It is well known that the unimodal maximum likelihood estimator of a density is consistent everywhere but at the mode. The authors review various ways to solve this problem and propose a new estimator that is concave over an interval containing the mode; this interval may be chosen by the user or through an algorithm. The authors show how to implement their solution and compare it to other approaches through simulations. They show that the new estimator is consistent everywhere and determine its rate of convergence in the Hellinger metric.     

Variance estimation when donor imputation is used to fill in missing values

Donor imputation is frequently used in surveys. However, very few variance estimation methods that take into account donor imputation have been developed in the literature. This is particularly true for surveys with high sampling fractions using nearest donor imputation, often called nearest‐neighbour imputation. In this paper, the authors develop a variance estimator for donor imputation based on the assumption that the imputed estimator of a domain total is approximately unbiased under an imputation model; that is, a model for the variable requiring imputation. Their variance estimator is valid, irrespective of the magnitude of the sampling fractions and the complexity of the donor imputation method, provided that the imputation model mean and variance are accurately estimated. They evaluate its performance in a simulation study and show that nonparametric estimation of the model mean and variance via smoothing splines brings robustness with respect to imputation model misspecifications. They also apply their variance estimator to real survey data when nearest‐neighbour imputation has been used to fill in the missing values. The Canadian Journal of Statistics 37: 400–416; 2009 © 2009 Statistical Society of Canada     

Nonparametric estimation of copula functions for dependence modelling

Copulas characterize the dependence among components of random vectors. Unlike marginal and joint distributions, which are directly observable, the copula of a random vector is a hidden dependence structure that links the joint distribution with its margins. Choosing a parametric copula model is thus a nontrivial task but it can be facilitated by relying on a nonparametric estimator. Here the authors propose a kernel estimator of the copula that is mean square consistent everywhere on the support. They determine the bias and variance of this estimator. They also study the effects of kernel smoothing on copula estimation. They then propose a smoothing bandwidth selection rule based on the derived bias and variance. After confirming their theoretical findings through simulations, they use their kernel estimator to formulate a goodness-of-fit test for parametric copula models.