Subsampling realised kernels

In a recent paper we have introduced the class of realised kernel estimators of the increments of quadratic variation in the presence of noise. We showed that this estimator is consistent and derived its limit distribution under various assumptions on the kernel weights. In this paper we extend our analysis, looking at the class of subsampled realised kernels and we derive the limit theory for this class of estimators. We find that subsampling is highly advantageous for estimators based on discontinuous kernels, such as the truncated kernel. For kinked kernels, such as the Bartlett kernel, we show that subsampling is impotent, in the sense that subsampling has no effect on the asymptotic distribution. Perhaps surprisingly, for the efficient smooth kernels, such as the Parzen kernel, we show that subsampling is harmful as it increases the asymptotic variance. We also study the performance of subsampled realised kernels in simulations and in empirical work.     

Variance estimation for unequal probability sampling

In this paper, we discuss the optimality of the variance estimator of the Horvitz-Thompson estimator proposed by Kott (1988) in the class of model-unbiased quadratic estimators. We also propose some improved estimators over Kott's estimator in the class of general quadratic estimators. Received: February 1999     

Minimax estimators of a normal variance

In the estimation problem of unknown variance of a multivariate normal distribution, a new class of minimax estimators is obtained. It is noted that a sequence of estimators in our class converges to the Stein's truncated estimator. Received: March 1998     

Calibration Estimation in Survey Sampling

Calibration estimation, where the sampling weights are adjusted to make certain estimators match known population totals, is commonly used in survey sampling. The generalized regression estimator is an example of a calibration estimator. Given the functional form of the calibration adjustment term, we establish the asymptotic equivalence between the functional-form calibration estimator and an instrumental variable calibration estimator where the instrumental variable is directly determined from the functional form in the calibration equation. Variance estimation based on linearization is discussed and applied to some recently proposed calibration estimators. The results are extended to the estimator that is a solution to the calibrated estimating equation. Results from a limited simulation study are presented.     

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.     

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.     

Simultaneous estimation of hazard rates of several exponential populations

Suppose independent random samples are drawn from k (2) populations with a common location parameter and unequal scale parameters. We consider the problem of estimating simultaneously the hazard rates of these populations. The analogues of the maximum likelihood (ML), uniformly minimum variance unbiased (UMVU) and the best scale equivariant (BSE) estimators for the one population case are improved using Rao‐Blackwellization. The improved version of the BSE estimator is shown to be the best among these estimators. Finally, a class of estimators that dominates this improved estimator is obtained using the differential inequality approach.     

Another look at the naive estimator in a regression model

We consider the linear regression model where only a particular linear function of the dependent variables is observed, Stahlecker and Schmidt (1987) proposed a naive least squares (LS) estimator for regression coefficients in such a case. In this note we represent their estimator as a general ridge estimator. This observation leads to a view different from the previous work and provides an easy way of obtaining many important properties of the naive LS estimator. Our approach also gives some insight into the relationship between the naive LS estimator and the generalized least squares estimator.     

Using Complex Surveys to Estimate the L1‐Median of a Functional Variable: Application to Electricity Load Curves

Mean profiles are widely used as indicators of the electricity consumption habits of customers. Currently, in Électricité De France (EDF), class load profiles are estimated using point‐wise mean profiles. Unfortunately, it is well known that the mean is highly sensitive to the presence of outliers, such as one or more consumers with unusually high‐levels of consumption. In this paper, we propose an alternative to the mean profile: the L ₁ ‐ median profile which is more robust. When dealing with large data sets of functional data (load curves for example), survey sampling approaches are useful for estimating the median profile avoiding storing the whole data. We propose here several sampling strategies and estimators to estimate the median trajectory. A comparison between them is illustrated by means of a test population. We develop a stratification based on the linearized variable which substantially improves the accuracy of the estimator compared to simple random sampling without replacement. We suggest also an improved estimator that takes into account auxiliary information. Some potential areas for future research are also highlighted.     

Distributional and Inferential Properties of the Estimated Precision Cp Based on Multiple Samples

Process precision index C_p has been widely used in the manufacturing industry to provide numerical measures on process potential. Pearn et al. (1998) considered an unbiased estimator of C_p for one single sample. They showed that the unbiased estimator is the UMVUE. They also proposed an efficient test for C_p based on one single sample, and showed that the test is the UMP test. In this paper, we consider an unbiased estimator of C_p for multiple samples. We show that the unbiased estimator is the UMVUE of C_p, which is asymptotically efficient. We also consider an efficient test for C_p, and show that the test is the UMPtest for multiple samples. The practitioners can use the proposed test on theirin-plant applications to obtain reliable decisions.     

A Bayesian-like estimator of the process capability index Cpmk

Pearn et al. (1992) proposed the capability index C_pmk, and investigated the statistical properties of its natural estimator for stable normal processes with constant mean μ. Chen and Hsu (1995) showed that under general conditions the asymptotic distribution of is normal if μ≠m, and is a linear combination of the normal and the folded-normal distributions if μ=m, where m is the mid-point between the upper and the lower specification limits. In this paper, we consider a new estimator for stable processes under a different (more realistic) condition on process mean, namely, P (μ≥m)=p, 0≤p≤1. We obtain the exact distribution, the expected value, and the variance of under normality assumption. We show that for P (μ≥m)=0, or 1, the new estimator is the MLE of C_pmk, which is asymptotically efficient. In addition, we show that under general conditions is consistent and is asymptotically unbiased. We also show that the asymptotic distribution of is a mixture of two normal distributions. RID="*" ID="*"  The research was partially supported by National Science Council of the Republic of China (NSC-89-2213-E-346-003).     

Strong representation of the presmoothed quantile function estimator for censored data

We consider lifetime data subject to right random censorship. In this context, this paper deals with the topic of estimating the distribution function of the lifetime and the corresponding quantile function. As it has been shown that the classical Kaplan–Meier estimator of the distribution function can be improved by means of presmoothing ideas, we introduce a quantile function estimator via the presmoothed distribution function estimator studied by Cao et al. [Journal of Nonparametric statistics, Vol. 17 (2005) pp. 31–56.] The main result of this paper is an almost sure representation of this presmoothed estimator. As a consequence, its strong consistency and asymptotic normality are established. The performance of this new quantile estimator is analyzed in a simulation study and applied to a real data example.     

Cross-entropy estimation in technical efficiency analysis

Technical efficiency analysis is a fundamental tool to measure the performance of production activity. Recently, an increasing interest in the state-contingent approach has emerged in the literature although such interest has not yet been accompanied by an increase of empirical applications. This is largely due to the fact that empirical models with state-contingent production frontiers are usually ill-posed. In this work, a discussion on the role of the generalized cross-entropy estimator within the state-contingent production framework is presented. To the best of the authors’ knowledge, the example provided in this work is the first real-world empirical application on technical efficiency analysis with the state-contingent approach using the generalized cross-entropy estimator.     

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 of a survival curve with randomly censored data in the presence of a covariate

This paper deals with the estimation of a survival curve in models with random right censoring and dependent censoring mechanism. We consider a specific dependent censorship model in which conditional on a covariate, the survival and censoring times are assumed to be independent. We investigate asymptotic properties of a corrected version of a survival curve estimator introduced by Cheng (1989). In particular we show uniform strong consistency and weak convergence to a Gaussian process. Comparisons of this estimator with the well-known Kaplan-Meier-estimator are included. Finally, some examples illustrate how the estimator performs. Received: February 2000     

Model‐Assisted Estimation of a Spatial Population Mean

This paper deals with the estimation of the mean of a spatial population. Under a design‐based approach to inference, an estimator assisted by a penalized spline regression model is proposed and studied. Proof that the estimator is design‐consistent and has a normal limiting distribution is provided. A simulation study is carried out to investigate the performance of the new estimator and its variance estimator, in terms of relative bias, efficiency, and confidence interval coverage rate. The results show that gains in efficiency over standard estimators in classical sampling theory may be impressive.     

Small sample corrections for LTS and MCD

The least trimmed squares estimator and the minimum covariance determinant estimator [6] are frequently used robust estimators of regression and of location and scatter. Consistency factors can be computed for both methods to make the estimators consistent at the normal model. However, for small data sets these factors do not make the estimator unbiased. Based on simulation studies we therefore construct formulas which allow us to compute small sample correction factors for all sample sizes and dimensions without having to carry out any new simulations. We give some examples to illustrate the effect of the correction factor.     

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.     

Ridge regression revisited

In general ridge (GR) regression p ridge parameters have to be determined, whereas simple ridge regression requires the determination of only one parameter. In a recent textbook on linear regression, Jürgen Gross argues that this constitutes a major complication. However, as we show in this paper, the determination of these p parameters can fairly easily be done. Furthermore, we introduce a generalization of the GR estimator derived by Hemmerle and by Teekens and de Boer. This estimator, which is more conservative, performs better than the Hoerl and Kennard estimator in terms of a weighted quadratic loss criterion.     

Forecast-error-based estimation of forecast uncertainty when the horizon is increased

Past forecast errors are employed frequently in the estimation of the unconditional forecast uncertainty, and several institutions have increased their forecast horizons in recent times. This work addresses the question of how forecast-error-based estimation can be performed if there are very few errors available for the new forecast horizons. It extends the results of Knüppel (2014) in order to relax the condition on the data structure that is required for the SUR estimator to be independent of unknown quantities. It turns out that the SUR estimator of the forecast uncertainty, which estimates the forecast uncertainty for all horizons jointly, tends to deliver large efficiency gains relative to the OLS estimator (i.e., the sample mean of the squared forecast errors for each individual horizon) in the case of increased forecast horizons. The SUR estimator is applied to the forecast errors of the Bank of England, the US Survey of Professional Forecasters, and the FOMC.