Transformation and Smoothing in Sample Survey Data

Abstract. We consider model‐based prediction of a finite population total when a monotone transformation of the survey variable makes it appropriate to assume additive, homoscedastic errors. As the transformation to achieve this does not necessarily simultaneously produce an easily parameterized mean function, we assume only that the mean is a smooth function of the auxiliary variable and estimate it non‐parametrically. The back transformation of predictions obtained on the transformed scale introduces bias which we remove using smearing. We obtain an asymptotic expansion for the prediction error which shows that prediction bias is asymptotically negligible and the prediction mean‐squared error (MSE) using a non‐parametric model remains in the same order as when a parametric model is adopted. The expansion also shows the effect of smearing on the prediction MSE and can be used to compute the asymptotic prediction MSE. We propose a model‐based bootstrap estimate of the prediction MSE. The predictor produces competitive results in terms of bias and prediction MSE in a simulation study, and performs well on a population constructed from an Australian farm survey.     

Improved Prediction Intervals and Distribution Functions

Abstract.  The plug-in solution is usually not entirely adequate for computing prediction intervals, as their coverage probability may differ substantially from the nominal value. Prediction intervals with improved coverage probability can be defined by adjusting the plug-in ones, using rather complicated asymptotic procedures or suitable simulation techniques. Other approaches are based on the concept of predictive likelihood for a future random variable. The contribution of this paper is the definition of a relatively simple predictive distribution function giving improved prediction intervals. This distribution function is specified as a first-order unbiased modification of the plug-in predictive distribution function based on the constrained maximum likelihood estimator. Applications of the results to the Gaussian and the generalized extreme-value distributions are presented.     

PREDICTION FOR EXPONENTIAL LIFETIMES BASED ON STEP-STRESS TESTING

ABSTRACT

This paper presents methods for constructing prediction limits for a step-stress model in accelerated life testing. An exponential life distribution with a mean that is a log-linear function of stress, and a cumulative exposure model are assumed. Two prediction problems are discussed. One concerns the prediction of the life at a design stress, and the other concerns the prediction of a future life during the step-stress testing. Both predictions require the knowledge of some model parameters. When estimates for the model parameters are available, a calibration method based on simulations is proposed for correcting the prediction intervals (regions) obtained by treating the parameter estimates as the true parameter values. Finally, a numerical example is given to illustrate the prediction procedure.     

A note on tolerance regions for random vectors and best linear predictors

When a (p+q)-variate column vector (x′,y′)′ has a (p+q)-variate normal density with mean vector (μ₁,μ₂) and covariance matrix Ω, unknown, Schervish (1980) obtains prediction intervals for the linear functions of a future y, given x. He bases the prediction interval on the F-distribution. However, for a specified linear function the statistic to be used is Student's t, since the prediction intervals based on t are shorter than those based on F. Similar results hold for the multivariate linear regression model.     

Statistical Problem Classes and Their Links to Information Theory

We begin by recalling the tripartite division of statistical problems into three classes, M-closed, M-complete, and M-open and then reviewing the key ideas of introductory Shannon theory. Focusing on the related but distinct goals of model selection and prediction, we argue that different techniques for these two goals are appropriate for the three different problem classes. For M-closed problems we give relative entropy justification that the Bayes information criterion (BIC) is appropriate for model selection and that the Bayes model average is information optimal for prediction. For M-complete problems, we discuss the principle of maximum entropy and a way to use the rate distortion function to bypass the inaccessibility of the true distribution. For prediction in the M-complete class, there is little work done on information based model averaging so we discuss the Akaike information criterion (AIC) and its properties and variants.

For the M-open class, we argue that essentially only predictive criteria are suitable. Thus, as an analog to model selection, we present the key ideas of prediction along a string under a codelength criterion and propose a general form of this criterion. Since little work appears to have been done on information methods for general prediction in the M-open class of problems, we mention the field of information theoretic learning in certain general function spaces.     

M-estimation for functional linear regression

This paper studies M-estimation in functional linear regression in which the dependent variable is scalar while the covariate is a function. An estimator for the slope function is obtained based on the functional principal component basis. The global convergence rate of the M-estimator of unknown slope function is established. The convergence rate of the mean-squared prediction error for the proposed estimators is also established. Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed procedure. Finally, the proposed method is applied to analyze the Berkeley growth data.     

递归预测器神经网络在中国金融市场的实证研究

金融时间序列预测是金融理论领域的研究热点之一。以金融市场中普遍存在的弱混沌为基础,对递归预测器神经网络在中国金融市场的预测应用进行实证研究。在网络训练上,提出用遗传算法优化网络的阈值、权值以及激发函数的幅值和斜率。对国内股票、期货和黄金市场中几个有代表性的品种进行实证检验,计算了预测均方根误差（RMSE）和预测精度（PA）,并和两种典型的神经网络预测模型——BP神经网络、径向基函数神经网络做了比较,结果表明该模型有较好的预测效果。     

Higher Moments and Prediction‐Based Estimation for the COGARCH(1,1) Model

COGARCH models are continuous time versions of the well‐known GARCH models of financial returns. The first aim of this paper is to show how the method of prediction‐based estimating functions can be applied to draw statistical inference from observations of a COGARCH(1,1) model if the higher‐order structure of the process is clarified. A second aim of the paper is to provide recursive expressions for the joint moments of any fixed order of the process. Asymptotic results are given, and a simulation study shows that the method of prediction‐based estimating function outperforms the other available estimation methods.     

Expansions of the coverage probabilities of prediction region based on a shrinkage estimator

An expansion formula for the coverage probability of prediction region based on a shrinkage estimator proposed by Joshi [Joshi, V. M. (1967). Inadmissibility of the usual confidence sets for the mean of a multivariate normal population. Ann. Math. Statist., 38, 1868–1875.] is obtained. Its error bound is evaluated in terms of a function of an unknown parameter. Applying this result, three types of asymptotic expansions are derived. These expansions show inadmissibility of the usual prediction region.     

Design and analysis of computer experiments when the output is highly correlated over the input space

To build a predictor, the output of a deterministic computer model or “code” is often treated as a realization of a stochastic process indexed by the code's input variables. The authors consider an asymptotic form of the Gaussian correlation function for the stochastic process where the correlation tends to unity. They show that the limiting best linear unbiased predictor involves Lagrange interpolating polynomials; linear model terms are implicitly included. The authors then develop optimal designs based on minimizing the limiting integrated mean squared error of prediction. They show through several examples that these designs lead to good prediction accuracy.     

Finite nonparametric grach model for foreign exchange volatility

GARCH model has been commonly used to describe the volatility of foreign exchange returns, which typically depends on returns many lags before, While the GARCH model provides a simple geometric decaying structure for persistence in time, it restricts tiie impact of variables to Quadratic functions. A finite nonparametric GARCH model is proposed that allows the variables' impact to be a smooth function of any form. A direct local polynomial estimation method for this finite GARCH model is proposed based on results on proportional additive model, and is applied to the German Mark (DEM)/US Dollar (USD) daily returns data. Estimators uf both the decaying rate and the impact function are obtained. Diagnostics show satisfactory out-of-sampie prediction based on the proposed model, which helps to better understand the dynamics of foreign exchange volatility.     

Bayesian estimation and prediction intervals for a Rayleigh distribution under a conjugate prior

This paper is an effort to obtain Bayes estimators of Rayleigh parameter and its associated risk based on a conjugate prior (square root inverted gamma prior) with respect to both symmetric loss function (squared error loss), and asymmetric loss function (precautionary loss function). We also derive the highest posterior density (HPD) interval for the Rayleigh parameter as well as the HPD prediction intervals for a future observation from this distribution. An illustrative example to test how the Rayleigh distribution fits a real data set is presented. Finally, Monte Carlo simulations are performed to compare the performances of the Bayes estimates under different conditions.     

Prediction in trend-renewal processes for repairable systems

Some problems of point and interval prediction in a trend-renewal process (TRP) are considered. TRP’s, whose realizations depend on a renewal distribution as well as on a trend function, comprise the non-homogeneous Poisson and renewal processes and serve as useful reliability models for repairable systems. For these processes, some possible ideas and methods for constructing the predicted next failure time and the prediction interval for the next failure time are presented. A method of constructing the predictors is also presented in the case when the renewal distribution of a TRP is unknown (and consequently, the likelihood function of this process is unknown). Using the prediction methods proposed, simulations are conducted to compare the predicted times and prediction intervals for a TRP with completely unknown renewal distribution with the corresponding results for the TRP with a Weibull renewal distribution and power law type trend function. The prediction methods are also applied to some real data.     

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.     

Bayesian validation assessment of multivariate computational models

Multivariate model validation is a complex decision-making problem involving comparison of multiple correlated quantities, based upon the available information and prior knowledge. This paper presents a Bayesian risk-based decision method for validation assessment of multivariate predictive models under uncertainty. A generalized likelihood ratio is derived as a quantitative validation metric based on Bayes’ theorem and Gaussian distribution assumption of errors between validation data and model prediction. The multivariate model is then assessed based on the comparison of the likelihood ratio with a Bayesian decision threshold, a function of the decision costs and prior of each hypothesis. The probability density function of the likelihood ratio is constructed using the statistics of multiple response quantities and Monte Carlo simulation. The proposed methodology is implemented in the validation of a transient heat conduction model, using a multivariate data set from experiments. The Bayesian methodology provides a quantitative approach to facilitate rational decisions in multivariate model assessment under uncertainty.     

Comparison of designs for generalized linear models under model misspecification

The purpose of this article is to demonstrate the use of the quantile dispersion graphs (QDGs) approach for comparing candidate designs for generalized linear models in the presence of model misspecification in the linear predictor. The proposed design criterion is based on the mean-squared error of prediction which incorporates the prediction variance and the bias caused by fitting the wrong model. The method of kriging is used to estimate the unknown function assumed to be the cause of model misspecification. The QDGs approach is also useful in assessing the robustness of a given design to values of the unknown parameters in the linear predictor. Three numerical examples are presented to illustrate the application of the proposed methodology.     

Estimating prediction error for complex samples

With a growing interest in using non-representative samples to train prediction models for numerous outcomes it is necessary to account for the sampling design that gives rise to the data in order to assess the generalized predictive utility of a proposed prediction rule. After learning a prediction rule based on a non-uniform sample, it is of interest to estimate the rule's error rate when applied to unobserved members of the population. Efron (1986) proposed a general class of covariance penalty inflated prediction error estimators that assume the available training data are representative of the target population for which the prediction rule is to be applied. We extend Efron's estimator to the complex sample context by incorporating Horvitz–Thompson sampling weights and show that it is consistent for the true generalization error rate when applied to the underlying superpopulation. The resulting Horvitz–Thompson–Efron estimator is equivalent to dAIC, a recent extension of Akaike's information criteria to survey sampling data, but is more widely applicable. The proposed methodology is assessed with simulations and is applied to models predicting renal function obtained from the large-scale National Health and Nutrition Examination Study survey. The Canadian Journal of Statistics 48: 204–221; 2020 © 2019 Statistical Society of Canada     

A Bayesian approach for non-homogeneous gamma degradation processes

A Bayesian approach based on the Markov Chain Monte Carlo technique is proposed for the non-homogeneous gamma process with power-law shape function. Vague and informative priors, formalized on some quantities having a “physical” meaning, are provided. Point and interval estimation of process parameters and some functions thereof are developed, as well as prediction on some observable quantities that are useful in defining the maintenance strategy is proposed. Some useful approximations are derived for the conditional and unconditional mean and median of the residual life to reduce computational time. Finally, the proposed approach is applied to a real dataset.     

Adaptively transformed mixed‐model prediction of general finite‐population parameters

For estimating area‐specific parameters (quantities) in a finite population, a mixed‐model prediction approach is attractive. However, this approach strongly depends on the normality assumption of the response values, although we often encounter a non‐normal case in practice. In such a case, transforming observations to make them suitable for normality assumption is a useful tool, but the problem of selecting a suitable transformation still remains open. To overcome the difficulty, we here propose a new empirical best predicting method by using a parametric family of transformations to estimate a suitable transformation based on the data. We suggest a simple estimating method for transformation parameters based on the profile likelihood function, which achieves consistency under some conditions on transformation functions. For measuring the variability of point prediction, we construct an empirical Bayes confidence interval of the population parameter of interest. Through simulation studies, we investigate the numerical performance of the proposed methods. Finally, we apply the proposed method to synthetic income data in Spanish provinces in which the resulting estimates indicate that the commonly used log transformation would not be appropriate.     

Theory & Methods: Krige,Smooth, Both or Neither?

Both kriging and non-parametric regression smoothing can model a non-stationary regression function with spatially correlated errors. However comparisons have mainly been based on ordinary kriging and smoothing with uncorrelated errors. Ordinary kriging attributes smoothness of the response to spatial autocorrelation whereas non-parametric regression attributes trends to a smooth regression function. For spatial processes it is reasonable to suppose that the response is due to both trend and autocorrelation. This paper reviews methodology for non-parametric regression with autocorrelated errors which is a natural compromise between the two methods. Re-analysis of the one-dimensional stationary spatial data of Laslett (1994) and a clearly non-stationary time series demonstrates the rather surprising result that for these data, ordinary kriging outperforms more computationally intensive models including both universal kriging and correlated splines for spatial prediction. For estimating the regression function, non-parametric regression provides adaptive estimation, but the autocorrelation must be accounted for in selecting the smoothing parameter.