LOCAL LINEAR FORECASTS USING CUBIC SMOOTHING SPLINES

This paper shows how cubic smoothing splines fitted to univariate time series data can be used to obtain local linear forecasts. The approach is based on a stochastic state‐space model which allows the use of likelihoods for estimating the smoothing parameter, and which enables easy construction of prediction intervals. The paper shows that the model is a special case of an ARIMA(0, 2, 2) model; it provides a simple upper bound for the smoothing parameter to ensure an invertible model; and it demonstrates that the spline model is not a special case of Holt's local linear trend method. The paper compares the spline forecasts with Holt's forecasts and those obtained from the full ARIMA(0, 2, 2) model, showing that the restricted parameter space does not impair forecast performance. The advantage of this approach over a full ARIMA(0, 2, 2) model is that it gives a smooth trend estimate as well as a linear forecast function.     

Holt-Winters Forecasting: An Alternative Formulation Applied to UK Air Passenger Data

This paper provides a formulation for the additive Holt-Winters forecasting procedure that simplifies both obtaining maximum likelihood estimates of all unknowns, smoothing parameters and initial conditions, and the computation of point forecasts and reliable predictive intervals. The stochastic component of the model is introduced by means of additive, uncorrelated, homoscedastic and Normal errors, and then the joint distribution of the data vector, a multivariate Normal distribution, is obtained. In the case where a data transformation was used to improve the fit of the model, cumulative forecasts are obtained here using a Monte-Carlo approximation. This paper describes the method by applying it to the series of monthly total UK air passengers collected by the Civil Aviation Authority, a long time series from 1949 to the present day, and compares the resulting forecasts with those obtained in previous studies.     

Maximum likelihood estimates in the multivariate normal with patterned mean and covariance via the em algorithm

The maximum likelihood equations for a multivariate normal model with structured mean and structured covariance matrix may not have an explicit solution. In some cases the model's error term may be decomposed as the sum of two independent error terms, each having a patterned covariance matrix, such that if one of the unobservable error terms is artificially treated as "missing data", the EM algorithm can be used to compute the maximum likelihood estimates for the original problem. Some decompositions produce likelihood equations which do not have an explicit solution at each iteration of the EM algorithm, but within-iteration explicit solutions are shown for two general classes of models including covariance component models used for analysis of longitudinal data.     

Inference based on progressive Type I interval censored data from log-normal distribution

This article considers inference for the log-normal distribution based on progressive Type I interval censored data by both frequentist and Bayesian methods. First, the maximum likelihood estimates (MLEs) of the unknown model parameters are computed by expectation-maximization (EM) algorithm. The asymptotic standard errors (ASEs) of the MLEs are obtained by applying the missing information principle. Next, the Bayes’ estimates of the model parameters are obtained by Gibbs sampling method under both symmetric and asymmetric loss functions. The Gibbs sampling scheme is facilitated by adopting a similar data augmentation scheme as in EM algorithm. The performance of the MLEs and various Bayesian point estimates is judged via a simulation study. A real dataset is analyzed for the purpose of illustration.     

Robust skew-<Emphasis Type="Italic">t</Emphasis> factor analysis models for handling missing data

This paper presents a novel framework for maximum likelihood (ML) estimation in skew-t factor analysis (STFA) models in the presence of missing values or nonresponses. As a robust extension of the ordinary factor analysis model, the STFA model assumes a restricted version of the multivariate skew-t distribution for the latent factors and the unobservable errors to accommodate non-normal features such as asymmetry and heavy tails or outliers. An EM-type algorithm is developed to carry out ML estimation and imputation of missing values under a missing at random mechanism. The practical utility of the proposed methodology is illustrated through real and synthetic data examples.     

AN ALTERNATIVE PARAMETRIC APPROACH FOR DISCRETE MISSING DATA PROBLEMS

We propose an iterative method of estimation for discrete missing data problems that is conceptually different from the Expectation–Maximization (EM) algorithm and that does not in general yield the observed data maximum likelihood estimate (MLE). The proposed approach is based conceptually upon weighting the set of possible complete-data MLEs. Its implementation avoids the expectation step of EM, which can sometimes be problematic. In the simple case of Bernoulli trials missing completely at random, the iterations of the proposed algorithm are equivalent to the EM iterations. For a familiar genetics-oriented multinomial problem with missing count data and for the motivating example with epidemiologic applications that involves a mixture of a left censored normal distribution with a point mass at zero, we investigate the finite sample performance of the proposed estimator and find it to be competitive with that of the MLE. We give some intuitive justification for the method, and we explore an interesting connection between our algorithm and multiple imputation in order to suggest an approach for estimating standard errors.     

Likelihood estimation of missing cell means in the fixed model analysis of variance

This paper examines the formation of maximum likelihood estimates of cell means in analysis of variance problems for cells with missing observations. Methods of estimating the means for missing cells has a long history which includes iterative maximum likelihood techniques, approximation techniques and ad hoc techniques. The use of the EM algorithm to form maximum likelihood estimates has resolved most of the issues associated with this problem. Implementation of the EM algorithm entails specification of a reduced model. As demonstrated in this paper, when there are several missing cells, it is possible to specify a reduced model that results in an unidentifiable likelihood. The EM algorithm in this case does not converge, although the slow divergence may often be mistaken by the unwary as convergence. This paper presents a simple matrix method of determining whether or not the reduced model results in an identifiable likelihood, and consequently in an EM algorithm that converges. We also show the EM algorithm in this case to be equivalent to a method which yields a closed form solution.     

Parametric estimation of location and scale parameters in ranked set sampling

This paper develops parametric inference for the parameters of location-scale family of distributions based on a ranked set sample. Likelihood function incorporates within-set ranking errors into the model through a missing data mechanism. The maximum likelihood estimators of the location-scale and missing data model parameters are constructed and an EM-algorithm is provided. It is shown that the proposed estimator is robust against imperfect ranking error and provides higher efficiency over its competitors.     

Robust methods for generalized linear models with nonignorable missing covariates

The EM algorithm is often used for finding the maximum likelihood estimates in generalized linear models with incomplete data. In this article, the author presents a robust method in the framework of the maximum likelihood estimation for fitting generalized linear models when nonignorable covariates are missing. His robust approach is useful for downweighting any influential observations when estimating the model parameters. To avoid computational problems involving irreducibly high‐dimensional integrals, he adopts a Metropolis‐Hastings algorithm based on a Markov chain sampling method. He carries out simulations to investigate the behaviour of the robust estimates in the presence of outliers and missing covariates; furthermore, he compares these estimates to the classical maximum likelihood estimates. Finally, he illustrates his approach using data on the occurrence of delirium in patients operated on for abdominal aortic aneurysm.     

The asymptotic covariance structure of log-linear model estimated parameters for the multiple recapture census

This paper presents a matrix formulation for log-linear model analysis of the incomplete contingency table which arises from multiple recapture census data. Explicit matrix product expressions are given for the asymptotic covariance structure of the maximum likelihood estimators of both the log-linear model parameter vector and the predicted value vector for the observed and missing cells. These results are illustrated for data pertaining to a population of children possessing a common congenital anomaly.     

The Monte Carlo EM method for estimating multivariate tobit latent variable models

We propose a multivariate tobit (MT) latent variable model that is defined by a confirmatory factor analysis with covariates for analysing the mixed type data, which is inherently non-negative and sometimes has a large proportion of zeros. Some useful MT models are special cases of our proposed model. To obtain maximum likelihood estimates, we use the expectation maximum algorithm with its E-step via the Gibbs sampler made feasible by Monte Carlo simulation and its M-step greatly simplified by a sequence of conditional maximization. Standard errors are evaluated by inverting a Monte Carlo approximation of the information matrix using Louis's method. The methodology is illustrated with a simulation study and a real example.     

Empirical Likelihood Inference of the Partial Linear Isotonic Errors-in-variables Regression Models with Missing Data

This article is concerned with statistical inference of the partial linear isotonic regression model missing response and measurement errors in covariates. We proposed an empirical likelihood ratio test statistics and show that it has a limiting weighted chi-square distribution. An adjusted empirical likelihood ratio statistic, which is shown to have a limiting standard central chi-square distribution, is then proposed further. A maximum empirical likelihood estimator is also developed. A simulation study is conducted to examine the finite-sample property of proposed procedure.     

Estimation and prediction for the generalized inverted exponential distribution based on progressively first-failure-censored data with application

In this paper, the estimation of parameters for a generalized inverted exponential distribution based on the progressively first-failure type-II right-censored sample is studied. An expectation–maximization (EM) algorithm is developed to obtain maximum likelihood estimates of unknown parameters as well as reliability and hazard functions. Using the missing value principle, the Fisher information matrix has been obtained for constructing asymptotic confidence intervals. An exact interval and an exact confidence region for the parameters are also constructed. Bayesian procedures based on Markov Chain Monte Carlo methods have been developed to approximate the posterior distribution of the parameters of interest and in addition to deduce the corresponding credible intervals. The performances of the maximum likelihood and Bayes estimators are compared in terms of their mean-squared errors through the simulation study. Furthermore, Bayes two-sample point and interval predictors are obtained when the future sample is ordinary order statistics. The squared error, linear-exponential and general entropy loss functions have been considered for obtaining the Bayes estimators and predictors. To illustrate the discussed procedures, a set of real data is analyzed.     

The generalized Gudermannian distribution: inference and volatility modelling

In this paper, we introduce a new distribution, called generalized Gudermannian (GG) distribution, and its skew extension for GARCH models in modelling daily Value-at-Risk (VaR). Basic structural properties of the proposed distribution are obtained including probability density and cumulative distribution functions, moments, and stochastic representation. The maximum likelihood method is used to estimate unknown parameters of the proposed model and finite sample performance of maximum likelihood estimates are evaluated by means of Monte-Carlo simulation study. The real data application on Nikkei 225 index is given to demonstrate the performance of GARCH model specified under skew extension of GG innovation distribution against normal, Student's-t, skew normal and generalized error and skew generalized error distributions in terms of the accuracy of VaR forecasts. The empirical results show that the GARCH model with GG innovation distribution produces the most accurate VaR forecasts for all confidence levels.     

Information attainable in some randomly incomplete multivariate response models

In a general parametric setup, a multivariate regression model is considered when responses may be missing at random while the explanatory variables and covariates are completely observed. Asymptotic optimality properties of maximum likelihood estimators for such models are linked to the Fisher information matrix for the parameters. It is shown that the information matrix is well defined for the missing-at-random model and that it plays the same role as in the complete-data linear models. Applications of the methodologic developments in hypothesis-testing problems, without any imputation of missing data, are illustrated. Some simulation results comparing the proposed method with Rubin's multiple imputation method are presented.     

INCOMPLETE DATA IN GENERALIZED LINEAR MODELS WITH CONTINUOUS COVARIATES

This paper proposes a method for estimating the parameters in a generalized linear model with missing covariates. The missing covariates are assumed to come from a continuous distribution, and are assumed to be missing at random. In particular, Gaussian quadrature methods are used on the E-step of the EM algorithm, leading to an approximate EM algorithm. The parameters are then estimated using the weighted EM procedure given in Ibrahim (1990). This approximate EM procedure leads to approximate maximum likelihood estimates, whose standard errors and asymptotic properties are given. The proposed procedure is illustrated on a data set.     

Comparison of spatial interpolation methods in the first order stationary multiplicative spatial autoregressive models

Three linear prediction methods of a single missing value for a stationary first order multiplicative spatial autoregressive model are proposed based on the quarter observations, observations in the first neighborhood, and observations in the nearest neighborhood. Three different types of innovations including Gaussian (symmetric and thin tailed), exponential (skew to right), and asymmetric Laplace (skew and heavy tailed) are considered. In each case, the proposed predictors are compared based on the two well-known criteria: mean square prediction and Pitman's measure of closeness. Parameter estimation is performed by maximum likelihood, least square errors, and Markov chain Monte Carlo (MCMC).     

Analysis of incomplete data in the presence of dependent competing risks from Marshall–Olkin bivariate Weibull distribution under progressive hybrid censoring

This article considers the statistical analysis of dependent competing risks model with incomplete data under Type-I progressive hybrid censored condition using a Marshall–Olkin bivariate Weibull distribution. Based on the expectation maximum algorithm, maximum likelihood estimators for the unknown parameters are obtained, and the missing information principle is used to obtain the observed information matrix. As the maximum likelihood approach may fail when the available information is insufficient, Bayesian approach incorporated with auxiliary variables is developed for estimating the parameters of the model, and Monte Carlo method is employed to construct the highest posterior density credible intervals. The proposed method is illustrated through a numerical example under different progressive censoring schemes and masking probabilities. Finally, a real data set is analyzed for illustrative purposes.     

Analysis of ordinal outcomes with longitudinal covariates subject to missingness

We propose a mixture model for data with an ordinal outcome and a longitudinal covariate that is subject to missingness. Data from a tailored telephone delivered, smoking cessation intervention for construction laborers are used to illustrate the method, which considers as an outcome a categorical measure of smoking cessation, and evaluates the effectiveness of the motivational telephone interviews on this outcome. We propose two model structures for the longitudinal covariate, for the case when the missing data are missing at random, and when the missing data mechanism is non-ignorable. A generalized EM algorithm is used to obtain maximum likelihood estimates.     

Maximum likelihood estimation of bivariate circular hidden Markov models from incomplete data

In this paper, we propose a hidden Markov model for the analysis of the time series of bivariate circular observations, by assuming that the data are sampled from bivariate circular densities, whose parameters are driven by the evolution of a latent Markov chain. The model segments the data by accounting for redundancies due to correlations along time and across variables. A computationally feasible expectation maximization (EM) algorithm is provided for the maximum likelihood estimation of the model from incomplete data, by treating the missing values and the states of the latent chain as two different sources of incomplete information. Importance-sampling methods facilitate the computation of bootstrap standard errors of the estimates. The methodology is illustrated on a bivariate time series of wind and wave directions and compared with popular segmentation models for bivariate circular data, which ignore correlations across variables and/or along time.