Semiparametric regression estimation for longitudinal data in models with martingale difference error's structure

An inherent characteristic of longitudinal data is the dependence among the observations within the same subject. For exhibiting dependencies among the observations within the same subject, this paper considers a semiparametric partially linear regression model for longitudinal data based on martingale difference error's structure. We establish a strong consistency for the least squares estimator of a parametric component and the estimator of a non-parametric function under some mild conditions. A simulation study shows the performance of the proposed estimator in finite samples.     

Convergence results for strict C-sequences

We examine properties of strict C-sequences. These sequences are a generalization both of eventual martingales and of quasimartingales. Several global and local convergence results are proved. The transform and the quadratic variation of a strict C-sequence are also studied. A comparison with various martingale generalizations is established.     

Asymptotic properties for estimates of nonparametric regression model with martingale difference errors

Consider the nonparametric regression model with martingale difference errors. Nonparametric estimator g _n (x) of regression function g(x) will be introduced, and its asymptotic properties are studied. In particular, the pointwise and uniform convergence of g _n (x) and its asymptotic normality will be investigated. This extends the earlier work on independent random errors.     

Effect of plant-capture in a capture-recapture experiment

An efficiency study is made of a plant-capture approach for estimating population size in recapture studies. The population is augmented by the insertion of a known number of planted individuals who have already been marked and whose behaviour is identical to that of other members. A comparison is made with the case where no plants are used by considering asymptotic efficiency and a simulation study.     

A goodness-of-fit test for the stratified proportional hazards model for survival data

Abstract

Goodness-of-fit testing is addressed in the stratified proportional hazards model for survival data. A test statistic based on within-strata cumulative sums of martingale residuals over covariates is proposed and its asymptotic distribution is derived under the null hypothesis of model adequacy. A Monte Carlo procedure is proposed to approximate the critical value of the test. Simulation studies are conducted to examine finite-sample performance of the proposed statistic.     

Sequential selectiok procedures based on confidence sequences for normal populations

In this paper confidence sequences are used to construct sequential procedures for selecting the population with the a common variance. These procedures are shown to provide substantial saving, particularly in the expected samplw sizes of the inferior populations,over various procedures in the literature. A new “indifference zone” formulation is given for the correct selection probability requirement, and confidence sequences are also applied to construct sequential procedures for this new selection goal.     

Sales Forecasting with Financial Indicators and Experts' Input

We present a method for forecasting sales using financial market information and test this method on annual data for US public retailers. Our method is motivated by the permanent income hypothesis in economics, which states that the amount of consumer spending and the mix of spending between discretionary and necessity items depend on the returns achieved on equity portfolios held by consumers. Taking as input forecasts from other sources, such as equity analysts or time‐series models, we construct a market‐based forecast by augmenting the input forecast with one additional variable, lagged return on an aggregate financial market index. For this, we develop and estimate a martingale model of joint evolution of sales forecasts and the market index. We show that the market‐based forecast achieves an average 15% reduction in mean absolute percentage error compared with forecasts given by equity analysts at the same time instant on out‐of‐sample data. We extensively analyze the performance improvement using alternative model specifications and statistics. We also show that equity analysts do not incorporate lagged financial market returns in their forecasts. Our model yields correlation coefficients between retail sales and market returns for all firms in the data set. Besides forecasting, these results can be applied in risk management and hedging.     

THE CONVERGENCE OF AUTOCORRELATIONS AND AUTOREGRESSIONS1

For a general class of scalar stationary processes, essentially those for which the best linear predictor is the best predictor (in the mean square sense), it is shown that, under fairly minor additional conditions, the sample autocorrelations converge to the true values almost surely and hniformly in the lag, t, at a rate (T^-1log T)^1/2, where T is the sample size. For ARMA processes, if |t|(log T)^a, a < ∞, the rate is the best possible, namely (T^-1log log T)^1/2. In particular the somewhat implausible condition, on the innovations, that E{ε(t)²| F_t-l} is constant is avoided in these results. The theorems are used to discuss autoregressive approximation. When the stationary process is a vector process the condition on the innovation sequence, ε(t), that E{ε(t)ε(t)| F_t-l} be constant, cannot be entirely avoided in relation to autoregressive approximation. This is also discussed.     

Checking a Semiparametric Additive Risk Model

McKeague and Sasieni [A partly parametric additive risk model. Biometrika 81 (1994) 501] propose a restriction of Aalen’s additive risk model by the additional hypothesis that some of the covariates have time-independent influence on the intensity of the observed counting process. We introduce goodness-of-fit tests for this semiparametric Aalen model. The asymptotic distribution properties of the test statistics are derived by means of martingale techniques. The tests can be adjusted to detect particular alternatives. As one of the most important alternatives we consider Cox’s proportional hazards model. We present simulation studies and an application to a real data set.     

Standardized Martingale Residuals Applied to Grouped Left Truncated Observations of Dementia Cases

The use of martingale residuals have been proposed for modelchecking and also to get a non-parametric estimate of the effectof an explanatory variable. We apply this approach to an epidemiologicalproblem which presents two characteristics: the data are lefttruncated due to delayed entry in the cohort; the data are groupedinto geographical units (parishes). This grouping suggests anatural way of smoothing the graph of residuals which is to computethe sum of the residuals for each parish. It is also naturalto present a graph with standardized residuals. We derive thevariances of the estimated residuals for left truncated datawhich allows computing the standardized residuals. This methodis applied to the study of dementia in a cohort of old people,and to the possible effect of the concentration of aluminum andsilica in drinking water on the risk of developing dementia.