On confidence regions of semiparametric nonlinear reproductive dispersion models

This article investigates the confidence regions for semiparametric nonlinear reproductive dispersion models (SNRDMs), which is an extension of nonlinear regression models. Based on local linear estimate of nonparametric component and generalized profile likelihood estimate of parameter in SNRDMs, a modified geometric framework of Bates and Wattes is proposed. Within this geometric framework, we present three kinds of improved approximate confidence regions for the parameters and parameter subsets in terms of curvatures. The work extends the previous results of Hamilton et al. [in Accounting for intrinsic nonlinearity in nonlinear regression parameter inference regions, Ann. Statist. 10, pp. 386–393, 1982], Hamilton [in Confidence regions for parameter subset in nonlinear regression, Biometrika, 73, pp. 57–64, 1986], Wei [in On confidence regions of embedded models in regular parameter families (a geometric approch), Austral. J. Statist. 36, pp. 327–338, 1994], Tang et al. [in Confidence regions in quasi-likelihood nonlinear models: a geometric approach, J. Biomath. 15, pp. 55–64, 2000b] and Zhu et al. [in On confidence regions of semiparametric nonlinear regression models, Acta. Math. Scient. 20, pp. 68–75, 2000].     

Variance bounds for estimators in autoregressive models with constraints

We consider nonlinear and heteroscedastic autoregressive models whose residuals are martingale increments with conditional distributions that fulfil certain constraints. We treat two classes of constraints: residuals depending on the past through some function of the past observations only, and residuals that are invariant under some finite group of transformations. We determine the efficient influence function for estimators of the autoregressive parameter in such models, calculate variance bounds, discuss information gains, and suggest how to construct efficient estimators. Without constraints, efficient estimators can be given by weighted least squares estimators. With the constraints considered here, efficient estimators are obtained differently, as one-step improvements of some initial estimator, similarly as in autoregressive models with independent increments.     

Fitting of self-exciting threshold autoregressive moving average nonlinear time-series model through genetic algorithm and development of out-of-sample forecasts

Self-exciting threshold autoregressive moving average (SETARMA) nonlinear time-series model is considered here. Sufficient conditions for invertibility and stationarity are derived. Parameter estimation algorithm is developed by employing real-coded genetic algorithm stochastic optimization procedure. A significant feature of the work done is that optimal out-of-sample forecasts up to three-step ahead and their forecast error variances are derived analytically. Relevant computer programs are written in statistical analysis system (SAS) and C. As an illustration, annual mackerel catch time-series data are considered. Forecast performance of the fitted model for hold-out data is evaluated by using Naive and Monte Carlo approaches. It is found that optimal out-of-sample forecast values are quite close to actual values and estimated variances are quite close to theoretical values. Superiority of the SETARMA model over the SETAR model for equal predictive ability through Diebold–Mariano test is also established.     

Classes of diffusion-type processes with a sufficient reduction

Classes of processes of the diffusion type permitting a sufficient data reduction are derived. None of these classes are exponential families in the usual sense. For one type of such classes the sufficient statistic equals that of a curved exponential family of diffusion-type processes. For a second type the last observation is sufficient. In particular cases both types of classes are defined by means of a RICCATI equation     

Minimum average risk estimators for coefficients in linear models

We consider the problem of estimating the coefficient vector β of a linear regression model with quadratic loss function. Some biased estimators which utilize the prior information about β are considered. Also studied is the problem of estimating the parameters of an over-identified structural equation from undersized samples.     

Using regression mixture models with non-normal data: examining an ordered polytomous approach

Mild to moderate skew in errors can substantially impact regression mixture model results; one approach for overcoming this includes transforming the outcome into an ordered categorical variable and using a polytomous regression mixture model. This is effective for retaining differential effects in the population; however, bias in parameter estimates and model fit warrant further examination of this approach at higher levels of skew. The current study used Monte Carlo simulations; 3000 observations were drawn from each of two subpopulations differing in the effect of X on Y. Five hundred simulations were performed in each of the 10 scenarios varying in levels of skew in one or both classes. Model comparison criteria supported the accurate two-class model, preserving the differential effects, while parameter estimates were notably biased. The appropriate number of effects can be captured with this approach but we suggest caution when interpreting the magnitude of the effects.     

An Improved Boxplot for Univariate Data

The boxplot is an effective data-visualization tool useful in diverse applications and disciplines. Although more sophisticated graphical methods exist, the boxplot remains relevant due to its simplicity, interpretability, and usefulness, even in the age of big data. This article highlights the origins and developments of the boxplot that is now widely viewed as an industry standard as well as its inherent limitations when dealing with data from skewed distributions, particularly when detecting outliers. The proposed Ratio-Skewed boxplot is shown to be practical and suitable for outlier labeling across several parametric distributions.     

The collected works of john w. tukey

We describe novel, analytical, data-analysis, and Monte-Carlo-simulation studies of strongly heteroscedastic data of both small and wide range.Many different types of heteroscedasticity and fixed or variable weighting are incorporated through error-variance models.Attention is given to parameter bias determinations, evaluations of their significances, and to new ways to correct for bias.The error-variance models allow for both additive and independent power-law errors, and the power exponent is shown to be able to be well determined for typical physicalsciences data by the rapidly-converging, general-purpose, extended-least-squares program we use.The fitting and error-variance models are applied to both low-and high-heteroscedasticity situations, including single-response data from radioactive decay.Monte-Carlo simulations of data with similar parameters are used to evaluate the analytical models developed and the various minimization methods em-ployed, such as extended and generalized least squares.Logarithmic and inversion transformations are investigated in detail, and it is shown analytically and by simulations that exponential data with constant percentage errors can be logarithmically transformed to allow a simple parameter-bias-removal procedure.A more-general bias-reduction approach combining direct and inversion fitting is also developed.Distributions of fitting-model and error-variance-model parameters are shown to be typically non-normal, thus invalidating the usual estimates of parameter bias and precision.Errors in conventional confidence-interval estimates are quantified by comparison with accurate simulation results.     

Notes on likelihood intervals and profiling

In many applications, decisions are made on the basis of function of parameters g(θ). When the value of g(theta;) is calculated using estimated values for te parameters, its is important to have a measure of the uncertainty associated with that value of g(theta;). Likelihood ratio approaches to finding likelihood intervals for functions of parameters have been shown to be more reliable, in terms of coverage probability, than the linearization approach. Two approaches to the generalization of the profiling algorithm have been proposed in the literature to enable construction of likelihood intervals for a function of parameters (Chen and Jennrich, 1996; Bates and Watts, 1988). In this paper we show the equivalence of these two methods. We also provide and analysis of cases in which neither profiling algorithm is appropriate. For one of these cases an alternate approach is suggested Whereas generalized profiling is based on maximizing the likelihood function given a constraint on the value of g(θ), the alternative algorithm is based on optimizing g(θ) given a constraint on the value of the likelihood function.     

Almost Periodic Solutions for Stochastic Differential Equations Driven By G-Brownian Motion

In this paper, we introduce the concept of the p-mean almost periodicity for stochastic processes in non linear expectation spaces. The existence and uniqueness of square-mean almost periodic solutions to some non linear stochastic differential equations driven by G-Brownian motion are established under some assumptions for the coefficients. The asymptotic stability of the unique square-mean almost periodic solution in the square-mean sense is also discussed.