A trinomial difference autoregressive model and its applications

This paper introduces a non-negative integer-valued autoregressive (INAR) process with seasonal structure of first order, which is an extension of the standard INAR(1) model proposed by Al-Osh and Alzaid [First-order integer-valued autoregressive (INAR(1)) process. J Time Ser Anal. 1987;8:261–275]. The main properties of the model are derived such as its stationarity and autocorrelation function (ACF), among others. The conditional least squares and conditional maximum likelihood estimators of the model parameters are studied and their asymptotic properties are established. Some detailed discussion is dedicated to the case where the marginal distribution of the process is Poisson. A Monte Carlo experiment is conducted to evaluate and compare the performances of these estimators for finite sample sizes. The standard Yule–Walker approach is also considered for comparison purposes. The empirical results indicate that, in general, the conditional maximum likelihood estimator presents much better performance in terms of bias and mean square error. The model is illustrated using a real data set.

In cluster-specific studies, ordinary logistic regression and conditional logistic regression for binary outcomes provide maximum likelihood estimator (MLE) and conditional maximum likelihood estimator (CMLE), respectively. In this paper, we show that CMLE is approaching to MLE asymptotically when each individual data point is replicated infinitely many times. Our theoretical derivation is based on the observation that a term appearing in the conditional average log-likelihood function is the coefficient of a polynomial, and hence can be transformed to a complex integral by Cauchy’s differentiation formula. The asymptotic analysis of the complex integral can then be performed using the classical method of steepest descent. Our result implies that CMLE can be biased if individual weights are multiplied with a constant, and that we should be cautious when assigning weights to cluster-specific studies.

A new estimator of the common odds ratio in one-to-one matched case-control studies is proposed. The connection between this estimator and the James-Stein estimating procedure is highlighted through the argument of estimating functions. Comparisons are made between this estimator, the conditional maximum likelihood estimator, and the estimator ignoring the matching in terms of finite sample bias, mean squared error, coverage probability, and length of confidence interval. In many situations, the new estimator is found to be more efficient than the conditional maximum likelihood estimator without being as biased as the estimator that ignores matching. The extension to multiple risk factors is also outlined.

A new First-Order mixture integer-valued threshold autoregressive process based on binomial thinning and negative binomial thinning

In this article, we introduce a new random coefficients self-exciting threshold integer-valued autoregressive process. The autoregressive coefficients are driven by a logistic regression structure, so that the explanatory variables can be included. Basic probabilistic and statistical properties of this model are discussed. Conditional least squares and conditional maximum likelihood estimators, as well as the asymptotic properties of the estimators, are discussed. The nonlinearity test of the model and existence test of explanatory variables are also addressed. As an illustration, we evaluate our estimates through a simulation study. Finally, we apply our method to the data sets of sexual offences in Ballina, New South Wales (NSW), Australia, with two covariates of temperature and drug offences. The result reveals that the proposed model fits the data sets well.

In this paper, we introduce an integer-valued threshold process that incorporates a random perturbation term with a negative binomial distribution, based on a generalized binomial thinning operator. We discuss the fundamental probabilistic and statistical properties of the model. The conditional least squares estimator and the conditional maximum likelihood estimator are examined for two scenarios: the threshold variable is known and unknown. Additionally, we derive the asymptotic properties of these estimators. Finally, the paper presents a numerical analysis of the estimation results as well as an example with real data.

Flexible binomial AR(1) processes using copulas

It has been known for several years that conditional and mixture maximum likelihood estimates do agree for the dichotomous Rasch model (RM). This equivalence may be attained by a sufficient number of mixing components in a specifically restricted latent-class model. Because the principle of such a semiparametric approach is a general one (Kiefer & Wolfowitz, 1956), it also applies to some other models of the Rasch type having simple sufficient statistics. Exact equivalence regarding the parameter estimates can be shown, for example, for the linear logistic test model, a RM with linearly constrained item parameters, and for the polytomous RM; only almost equivalence is found for the mixed RM. As formal proofs of these results are difficult, the presentation focuses on numerical examples demonstrating the said (almost) equivalence.

BackgroundGroup sequential designs are increasingly employed to allow trials to stop early with statistical rigor. While existing work focuses on intention-to-treat effect on clinical endpoints, the properties of mediation analysis (commonly conducted in psychological trials to understand a causal mechanism) remain unknown under group sequential designs.MethodsConsidering a group sequential design with one interim analysis for early stopping for efficacy, we conduct a simulation study to evaluate existing analysis techniques when the treatment effect on a continuous outcome is partially or fully mediated by a continuous intermediate variable measuring a casual mechanism. We study the probability of rejecting the null hypotheses on the total effect (i.e., intention-to-treat effect), direct effect and indirect effect, respectively. We examine the bias of maximum likelihood estimator for these effects. We investigate if the penalized (and conditional) maximum likelihood estimator has smaller bias than the maximum likelihood estimator when a trial stopped (did not stop) early.ResultsThe presence of an intermediate variable reduces the power of a group sequential design when sample size calculation ignores the causal mechanism, though type I error control remains unaffected. The maximum likelihood estimator is unbiased only for the mediator-outcome path, impacting the properties of mediation analysis since existing methods typically rely on it to estimate the pathways. The penalized maximum likelihood estimator for other pathways has similar bias to the stage-one maximum likelihood estimator, while the conditional maximum likelihood estimator shows negligible or smaller bias than the usual maximum likelihood estimator for estimating the total and the direct effects only.ConclusionsMediation analysis needs additional consideration in group sequential designs. As with fixed trial designs, the sample size calculation of group sequential designs should account for the total variability underlying a causal mechanism when the treatment effect is hypothesized to be mediated by an intermediate variable, or risk the overall power to detect an intention-to-treat (total) effect being lower than the nominal value. We suggest reporting several estimators and acknowledging that they may be biased for some mediation pathways. More research is needed to develop methods for the analysis of indirect effect under group sequential designs.Supplementary InformationThe online version contains supplementary material available at 10.1186/s12874-025-02714-y.

We propose a generalized mixture integer-valued generalized autoregressive conditional heteroscedastic model to provide a more flexible modeling framework. This model includes many mixture integer-valued models with different distributions already studied in the literature. The conditional and unconditional moments are discussed and the necessary and sufficient first- and second-order stationary conditions are derived. We also investigate the theoretical properties such as strict stationarity and ergodicity for the mixture process. The conditional maximum likelihood estimators via the EM algorithm are derived and the performances of the estimators are studied via simulation. The model can be selected in terms of both the number of mixture regimes and the number of orders in each regime by several different criteria. A real-life data example is also given to assess the performance of the model.

Conditional likelihood inference in a heteroscedastic functional measurement error model

SUMMARY This paper concerns the efficiency of the conditional likelihood method for inference in models which include nuisance parameters. A new concept of ancillarity, asymptotic weak ancillarity, is introduced. It is shown that the conditional maximum likelihood estimator and the conditional score test of 0, the parameter of interest, are asymptotically equivalent to their unconditional counterparts, and hence are asymptotically efficient, provided that the conditioning statistic is asymptotically weakly ancillary. The key assumption that the conditioning statistic is asymptotically weakly ancillary is verified when the underlying distribution is from exponential families. Some illustrative examples are given.

Observations on N cross-section units at T time points are used to estimate a simple statistical model involving an autoregressive process with an additive term specific to the unit. Different assumptions about the initial conditions are (a) initial state fixed, (b) initial state random, (c) the unobserved individual effect independent of the unobserved dynamic process with the initial value fixed, and (d) the unobserved individual effect independent of the unobserved dynamic process with initial value random. Asymptotic properties of the maximum likelihood and "covariance" estimators are obtained when T → ∞ and when N → ∞. The relationship between the pseudo and conditional maximum likelihood estimators is clarified. A simple consistent estimator that is independent of the initial conditions and the way in which T or N → ∞ is also suggested.

In 1985, Miettinen and Nurminen proposed asymptotic likelihood‐based methods for the interval estimation of risk differences, risk ratios and risk‐odds ratios. In the present study, with the use of simulated coverage probabilities in small samples, the actual confidence coefficients of all three of the proposed interval estimates for unstratified data were determined in a comparison with the usual ones. Rated against the 95% coverage probability, the likelihood methods that use restricted maximum likelihood estimators accurately attained the nominal level. The recommended limits also apportioned fairly equal error rates for both tails. The customary intervals performed less successfully. For simple risk differences the estimated confidence probabilities failed to reach the 95% level. For the differences in log‐transformed risks, the tail probabilities of the confidence intervals were disparately allocated, and the method was occasionally incomputable. For the difference in logit‐transformed risks, the uncorrected Woolf intervals were at times fallible and somewhat conservative in comparison to the likelihood ratio‐based intervals. Modification of the conservative Cornfield limits for the risk‐odds ratio greatly improved the accuracy of the likelihood score‐based procedure. With sparse‐stratified data the likelihood score approach produced a biased estimate of the risk‐odds ratio, which for matched pairs equals the square of the exact, conditional maximum likelihood estimate.

Repeated testing in a group sequential trial can result in bias in the maximum likelihood estimate of the unknown parameter of interest. Many authors have therefore proposed adjusted point estimation procedures, which attempt to reduce such bias. Here, we describe nine possible point estimators within a common general framework for a two-stage group sequential trial. We then contrast their performance in five example trial settings, examining their conditional and marginal biases and residual mean square error. By focusing on the case of a trial with a single interim analysis, additional new results aiding the determination of the estimators are given. Our findings demonstrate that the uniform minimum variance unbiased estimator, whilst being marginally unbiased, often has large conditional bias and residual mean square error. If one is concerned solely about inference on progression to the second trial stage, the conditional uniform minimum variance unbiased estimator may be preferred. Two estimators, termed mean adjusted estimators, which attempt to reduce the marginal bias, arguably perform best in terms of the marginal residual mean square error. In all, one should choose an estimator accounting for its conditional and marginal biases and residual mean square error; the most suitable estimator will depend on relative desires to minimise each of these factors. If one cares solely about the conditional and marginal biases, the conditional maximum likelihood estimate may be preferred provided lower and upper stopping boundaries are included. If the conditional and marginal residual mean square error are also of concern, two mean adjusted estimators perform well.