Cox-McFadden Partial and Marginal Likelihoods for the Proportional Hazard Model with Random Effects

BackgroundWhen boosting algorithms are used for building survival models from high-dimensional data, it is common to fit a Cox proportional hazards model or to use least squares techniques for fitting semiparametric accelerated failure time models. There are cases, however, where fitting a fully parametric accelerated failure time model is a good alternative to these methods, especially when the proportional hazards assumption is not justified. Boosting algorithms for the estimation of parametric accelerated failure time models have not been developed so far, since these models require the estimation of a model-specific scale parameter which traditional boosting algorithms are not able to deal with.ResultsWe introduce a new boosting algorithm for censored time-to-event data which is suitable for fitting parametric accelerated failure time models. Estimation of the predictor function is carried out simultaneously with the estimation of the scale parameter, so that the negative log likelihood of the survival distribution can be used as a loss function for the boosting algorithm. The estimation of the scale parameter does not affect the favorable properties of boosting with respect to variable selection.ConclusionThe analysis of a high-dimensional set of microarray data demonstrates that the new algorithm is able to outperform boosting with the Cox partial likelihood when the proportional hazards assumption is questionable. In low-dimensional settings, i.e., when classical likelihood estimation of a parametric accelerated failure time model is possible, simulations show that the new boosting algorithm closely approximates the estimates obtained from the maximum likelihood method.

For competing risks data, the Fine-Gray proportional hazards model for subdistribution has gained popularity for its convenience in directly assessing the effect of covariates on the cumulative incidence function. However, in many important applications, proportional hazards may not be satisfied, including multicenter clinical trials, where the baseline subdistribution hazards may not be common due to varying patient populations. In this article, we consider a stratified competing risks regression, to allow the baseline hazard to vary across levels of the stratification covariate. According to the relative size of the number of strata and strata sizes, two stratification regimes are considered. Using partial likelihood and weighting techniques, we obtain consistent estimators of regression parameters. The corresponding asymptotic properties and resulting inferences are provided for the two regimes separately. Data from a breast cancer clinical trial and from a bone marrow transplantation registry illustrate the potential utility of the stratified Fine-Gray model.

The expression of X-chromosome undergoes three possible biological processes: X-chromosome inactivation (XCI), escape of the X-chromosome inactivation (XCI-E), and skewed X-chromosome inactivation (XCI-S). Although these expressions are included in various predesigned genetic variation chip platforms, the X-chromosome has generally been excluded from the majority of genome-wide association studies analyses; this is most likely due to the lack of a standardized method in handling X-chromosomal genotype data. To analyze the X-linked genetic association for time-to-event outcomes with the actual process unknown, we propose a unified approach of maximizing the partial likelihood over all of the potential biological processes. The proposed method can be used to infer the true biological process and derive unbiased estimates of the genetic association parameters. A partial likelihood ratio test statistic that has been proved asymptotically chi-square distributed can be used to assess the X-chromosome genetic association. Furthermore, if the X-chromosome expression pertains to the XCI-S process, we can infer the correct skewed direction and magnitude of inactivation, which can elucidate significant findings regarding the genetic mechanism. A population-level model and a more general subject-level model have been developed to model the XCI-S process. Finite sample performance of this novel method is examined via extensive simulation studies. An application is illustrated with implementation of the method on a cancer genetic study with survival outcome.

Abstract. The increasing availability of synthetic aperture radar (SAR) data from a range of different sensors necessitates efficient methods for semi-automated information extraction at multiple spatial scales for different fields of application. The focus of the presented study is two-fold: 1) to evaluate the applicability of multi-temporal TerraSAR-X imagery for multiresolution segmentation, and 2) to identify suitable Scale Parameters through different weighing of different homogeneity criteria, mainly colour variance. Multiresolution segmentation was used for segmentation of multi-temporal TerraSAR-X imagery, and the ESP (Estimation of Scale Parameter) tool was used to identify suitable Scale Parameters for image segmentation. The validation of the segmentation results was performed using very high resolution WorldView-2 imagery and a reference map, which was created by an ecological expert. The results of multiresolution segmentation revealed that in the context of object-based image analysis the TerraSAR-X images are applicable for generating optimal image objects. Furthermore, ESP tool can be used as an indicator for estimation of Scale Parameter for multiresolution segmentation of TerraSAR-X imagery. Additionally, for more reliable results, this study suggests that the homogeneity criterion of colour, in a variance based segmentation algorithm, needs to be set to high values. Setting the shape/colour criteria to 0.005/0.995 or 0.00/1 led to the best results and to the creation of adequate image objects.

This study investigates the performance of four parameter estimation methods for the Laplace distribution: Method of Moments (MM), Maximum Likelihood Estimation (MLE), Minimum Chi-Square Estimation using equiprobable cells (MCE-EQ), and Minimum Chi-Square Estimation using Representative Points (MCE-RP). Through comprehensive Monte Carlo simulations with sample sizes ranging from 50 to 400, we compare the root mean squared error (RMSE) of the location (μ) and scale (b) parameter estimates. Our results demonstrate that while MLE remains robust for location estimation, the MCE-RP method consistently outperforms other estimators—including MLE—for the scale parameter, particularly in small to moderate samples. The use of Representative Points, which provide an optimal discretization of the distribution, significantly enhances estimation precision. These findings are especially relevant for medical research, where accurate estimation of variability—such as in biomarker concentration levels or physiological response times—is critical for reliable sample size determination, risk assessment, and clinical decision-making. MCE-RP thus offers a superior, reliable estimator for the Laplace scale parameter, with direct implications for improving statistical inference in applied biomedical studies. Purpose: The purpose of this research is to empirically evaluate and compare the finite-sample performance of four estimation methods for the Laplace distribution’s parameters, with a focus on the novel application of Representative Points in minimum chi-square estimation. This work seeks to bridge the gap between theoretical estimation methods and practical applications, providing applied researchers with a more robust estimation tool when modeling data with Laplace characteristics, such as those commonly encountered in medical and biomedical studies. Methods: We conducted an extensive Monte Carlo simulation study to compare the four estimation methods: MM, MLE, MCE-EQ, and MCE-RP. For each method, we generated independent and identically distributed samples from a standard Laplace distribution (μ=0, b=1) with sample sizes n = 50, 100, 200, and 400. Each scenario was replicated 1,000 times. The performance of each estimator was assessed using the root mean squared error (RMSE) for both μ and b. The MCE-RP method utilized pre-computed Representative Points for the standard Laplace distribution, which were transformed according to preliminary MLE estimates to form an optimal cell structure for chi-square minimization. All nonlinear optimizations required for MCE-EQ and MCE-RP were implemented programmatically. Results: The simulation results indicate that MLE performs best for estimating the location parameter μ across all sample sizes. However, for the scale parameter b, the MCE-RP method consistently yields lower RMSE values compared to MLE, MM, and MCE-EQ. In many cases, particularly for smaller samples, the RMSE of MCE-RP is approximately half that of MLE for b. The advantage of MCE-RP is evident across varying numbers of Representative Points (m = 5, 10, 15, 20), with optimal performance often observed at m = 10 or 15. These findings confirm that MCE-RP provides a more precise and reliable estimator for the scale parameter, making it particularly advantageous in small-sample settings. Contribution: This paper contributes to the statistical methodology for the Laplace distribution by introducing and validating the use of Representative Points within a minimum chi-square estimation framework. The key contributions are: (1) demonstrating that MCE-RP significantly outperforms established methods for estimating the scale parameter; (2) providing empirical evidence that RP-based discretization enhances estimation efficiency, especially in finite samples; (3) offering practical guidance for applied researchers in fields such as medical statistics, where accurate scale estimation is crucial for variability assessment, power analysis, and reliable inference; and (4) laying a methodological foundation for extending the RP approach to other location-scale distributions.

This study examines the estimation of the scale parameter of the Weibull-Power Function Distribution (WPFD) using both maximum likelihood and Bayesian approaches. Bayesian estimation is conducted under one informative Gamma prior with hyperparameters , as well as two non-informative priors: the uniform prior and Jeffreys’ prior. For each prior specification, Bayes estimators are derived under squared error, quadratic, and precautionary loss functions. Closed-form expressions for the posterior distributions and corresponding Bayes estimators are obtained. The finite-sample performance of the competing estimators is evaluated through a Monte Carlo simulation study based on 1000 replications. Estimator performance is assessed using mean squared error (MSE), bias, and coverage probability. The results indicate that all estimators are consistent, with bias and MSE decreasing as sample size increases. Across different prior specifications and parameter settings, the Bayesian estimator under the quadratic loss function consistently attains the lowest MSE, yielding reductions of approximately 10-20% relative to the maximum likelihood estimator in small and moderate samples. These findings suggest that Bayesian estimation under quadratic loss provides improved finite-sample efficiency for estimating the WPFD scale parameter, while maintaining asymptotic comparability with the maximum likelihood approach.

As the amount of data increases, the data can obey either a single distribution, two distributions or even multiple distributions. If the distribution parameters and mixing ratio can be estimated for mixed data, it will be beneficial to further analysis and research in practical applications. In this paper, the EM algorithm and the Nelder-Mead(NM) algorithm are applied to the mixed parameter estimation of the two distributions, including mixing two identical distributions, mixing two different distributions, and comparing the EM algorithm and the Nelder-Mead algorithm to estimate the accuracy of the mixed distribution parameter estimation and its initial comparison. The stability of the value and other advantages and disadvantages. A large amount of data simulation results found that the EM algorithm has a good effect on the estimation of mixed distribution parameters, with high accuracy and fast convergence. The initial value selection has little effect on the results, and the preliminary derivation process is more complicated; the Nelder-Mead algorithm has a good effect on the estimation of mixed distribution parameters. High precision, fast convergence speed, the initial value selection has a greater impact on the result, the preliminary derivation process is relatively simple, and the application range is wide. When the initial value is close to the peak value of the real data, the parameter estimation effect is better; the two algorithms have poorer effect on the unimodal distribution parameter estimation for the data, and the bimodal distribution parameter estimation effect is better.

Improved point and interval estimation of the smallest scale parameter of n independent populations following two-parameter exponential distributions are studied. The model is formulated in such a way that allows for treating the estimation of the smallest scale parameter as a problem of estimating an unrestricted scale parameter in the presence of a nuisance parameter. The classes of improved point and interval estimators are enriched with Stein-type, Brewster and Zidek-type, Maruyama-type and Strawderman-type improved estimators under both quadratic and entropy losses, whereas using as a criterion the coverage probability, with Stein-type, Brewster and Zidek-type, and Maruyama-type improved intervals. The sampling framework considered incorporates important life-testing schemes such as i.i.d. sampling, type-II censoring, progressive type-II censoring, adaptive progressive type-II censoring, and record values.

1 Order statistics in exponential distribution

Underdetermined blind source separation (UBSS) deals with the problem of estimating n source signals from m measurements (n > m), with an unknown mixing process. Most researches pay attention to the sparsity of speech to recover source signals, such as the DUET (degenerate unmixing estimation technique) algorithm, which can separate any number of sources using only two mixtures with the help of estimation of mixing parameters but cant be used in the time-varying system. We develop a block-online algorithm for a dynamic environment that is based on mixing parameter estimation from reliable time-frequency points determined by the total power and FCM (Fuzzy C-Means) clustering of parameter estimates for every time-frame. To track the variable mixing parameters, the estimated ones from the previous frame can be used to initialize FCM in the current frame with changed mixing measure instead of random initialization in case of the conventional FCM. Thus, an online adaptive algorithm of UBSS with the time-varying matrix is realized.

<p xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" dir="auto" id="d620556e60"> <b>Objectives:</b> The Cox proportional hazard model had been used extensively to evaluate time-to-event (TTE) endpoints in drug development. The model assumes that event times are unique and distinct across subjects. When tied times occur, correction methods can be used to minimize bias in parameter estimation. Efron’s correction method is considered a balanced choice between the accuracy of estimate (over Breslow method) and computational intensity (over Exact method) and is used as the default option in R, however, its implementation in NONMEM has not been developed. This method introduces Efron correction in NONMEM for evaluating the effects of time-variant and time-invariant covariates on TTE endpoints in Cox regression. <p xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" dir="auto" id="d620556e65"> <b>Methods:</b> To implement the Cox Regression with Efron Correction in NONMEM, simulation datasets with 500 subjects were created, consisting of a time-variant covariate, a time-invariant covariate, and a TTE response variable. The event times were generated based on a bathtubs-like baseline hazard function and prespecified covariates’ effects on the baseline hazard for the events. Censoring times were generated from a constant hazard. The time records were obtained from either the event time or censoring time, whichever occurred first. The tied-time records were created by rounding up the times to the closest day, week, or month, mimicking scenarios when the time records are collected by day, week, or month. To implement Cox regression in NONMEM, the datasets were rearranged such that the subjects were organized by event/censoring times in descending order. For tied-time records, subjects with tied censoring times arranged before subjects with tied event times. The partial likelihood calculation for Efron correction employed a do-while loop and two variables defined in abbreviation block to facilitate calculations for the numerators and denominators of the partial likelihoods across the events over time. Lastly, model parameters were estimated in NONMEM and R for validation and comparison. <p xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" dir="auto" id="d620556e70"> <b>Results:</b> Tie-time records of 305, 444, and 484 were generated when rounding up the event/censoring times to the nearest day, week, and month, respectively, over 195 distinct days, 56 distinct weeks, and 16 distinct months. Estimated coefficients and the corresponding standard errors for the effects of time-invariant and time-variant covariates were found to be similar between NONMEM and R across all cases, when applying Efron correction in the Cox regression analyses (differences &lt; 0.2%). This indicates that the newly proposed workflow in NONMEM is comparable to R and can be effectively adopted for TTE analysis with tie-records. <p xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" dir="auto" id="d620556e75"> <b>Conclusions:</b> The developed NONMEM workflow successfully implements Efron correction for tied-time records in Cox regression. Compared to traditional statistical software, this workflow can be extended to incorporate complex nonlinear time-variant covariates and pharmacokinetic and pharmacodynamic models with ordinary differential equations for TTE analysis, conveniently within NONMEM in the field of pharmacometrics.

SYMARFIMA: A dynamical model for conditionally symmetric time series with long range dependence mean structure

This paper is concerned with problem of comparing different estimators of the scale parameter of two parameters weibull distribution where is shape parameter and is scale parameter and then comparing the efficiency of the estimators which are maximum likelihood and minimax estimator with (quadratic loss function) ,median method and moment method. the comparison was done using simulation procedure with different sample size and values of parameters ,on the bases of simulation experiment we conclude that minimax method with quadratic loss function was the best method at all sample size and values of parameters.

In survival analyses that include a time-dependent covariate, the standard partial likelihood approach is often complicated by the fact that the necessary observations of the time-dependent covariate are not always available for the subjects in each risk set. In this article, we consider the situation of a binary time-dependent covariate. We specify a longitudinal model for the binary covariate with missing observations and a proportional hazards model for survival time conditional on fixed covariates and the completed time-dependent covariate history. The joint posterior distribution of the missing covariates and the parameters in the covariate and hazard models is simulated using Gibbs sampling. The joint modeling approach allows the missing covariates to be generated using all of the observed covariates as well as the disease outcome and failure time information. Simulation studies show that this approach often leads to lower bias of relative risk estimates associated with the time-dependent covariate and superior coverage of interval estimates as compared to variations of approaches based on imputing missing covariate values using previous values and then performing partial likelihood analysis. Our research is motivated by a study of survival of lung cancer patients after surgery in which current smoking status is a binary time-dependent covariate. We apply the methods to data from this study and explore the advantages of the joint modeling approach. The application demonstrates that the joint modeling approach can have improved efficiency in estimating the coefficients of both the fixed and the time-dependent covariates. Moreover, it can facilitate analyses using more complicated functions of the time-dependent covariate, such as cumulative smoking, that are difficult or inappropriate to use with the comparison methods.

A note on universal admissibility of scale parameter estimators