Bias Correction for a Generalized Log-Gamma Regression Model

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.

This paper investigates a method for fitting piecewise log-gamma regression models to censored survival data. Partitioning is performed using analysis of the distributions of residuals and cross-validation estimates of average squared error. Instead of evaluating the maximum likelihood estimates simultaneously for all the parameters, the regression coefficients and scale parameter are estimated for different fixed values of the shape parameter in each stratum. Likelihood ratio tests for discovering the best parametric regression model among the parametric families of lifetime distributions is performed in each stratum. Thus, the proposed method can identify the model for each subsample if the whole sample consists of subsamples having different regression models. The proposed method is illustrated with real and simulated data

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.

ملخص:- تحتوي نماذج الانحدار على مکونين أحدهما مکون عشوائي يمثل حد الخطأ (Error Term) والأخر مکون غير عشوائي يمثل متغير الاستجابة(Response Variable) وهذا المکون هو محل الاهتمام فيتم تقديره بإحدى الطريقتين هما: الطريقة المعلمية أو الطريقة اللامعلمية أو کلاهما. نموذج الانحدار المعلمي (Parametric Regression Model) هو النموذج الأکثر شيوعا حيث أنه يفترض أن متغير الاستجابة له صيغه دالية محدده مسبقا والتي تم الحصول عليها من معلومات سابقه حول هيکل دالة الانحدار من خلال ذلک يتم تقدير فئة من المعالم. ورغم أن النماذج المعلمية تعد أحد أهم أدوات تحليل البيانات؛ لأنها فعاله ويمکن تفسيرها بسهوله، إلا انها لم تستطع ان تکون کافية في بعض الحالات. أما نموذج الانحدار اللامعلمي (Nonparametric Regression Model) فانه يعتمد على تقدير دالة الانحدار مباشرة من البيانات کما أنه لا يفترض فروضا معينه لتقدير معالم النموذج مثل التي يفترضها النموذج المعلمي وبالتالي فهو يتمتع بمرونة أکثر لاکتشاف البيانات التي قد تکون مفقودة او في الحالات التي لا تتوافر فيها معلومات سابقة. يتم انشاء نموذج الانحدار شبه المعلمي (Semi parametric Regression Model) من خلال مزج الطريقتين السابقتين معا. فغالبا ما يتم استخدام هذا النموذج في المواقف التي تکون فيها الافتراضات المعلمية غير محددة وغير متسقة، أو أن النموذج اللامعلمي لا يعمل بشکل کامل. يشتمل هذا البحث على تطبيق نماذج الانحدار المعلمية وشبه المعلمية بالإضافة إلى النموذج اللامعلمى کحالة خاصة من نموذج الانحدار شبه المعلمى لتقدير الدالة الافتراضية وتحديد شکل العلاقة بين المتغير التابع والمتغيرات المستقلة بالاضافة الى شکل انتشار البيانات مع خط (أو منحني) التقدير وذلک على بيانات واقعية تم الحصول عليها من الإتحاد المصري لألعاب القوى والمعروفة بـبيانات لعبة العشاري . Regression models contain two components, one of which is a random component that represents the Error Term and the other is a non-random component that represents the response variable and this component is of interest and is estimated in one of two methods: the parametric method or the nonparametric method or both. Parametric regression model is the most common model since it assumes that the response variable has a predetermined functional formula which was obtained from previous information about the structure of the regression function by which a class of parameters is estimated. Although parameter models are one of the most important tools for analyzing data; Because it is effective and can be explained easily, but it could not be sufficient in some cases. As for the nonparametric regression model, it relies on estimating the regression function directly from the data, as it does not assume specific assumptions for estimating the model parameters, such as those assumed by the parametric model, and thus it has more flexibility to discover data that may be missing or in cases where previous information is not available. The semi-parametric regression model is constructed by blending the two previous methods together. This model is often used in situations where the parametric assumptions are indeterminate and inconsistent, or the nonparametric model is not fully functional. This research includes the application of parametric and semi-parametric regression models in addition to the nonparametric model as a special case of the semi-parametric regression model for estimating the hypothetical function and determining the form of the relationship between the dependent variable and the independent variables in addition to the data spread form with a line (or curve) estimation on realistic data On it from the Egyptian Athletics Federation, known as Decathlon data.

1 Order statistics in exponential distribution

Test-negative design (TND) studies are increasingly common in evaluating vaccine effectiveness (VE) for various infectious diseases. TND studies are susceptible to bias due to disease outcome misclassification caused by imperfect test sensitivity and specificity. Several bias correction methods have been proposed. However, sample size or power considerations for TND studies incorporating bias correction for such misclassification have not yet been investigated. We used Monte Carlo simulations to assess how bias correction affects the statistical power and sample size for VE estimation in TND studies. Simulations were conducted under varying levels of diagnostic test sensitivities (60%, 80%, and 95%). Bias correction was implemented using the multiple over-imputation method, which accounts for test misclassification through a parametric bootstrapping approach. Using a malaria vaccine as an example, we defined six vaccination status categories based on the time since receipt of the third or fourth vaccine dose. In the simulated target population, vaccination coverage was assumed to be low (< 10%) except for the group vaccinated more than 12 months after dose 4. We assumed relatively low VE (< 50%) against clinical malaria cases and a 30% malaria positivity rate among unvaccinated individuals presenting with malaria-related symptoms. Statistical power to detect VE was estimated for each vaccination status, both with and without bias correction. Estimated VEs based on observed data were consistently underestimated across all vaccination status groups due to diagnostic misclassification. In contrast, bias-corrected estimates were approximately unbiased but displayed wider confidence intervals, with their precision decreasing at lower test sensitivities. Statistical power to detect VE declined substantially when diagnostic test sensitivity was low. For instance, at 80% sensitivity, only three vaccination status groups reached 80% power with a sample size of 10,000, whereas the same power was achieved with just 6,000 individuals under a perfect test. Bias due to imperfect diagnostic testing can substantially reduce the power of TND studies. Power calculations should account for outcome misclassification and potential correction methods. Failure to do so may lead to underpowered studies and misleading VE estimates.

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.

We derive analytic expressions for the biases of the maximum likelihood estimators of the scale parameter in the half-logistic distribution with known location, and of the location parameter when the latter is unknown. Using these expressions to bias-correct the estimators is highly effective, without adverse consequences for estimation mean squared error. The overall performance of the first of these bias-corrected estimators is slightly better than that of a bootstrap bias-corrected estimator. The bias-corrected estimator of the location parameter significantly out-performs its bootstrapped-based counterpart. Taking computational costs into account, the analytic bias corrections clearly dominate the use of the bootstrap.

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.

This study develops a new bias-corrected estimator for the fixed-effects dynamic panel data model and derives its limiting distribution for fixed T and N large. The bias-corrected estimator is derived as a bias correction of the least-squares dummy variable (within) estimator. It does not share some of the drawbacks of recently developed IV and GMM estimators and is relatively easy to compute. Monte Carlo experiments provide evidence for the bias-corrected estimator to perform well even in small samples. The paper contains an application to a model of unemployment dynamics at the U.S. state level for the 1991-2000 period.

Implementing a generic method for bias correction in statistical models using random effects, with spatial and population dynamics examples

Regression analysis is one method in statistics that used to know the pattern of functional relationships between response variables and predictor variables. Combination of parametric and nonparametric regression is semi parametric regression. The most popular estimator for nonparametric or semi parametric regression is spline truncated estimator. Problems in everyday life often using regression modeling with longitudinal data. Longitudinal data is a combination of cross-section data and time-series data. In longitudinal data, between subjects are independent of each other but between observations in the subject are dependent. One of the most important parts of statistical inference is interval estimation. Interval estimation aims to determine predictor variables that have a significant effect on the response variable. This study aims to obtain the form of interval estimation for parameters of semi parametric regression models using spline truncated estimator in longitudinal data. To solve this problem, the Weighted Least Square method and a pivotal quantity method were used for unknown population variance cases. The result of the theoretical study was that pivotal quantity distributed student-t. The shortest parameter interval estimation of semi parametric spline truncated regression model was obtained through the optimization process using the method of Lagrange.Regression analysis is one method in statistics that used to know the pattern of functional relationships between response variables and predictor variables. Combination of parametric and nonparametric regression is semi parametric regression. The most popular estimator for nonparametric or semi parametric regression is spline truncated estimator. Problems in everyday life often using regression modeling with longitudinal data. Longitudinal data is a combination of cross-section data and time-series data. In longitudinal data, between subjects are independent of each other but between observations in the subject are dependent. One of the most important parts of statistical inference is interval estimation. Interval estimation aims to determine predictor variables that have a significant effect on the response variable. This study aims to obtain the form of interval estimation for parameters of semi parametric regression models using spline truncated estimator in longitudinal data. To solve this problem, t...

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

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.

Consider an i.i.d. sample X^*_1,X^*_2,...,X^*_n from a location-scale family, and assume that the only available observations consist of the partial maxima (or minima)sequence, X^*_{1:1},X^*_{2:2},...,X^*_{n:n}, where X^*_{j:j}=max{X^*_1,...,X^*_j}. This kind of truncation appears in several circumstances, including best performances in athletics events. In the case of partial maxima, the form of the BLUEs (best linear unbiased estimators) is quite similar to the form of the well-known Lloyd's (1952, Least-squares estimation of location and scale parameters using order statistics, Biometrika, vol. 39, pp. 88-95) BLUEs, based on (the sufficient sample of) order statistics, but, in contrast to the classical case, their consistency is no longer obvious. The present paper is mainly concerned with the scale parameter, showing that the variance of the partial maxima BLUE is at most of order O(1/log n), for a wide class of distributions.