Technical Note—When Does the βth Percentile Residual Life Function Determine the Distribution?

In survival analysis, median residual lifetime is often used as a summary measure to assess treatment effectiveness; it is not clear, however, how such a quantity could be estimated for a given dynamic treatment regimen using data from sequential randomized clinical trials. We propose a method to estimate a dynamic treatment regimen-specific median residual life (MERL) function from sequential multiple assignment randomized trials. We present the MERL estimator, which is based on inverse probability weighting, as well as, two variance estimates for the MERL estimator. One variance estimate follows from Lunceford, Davidian and Tsiatis' 2002 survival function-based variance estimate and the other uses the sandwich estimator. The MERL estimator is evaluated, and its two variance estimates are compared through simulation studies, showing that the estimator and both variance estimates produce approximately unbiased results in large samples. To demonstrate our methods, the estimator has been applied to data from a sequentially randomized leukemia clinical trial.

An important property that any lifetime model should satisfy is scale invariance. In this paper, a new scale-invariant quasi-inverse Lindley (QIL) model is presented and studied. Its basic properties, including moments, quantiles, skewness, kurtosis, and Lorenz curve, have been investigated. In addition, the well-known dynamic reliability measures, such as failure rate (FR), reversed failure rate (RFR), mean residual life (MRL), mean inactivity time (MIT), quantile residual life (QRL), and quantile inactivity time (QIT) are discussed. The FR function considers the decreasing or upside-down bathtub-shaped, and the MRL and median residual lifetime may have a bathtub-shaped form. The parameters of the model are estimated by applying the maximum likelihood method and the expectation-maximization (EM) algorithm. The EM algorithm is an iterative method suitable for models with a latent variable, for example, when we have mixture or competing risk models. A simulation study is then conducted to examine the consistency and efficiency of the estimators and compare them. The simulation study shows that the EM approach provides a better estimation of the parameters. Finally, the proposed model is fitted to a reliability engineering data set along with some alternatives. The Akaike information criterion (AIC), Kolmogorov-Smirnov (K-S), Cramer-von Mises (CVM), and Anderson Darling (AD) statistics are used to compare the considered models.

In survival analysis interest lies in modeling data that describe the time to a particular event. Informative functions, namely the hazard function and mean residual life function, can be obtained from the model's distribution function. We focus on the mean residual life function which provides the expected remaining life given that the subject has survived (i.e., is event-free) up to a particular time. This function is of direct interest in reliability, medical, and actuarial fields. In addition to its practical interpretation, the mean residual life function characterizes the survival distribution. In terms of mean residual life function inference, there are two shortcomings present in the current literature. First off, the shape of the functional is often restricted, which forces the researcher to make an assumption that may not be appropriate. Secondly, in cases where the shape of the functional is not parametrically specified, full inference is not obtained. The aim of our research is to provide a modeling approach that yields full inference for the mean residual life function, and is not restrictive on the shape of the functional. In particular, we develop general Bayesian nonparametric modeling methods for inference for mean residual life functions built from a mixture model for the associated survival distribution. Although the prior model is not placed on the mean residual life function directly, our methods offer rich inference for the desired functional. We place a Dirichlet process mixture model on the survival function, and discuss the importance of careful kernel selection to ensure desirable properties for the mean residual life function. We advocate for a mixture model with a gamma kernel and dependent baseline distribution for the Dirichlet process prior. We extend our model to the regression setting by modeling the joint distribution for the survival response and random covariates. This approach provides a flexible method for obtaining inference for the regression functionals when the number of random covariates is small to moderate. We further extend our methods to the scenario where interest lies in comparison of survival between two experimental groups. Typically, we expect the range of survival in the two groups to be the same, but exhibiting different characteristics over that range. Here, we develop a dependent Dirichlet process prior for the mixing distributions having shared locations across the two groups and varying weights to incorporate dependency between populations and achieve richer inferential results. The final scenario we consider is the case in which the researcher believes two populations have ordered mean residual life functions. For such applications, a prior model that incorporates an ordering constraint on the mean residual life functions is attractive. We introduce a mixture of Erlang distributions with weights constructed using Dirichlet process priors that provides the mean residual life ordering result. We demonstrate the utility of our modeling methods through simulation and real data examples. In addition, we draw comparisons with both parametric and semiparametric models.

Given a negative plurisubharmonic function $\phi $ in a bounded pseudoconvex domain of $\mathbb C^n$, we introduce and study its <em>residual function</em> $g_\phi $ determined by the asymptotic behavior of $\phi $ near its singularity points, both inside

ABSTRACTThis article focuses on the estimation of percentile residual life function with left-truncated and right-censored data. Asymptotic normality and a pointwise confidence interval that does not require estimating the unknown underlying distribution function of the proposed empirical estimator are obtained. Some simulation studies and a real data example are used to illustrate our results.

Residual coefficient of variation and some characterization results

Fractional nonlinear partial differential equations are used in many scientific fields to model various processes, although most of these equations lack closed-form solutions. For this reason, methods for approximating solutions that occasionally yield closed-form solutions are crucial for solving these equations. This study introduces a novel technique that combines the residual function and a modified fractional power series with the Elzaki transform to solve various nonlinear problems within the Caputo derivative framework. The accuracy and effectiveness of our approach are validated through analyses of absolute, relative, and residual errors. We utilize the limit principle at zero to identify the coefficients of the series solution terms, while other methods, including variational iteration, homotopy perturbation, and Adomian, depend on integration. In contrast, the residual power series method uses differentiation, and both approaches encounter difficulties in fractional contexts. Furthermore, the effectiveness of our approach in addressing nonlinear problems without relying on Adomian and He polynomials enhances its superiority over various approximate series solution techniques.

Autotransplantation of spleen tissue is an attempt for maintenance of splenic functions when splenectomy is indicated in children. It minimizes the risks of overwhelming postsplenectomy infection and it has been done in children with severe portal hypertension due to hepatosplenic mansonic schistosomiasis that underwent splenectomy. The purposes of this investigation were to study the morphology of the residual splenic tissue; to evaluate the residual filtration function of this splenosis; and to assess the immune response to polyvalent pneumococcal vaccine of these patients. Twenty-three children with portal hypertension from mansonic schistosomiasis who underwent splenectomy, ligature of the left gastric vein, autotransplantation of spleen tissue into an omental pouch were evaluated for residual splenic parenchyma and functions. Tc-99m sulfur colloid liver-spleen scans were used for detection of splenic nodules. The search for Howell Jolly bodies were used for assessing the filtration function and Enzyme-linked immunosorbent assay was used for measuring the relative rise in titter of specific pneumococcal antibodies. Splenosis was evident in all children; however, in two there were less than five splenic nodules in the greater omentum, which was considered insufficient. Howell-Jolly bodies were found in the peripheral blood only in these two patients with less evident splenosis. The immune response was adequate in 15 patients; it was intermediate in 4 patients and inadequate in 4 patients. Autotransplantation of spleen tissue into an omental pouch is efficient in maintaining the filtration splenic function in more than 90% of the cases and the immune response to pneumococcal vaccination in approximately 65% of the children.

Vision restoration training (VRT) in hemianopia patients leads to visual field enlargements, but the mechanisms of this vision restoration are not known. To investigate the role of residual vision in recovery, we studied topographic features of visual field charts and determined residual functions in local regions and their immediate surround. We analyzed High Resolution Perimetry visual field charts of hemianopic stroke patients (n = 23) before and after 6 months of VRT and identified all local visual field regions with ("hot spots", n = 688) or without restoration ("cold spots", n = 3426). Topographic features of these spots at baseline where then related to (i) their respective local residual function, (ii) residual activity in their spatial neighbourhood, and (iii) their distance to the scotoma border estimated in cortical coordinates following magnification factor transformation. Visual field areas had a greater probability of becoming vision restoration hot spots if they had more residual activity in both local areas and in a spatially limited surround of 5° of visual angle. Hot spots were typically also located closer than 4 mm from the scotoma border in cortical coordinates. Thus, restoration depended on residual activity in both the local region and its immediate surround. Our findings confirm the special role of residual structures in visual field restoration which is likely mediated by partially surviving neuronal elements. Because the immediate but not distant surround influenced outcome of individual spots, we propose that lateral interactions, known to play a role in perceptual learning and receptive field plasticity, also play a major role in vision restoration.

Life expectancy of beagles was calculated from 303 beagles (162 female and 141 male) with known birth and death dates. Most of the dogs were either spayed or castrated during the first year of their life. Both natural death and euthanasia due to terminal illness were considered as a death event. These beagles received regular veterinary medical care during their life.Gender had no effect on median survival of beagles (male = 12.82 years; female = 12.87 years; p = 0.610). Therefore, combined data of both male and female beagles were used for the calculation of the life expectancy. The age specific median residual lifetime (age in years/median residual lifetime in years) was 0.5/12.8, 1.5/12.1, 2.5/11.1, 3.5/10.1, 4.5/9.1, 5.5/8.2, 6.5/7.2, 7.5/6.3, 8.5/5.4, 9.5/4.5, 10.5/3.7, 11.5/2.8, 12.5/2.1, 13.5/1.6, 14.5/1.2, and 15.5/0.6. Mortality increased dramatically in beagles after 9 years of age (95% CI: 8.7, 10.0 years). The results of this study suggest that earlier diagnostic or therapeutic intervention is warranted in beagles beginning at 9 years of age. This study was supported by Hill's Pet Nutrition, Inc.

Motivation: Microstructural changes of white matter tracts on ventral stream function deficits after posterior cerebral artery (PCA) stroke remain poorly understood. Goal(s): This study investigates the association between the integrity of six essential white matter tracts and residual ventral stream functions in PCA stroke survivors. Approach: We used tractography and diffusion tensor imaging (DTI) to assess microstructural integrity along tracts of interest, with principal component analysis to capture significant spatial variations for white matter microstructure. Results: White matter integrity correlates with residual ventral stream functions, specifically in colour perception, fine shape discrimination, and face recognition. Impact: This study demonstrates that applying principal component analysis to along-tract DTI metrics is an effective approach to evaluate white matter integrity, aiding in the assessment of ventral stream function loss after stroke.

Estimation of two ordered mean residual life functions

The neuronal processes that sustain consciousness, arousal and awareness interfered within disorders of consciousness remain poorly defined. As a consequence, subjects with DoC are classified (and the residual brain functions are assessed) based on vital and neurological signs and behavioural patterns. Efficient and reliable clinical scales have been designed and are commonly in use for the bedside evaluation of subjects with DoC. Problems in defining consciousness and its behavioural descriptors by unambiguous, common and applicable terms nevertheless still limit the classification of DoC as distinct diagnostic entities.

Biomass additivity is a desirable characteristic of a system of equations for predicting component as well as total tree biomass since it eliminates the inconsistency between the sum of predicted values for components such as stem, bark, branch and leaf and the prediction for the total tree. Besides logical consistency, a system of additive biomass equations when estimated by taking into account the inherent correlation among the biomass components has greater statistical efficiency than separately estimated equations for individual components. Using mostly small sample data from both published and unpublished sources, a system of non-linear additive biomass equations was developed for 15 native eucalypt forest tree species of temperate Australia. Diameter at breast height was used as the independent variable for all 15 species, while the combined variable of diameter and tree height was used for 14 species with height data. The system of additive equations provided more accurate biomass estimates than the common approach of separately fitting total tree and component biomass equations using log transformed data through least squares regression. Residual error variances were collectively estimated for each species by pooling small sample data across species and using indicator variables to represent the scale factor for each species in a residual variance function. This method overcame a common problem in estimating heteroscedastic error variance in non-linear biomass equations with additive error terms for small samples. From the estimated residual variance functions, approximate confidence bands containing about 95% of the observed data about the mean curve of predicted biomass were derived for all biomass components of each species. This system of additive biomass equations will prove to be useful for biomass estimation of native eucalypt forests of temperate Australia.

In this paper, a new five parameter extended Dagum model called new modified Dagum (NMD) distribution is proposed. The proposed distribution is flexible as its density contains important sub-models such as modified Dagum, new Dagum, new modified Burr III, modified Burr III, Frechet, modified Frechet and many other distributions. The NMD density function is symmetrical, left-skewed, right-skewed, J, reverse-J and arc. The NMD distribution can produce all types of failure rates such as modified bathtub, bathtub, inverted bathtub, increasing and decreasing. To show the importance of the proposed distribution, we derive mathematical properties such as random number generator, sub-models, ordinary moments, moment generating function, characteristic function, incomplete moments, inequality measures, residual life functions and reliability measures. We characterize the NMD distribution via innovative techniques. We address the maximum likelihood estimation technique for the model parameters. We evaluate the precision of the maximum likelihood estimators via simulation study on the basis of the graphical results. We consider an application to a real data set to clarify the potentiality and utility of the NMD model. We establish empirically that the proposed model is suitable for survival times of patient’s application. We apply goodness of fit statistics and graphical tools to examine the adequacy of the NMD distribution.