Lundberg inequalities for a Cox model with a piecewise constant intensity

1996 ◽  
Vol 33 (1) ◽  
pp. 196-210 ◽  
Author(s):  
Hanspeter Schmidli
Keyword(s):  
Risk Model ◽  
Cox Model ◽  
Risk Process ◽  
Constant Intensity ◽  
Small Claim ◽  
Martingale Approach ◽  
Discrete Component ◽  
Piecewise Constant ◽  
Jump Intensity ◽  
Lundberg Inequality

A Cox risk process with a piecewise constant intensity is considered where the sequence (Li) of successive levels of the intensity forms a Markov chain. The duration σi of the level Li is assumed to be only dependent via Li. In the small-claim case a Lundberg inequality is obtained via a martingale approach. It is shown furthermore by a Lundberg bound from below that the resulting adjustment coefficient gives the best possible exponential bound for the ruin probability. In the case where the stationary distribution of Li contains a discrete component, a Cramér–Lundberg approximation can be obtained. By way of example we consider the independent jump intensity model (Björk and Grandell 1988) and the risk model in a Markovian environment (Asmussen 1989).

Lundberg inequalities for a Cox model with a piecewise constant intensity

1996 ◽  
Vol 33 (01) ◽  
pp. 196-210 ◽  
Author(s):  
Hanspeter Schmidli
Keyword(s):  
Risk Model ◽  
Cox Model ◽  
Risk Process ◽  
Constant Intensity ◽  
Small Claim ◽  
Martingale Approach ◽  
Discrete Component ◽  
Piecewise Constant ◽  
Jump Intensity ◽  
Lundberg Inequality

A Cox risk process with a piecewise constant intensity is considered where the sequence (Li ) of successive levels of the intensity forms a Markov chain. The duration σi of the level Li is assumed to be only dependent via Li . In the small-claim case a Lundberg inequality is obtained via a martingale approach. It is shown furthermore by a Lundberg bound from below that the resulting adjustment coefficient gives the best possible exponential bound for the ruin probability. In the case where the stationary distribution of Li contains a discrete component, a Cramér–Lundberg approximation can be obtained. By way of example we consider the independent jump intensity model (Björk and Grandell 1988) and the risk model in a Markovian environment (Asmussen 1989).

scholarly journals Some results of ruin probability for the classical risk process

2003 ◽  
Vol 7 (3) ◽  
pp. 133-146
Author(s):  
He Yuanjiang ◽  
Li Xucheng ◽  
John Zhang
Keyword(s):  
Density Function ◽  
Ruin Probability ◽  
Counting Process ◽  
Risk Model ◽  
Cox Model ◽  
Random Sequence ◽  
Complete Solution ◽  
Classical Model ◽  
Risk Process ◽  
Classical Risk Model

The computation of ruin probability is an important problem in the collective risk theory. It has applications in the fields of insurance, actuarial science, and economics. Many mathematical models have been introduced to simulate business activities and ruin probability is studied based on these models. Two of these models are the classical risk model and the Cox model. In the classical model, the counting process is a Poisson process and in the Cox model, the counting process is a Cox process. Thorin (1973) studied the ruin probability based on the classical model with the assumption that random sequence followed the Γ distribution with density function f(x)=x1β−1β1βΓ(1/β)e−xβ, x>0, where β>1. This paper studies the ruin probability of the classical model where the random sequence follows the Γ distribution with density function f(x)=αnΓ(n)xn−1e−αx, x>0, where α>0 and n≥2 is a positive integer. An intermediate general result is given and a complete solution is provided for n=2. Simulation studies for the case of n=2 is also provided.

A novel prognostic model in elderly patients with acute myeloid leukemia: results of 909 patients entered into the prospective AML96 trial

Blood ◽  
2010 ◽  
Vol 116 (6) ◽  
pp. 971-978 ◽  
Author(s):  
Christoph Röllig ◽  
Christian Thiede ◽  
Martin Gramatzki ◽  
Walter Aulitzky ◽  
Heinrich Bodenstein ◽  
...  
Keyword(s):  
Acute Myeloid Leukemia ◽  
Myeloid Leukemia ◽  
Risk Model ◽  
Cox Model ◽  
Prognostic Significance ◽  
Disease Free Survival ◽  
Intermediate Risk ◽  
Npm1 Mutation ◽  
Prognostic Groups ◽  
Acute Myeloid

Abstract We present an analysis of prognostic factors derived from a trial in patients with acute myeloid leukemia older than 60 years. The AML96 trial included 909 patients with a median age of 67 years (range, 61-87 years). Treatment included cytarabine-based induction therapy followed by 1 consolidation. The median follow-up time for all patients is 68 months (5.7 years). A total of 454 of all 909 patients reached a complete remission (50%). Five-year overall survival (OS) and disease-free survival were 9.7% and 14%, respectively. Multivariate analyses revealed that karyotype, age, NPM1 mutation status, white blood cell count, lactate dehydrogenase, and CD34 expression were of independent prognostic significance for OS. On the basis of the multivariate Cox model, an additive risk score was developed that allowed the subdivision of the largest group of patients with an intermediate-risk karyotype into 2 groups. We are, therefore, able to distinguish 4 prognostic groups: favorable risk, good intermediate risk, adverse intermediate risk, and high risk. The corresponding 3-year OS rates were 39.5%, 30%, 10.6%, and 3.3%, respectively. The risk model allows further stratification of patients with intermediate-risk karyotype into 2 prognostic groups with implications for the therapeutic strategy. This study was registered at www.clinicaltrials.gov as #NCT00180115.

scholarly journals On a Class of Semi-Markov Risk Models Obtained as Classical Risk Models in a Markovian Environment

1984 ◽  
Vol 14 (1) ◽  
pp. 23-43 ◽  
Author(s):  
Jean-Marie Reinhard
Keyword(s):  
Risk Model ◽  
Risk Process ◽  
Ruin Probabilities ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Risk Models ◽  
Markovian Environment ◽  
Necessary And Sufficient ◽  
Natural Way ◽  
Quantities Of Interest

AbstractWe consider a risk model in which the claim inter-arrivals and amounts depend on a markovian environment process. Semi-Markov risk models are so introduced in a quite natural way. We derive some quantities of interest for the risk process and obtain a necessary and sufficient condition for the fairness of the risk (positive asymptotic non-ruin probabilities). These probabilities are explicitly calculated in a particular case (two possible states for the environment, exponential claim amounts distributions).

scholarly journals Machine Learning for Building Immune Genetic Model in Hepatocellular Carcinoma Patients

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Jun Liu ◽  
Zheng Chen ◽  
Wenli Li
Keyword(s):  
Hepatocellular Carcinoma ◽  
Risk Score ◽  
Risk Model ◽  
Cox Model ◽  
Cancer Genome ◽  
Algorithm Analysis ◽  
Immune Genes ◽  
Normal Tissues ◽  
Model Based ◽  
Tumor Stages

Background. Hepatocellular carcinoma (HCC) is the leading liver cancer with special immune microenvironment, which played vital roles in tumor relapse and poor drug responses. In this study, we aimed to explore the prognostic immune signatures in HCC and tried to construct an immune-risk model for patient evaluation. Methods. RNA sequencing profiles of HCC patients were collected from the cancer genome Atlas (TCGA), international cancer genome consortium (ICGC), and gene expression omnibus (GEO) databases (GSE14520). Differentially expressed immune genes, derived from ImmPort database and MSigDB signaling pathway lists, between tumor and normal tissues were analyzed with Limma package in R environment. Univariate Cox regression was performed to find survival-related immune genes in TCGA dataset, and in further random forest algorithm analysis, significantly changed immune genes were used to generate a multivariate Cox model to calculate the corresponding immune-risk score. The model was examined in the other two datasets with recipient operation curve (ROC) and survival analysis. Risk effects of immune-risk score and clinical characteristics of patients were individually evaluated, and significant factors were then used to generate a nomogram. Results. There were 52 downregulated and 259 upregulated immune genes between tumor and relatively normal tissues, and the final immune-risk model (based on SPP1, BRD8, NDRG1, KITLG, HSPA4, TRAF3, ITGAV and MAP4K2) can better differentiate patients into high and low immune-risk subpopulations, in which high score patients showed worse outcomes after resection ( p < 0.05 ). The differentially enriched pathways between the two groups were mainly about cell proliferation and cytokine production, and calculated immune-risk score was also highly correlated with immune infiltration levels. The nomogram, constructed with immune-risk score and tumor stages, showed high accuracy and clinical benefits in prediction of 1-, 3- and 5-year overall survival, which is useful in clinical practice. Conclusion. The immune-risk model, based on expression of SPP1, BRD8, NDRG1, KITLG, HSPA4, TRAF3, ITGAV, and MAP4K2, can better differentiate patients into high and low immune-risk groups. Combined nomogram, using immune-risk score and tumor stages, could make accurate prediction of 1-, 3- and 5-year survival in HCC patients.

scholarly journals A Lévy Insurance Risk Process with Tax

2008 ◽  
Vol 45 (02) ◽  
pp. 363-375 ◽  
Author(s):  
Hansjörg Albrecher ◽  
Jean-François Renaud ◽  
Xiaowen Zhou
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Risk Process ◽  
Fluctuation Theory ◽  
Insurance Risk ◽  
Exit Problem ◽  
Tax Payments ◽  
Aggregate Income

Using fluctuation theory, we solve the two-sided exit problem and identify the ruin probability for a general spectrally negative Lévy risk process with tax payments of a loss-carry-forward type. We study arbitrary moments of the discounted total amount of tax payments and determine the surplus level to start taxation which maximises the expected discounted aggregate income for the tax authority in this model. The results considerably generalise those for the Cramér-Lundberg risk model with tax.

DYNAMIC MEAN-VARIANCE OPTIMIZATION UNDER CLASSICAL RISK MODEL WITH FRACTIONAL BROWNIAN MOTION PERTURBATION

2008 ◽  
Vol 11 (04) ◽  
pp. 589-602 ◽  
Author(s):  
HUAYUE ZHANG ◽  
LIHUA BAI
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Risk Model ◽  
Efficient Frontier ◽  
Optimal Investment ◽  
Risk Process ◽  
Classical Risk Model ◽  
Classical Risk Process ◽  
Optimal Investment Problem ◽  
Mean Variance

In this paper, we apply the completion of squares method to study the optimal investment problem under mean-variance criteria for an insurer. The insurer's risk process is modelled by a classical risk process that is perturbed by a standard fractional Brownian motion with Hurst parameter H ∈ (1/2, 1). By virtue of an auxiliary process, the efficient strategy and efficient frontier are obtained. Moreover, when H → 1/2+ the results converge to the corresponding (known) results for standard Brownian motion.

scholarly journals Application of ensemble clustering and survival tree analysis for identifying prognostic clinicogenomic features in patients with colorectal cancer from the 100,000 Genomes Project

2021 ◽  
Vol 14 (1) ◽  
Author(s):  
Yuguo Wei ◽  
Nikolaos Papachristou ◽  
Stefanie Mueller ◽  
J. C. Ambrose ◽  
P. Arumugam ◽  
...  
Keyword(s):  
Colorectal Cancer ◽  
Risk Model ◽  
Cox Model ◽  
Risk Groups ◽  
Rank Test ◽  
Cancer Stage ◽  
Ensemble Clustering ◽  
Stage 4 ◽  
Cluster 2 ◽  
Survival Tree

Abstract Objective The objective of this study was to employ ensemble clustering and tree-based risk model approaches to identify interactions between clinicogenomic features for colorectal cancer using the 100,000 Genomes Project. Results Among the 2211 patients with colorectal cancer (mean age of diagnosis: 67.7; 59.7% male), 16.3%, 36.3%, 39.0% and 8.4% had stage 1, 2, 3 and 4 cancers, respectively. Almost every patient had surgery (99.7%), 47.4% had chemotherapy, 7.6% had radiotherapy and 1.4% had immunotherapy. On average, tumour mutational burden (TMB) was 18 mutations/Mb and 34.4%, 31.3% and 25.7% of patients had structural or copy number mutations in KRAS, BRAF and NRAS, respectively. In the fully adjusted Cox model, patients with advanced cancer [stage 3 hazard ratio (HR)  =  3.2; p  <  0.001; stage 4 HR  =  10.2; p  <  0.001] and those who had immunotherapy (HR  =  1.8; p  <  0.04) or radiotherapy (HR  =  1.5; p  <  0.02) treatment had a higher risk of dying. The ensemble clustering approach generated four distinct clusters where patients in cluster 2 had the best survival outcomes (1-year: 98.7%; 2-year: 96.7%; 3-year: 93.0%) while patients in cluster 3 (1-year: 87.9; 2-year: 70.0%; 3-year: 53.1%) had the worst outcomes. Kaplan–Meier analysis and log rank test revealed that the clusters were separated into distinct prognostic groups (p  <  0.0001). Survival tree or recursive partitioning analyses were performed to further explore risk groups within each cluster. Among patients in cluster 2, for example, interactions between cancer stage, grade, radiotherapy, TMB, BRAF mutation status were identified. Patients with stage 4 cancer and TMB  ≥  1.6 mutations/Mb had 4 times higher risk of dying relative to the baseline hazard in that cluster.

scholarly journals Level Set Segmentation: Active Contours without edge

2009 ◽  
Author(s):  
Kishore Mosaliganti ◽  
Benjamin Smith ◽  
Arnaud Gelas ◽  
Alexandre Gouaillard ◽  
sean megason
Keyword(s):  
Level Set ◽  
Level Sets ◽  
Active Contours ◽  
Level Set Methods ◽  
Constant Intensity ◽  
Piecewise Constant ◽  
Gradient Information ◽  
Cardiac Image ◽  
Contour Evolution ◽  
Processing Framework

An Insight Toolkit (ITK) processing framework for segmentation using active contours without edges is presented in this paper. Our algorithm is based on the work of Chan and Vese [1] that uses level- sets to accomplish region segmentation in images with poor or no gradient information. The basic idea is to partion the image into two piecewise constant intensity regions. This work is in contrast to the level-set methods currently available in ITK which necessarily require gradient information. Similar to those methods, the methods presented in this paper are also made efficient using a sparse implementation strategy that solves the contour evolution PDE at the level-set boundary. The framework consists of 6 new ITK filters that inherit in succession from itk::SegmentationFilter. We include 2D/3D example code, parameter settings and show the results generated on a 2D cardiac image.

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