Hazard rates play an important role in various areas, e.g., reliability theory, survival analysis, biostatistics, queueing theory and actuarial studies. Mixtures of distributions are also of a great preeminence in such areas as most populations of components are indeed heterogeneous. In this study we present a sufficient condition for mixtures of two elements\ of the same natural exponential family (NEF) to have an increasing hazard rate. We then apply this condition to some classical NEF's having either quadratic, or cubic variance functions (VF) and others as well. A particular attention is devoted to the hyperbolic cosine NEF having a quadratic VF, the Ressel NEF having a cubic VF and to the Kummer distributions of type 2 NEF. The application of such a sufficient condition is quite intricate and cumbersome, in particular when applied to the latter three NEF's. Various lemmas and propositions are needed then to verify this condition for these NEF's.
In this paper we introduce a mixture cure model with a linear hazard rate regression model for the event times. Cure models are statistical models for event times that take into account that a fraction of the population might never experience the event of interest, this fraction is said to be {`}cured{'}. The population survival function in a mixture cure model takes the form $S(t) = 1 - π+ π\exp(-\int_0^tα(s)\,d s)$, where $π$ is the probability of being susceptible to the event under study, and $α(s)$ is the hazard rate of the susceptible fraction. We let both $π$ and $α(s)$ depend on possib…
We develop a functional proportional hazards mixture cure (FPHMC) model with scalar and functional covariates measured at the baseline. The mixture cure model, useful in studying populations with a cure fraction of a particular event of interest is extended to functional data. We employ the EM algorithm and develop a semiparametric penalized spline-based approach to estimate the dynamic functional coefficients of the incidence and the latency part. The proposed method is computationally efficient and simultaneously incorporates smoothness in the estimated functional coefficients via roughness …
Google Research（谷歌研究院）；Carnegie Mellon University（卡内基梅隆大学）
Survival analysis is a challenging variation of regression modeling because of the presence of censoring, where the outcome measurement is only partially known, due to, for example, loss to follow up. Such problems come up frequently in medical applications, making survival analysis a key endeavor in biostatistics and machine learning for healthcare, with Cox regression models being amongst the most commonly employed models. We describe a new approach for survival analysis regression models, based on learning mixtures of Cox regressions to model individual survival distributions. We propose an…
Single-index Semiparametric Transformation Cure Models with Interval-censored Data
School of Economics, Jinan University（暨南大学经济学院）；International Center for Interdisciplinary Statistics, School of Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学学院交叉统计国际中心）
Semi-parametric survival analysis methods like the Cox Proportional Hazards (CPH) regression (Cox, 1972) are a popular approach for survival analysis. These methods involve fitting of the log-proportional hazard as a function of the covariates and are convenient as they do not require estimation of the baseline hazard rate. Recent approaches have involved learning non-linear representations of the input covariates and demonstrate improved performance. In this paper we argue against such deep parameterizations for survival analysis and experimentally demonstrate that more interpretable semi-par…
The proportional hazards (PH) model is arguably one of the most popular models used to analyze time to event data arising from clinical trials and longitudinal studies, among many others. In many such studies, the event time of interest is not directly observed but is known relative to periodic examination times; i.e., practitioners observe either current status or interval-censored data. The analysis of data of this structure is often fraught with many difficulties. Further exacerbating this issue, in some such studies the observed data also consists of instantaneous failures; i.e., the event…
Analysing subgroups defined by biomarkers is of increasing importance in clinical research. In some situations the biomarker is subject to misclassification error, meaning the true subgroups are identified with imperfect sensitivity and specificity. For time-to-event data, it is improper to assume the Cox proportional hazards model for the effects with respect to the true subgroups, since the survival distributions with respect to the diagnosed subgroups will not adhere to the proportional hazards assumption. This precludes the possibility of using simple adjustment procedures. Instead, we pre…
Survival models incorporating random effects to account for unmeasured heterogeneity are being increasingly used in biostatistical and applied research. Specifically, unmeasured covariates whose lack of inclusion in the model would lead to biased, inefficient results are commonly modelled by including a subject-specific (or cluster-specific) frailty term that follows a given distribution (e.g. Gamma or log-Normal). Despite that, in the context of parametric frailty models little is known about the impact of misspecifying the baseline hazard, the frailty distribution, or both. Therefore, our ai…
We obtain discrete mixture representations for parametric families of probability distributions on Euclidean spheres, such as the von Mises--Fisher, the Watson and the angular Gaussian families. In addition to several special results we present a general approach to isotropic distribution families that is based on density expansions in terms of special surface harmonics. We discuss the connections to stochastic processes on spheres, in particular random walks, discrete mixture representations derived from spherical diffusions, and the use of Markov representations for the mixing base to obtain…
We investigate existence and properties of discrete mixture representations $P_θ=\sum_{i\in E} w_θ(i) \, Q_i$ for a given family $P_θ$, $θ\inΘ$, of probability measures. The noncentral chi-squared distributions provide a classical example. We obtain existence results and results about geometric and statistical aspects of the problem, the latter including loss of Fisher information, Rao-Blackwellization, asymptotic efficiency and nonparametric maximum likelihood estimation of the mixing probabilities.
The mixture cure rate model is the most commonly used cure rate model in the literature. In the context of mixture cure rate model, the standard approach to model the effect of covariates on the cured or uncured probability is to use a logistic function. This readily implies that the boundary classifying the cured and uncured subjects is linear. In this paper, we propose a new mixture cure rate model based on interval censored data that uses the support vector machine (SVM) to model the effect of covariates on the uncured or the cured probability (i.e., on the incidence part of the model). Our…
The reversed aging intensity function is defined as the ratio of the instantaneous reversed hazard rate to the baseline value of the reversed hazard rate. It analyzes the aging property quantitatively, the higher the reversed aging intensity, the weaker the tendency of aging. In this paper, a family of generalized reversed aging intensity functions is introduced and studied. Those functions depend on a real parameter. If the parameter is positive they characterize uniquely the distribution functions of univariate positive absolutely continuous random variables, in the opposite case they charac…
For two vast families of mixture distributions and a given prior, we provide unified representations of posterior and predictive distributions. Model applications presented include bivariate mixtures of Gamma distributions labelled as Kibble-type, non-central Chi-square and F distributions, the distribution of $R^2$ in multiple regression, variance mixture of normal distributions, and mixtures of location-scale exponential distributions including the multivariate Lomax distribution. An emphasis is also placed on analytical representations and the relationships with a host of existing distribut…
Sparse models are desirable for many applications across diverse domains as they can perform automatic variable selection, aid interpretability, and provide regularization. When fitting sparse models in a Bayesian framework, however, analytically obtaining a posterior distribution over the parameters of interest is intractable for all but the simplest cases. As a result practitioners must rely on either sampling algorithms such as Markov chain Monte Carlo or variational methods to obtain an approximate posterior. Mean field variational inference is a particularly simple and popular framework t…
We investigate the geometry of the maximal a posteriori (MAP) partition in the Bayesian Mixture Model where the component and the base distributions are chosen from conjugate exponential families. We prove that in this case the clusters are separated by the contour lines of a linear functional of the sufficient statistic. As a particular example, we describe Bayesian Mixture of Normals with Normal-inverse-Wishart prior on the component mean and covariance, in which the clusters in any MAP partition are separated by a quadratic surface. In connection with results of Rajkowski (2018), where the …
In this paper we are interested in testing whether there are any signals hidden in high dimensional noise data. Therefore we study the family of goodness-of-fit tests based on $Φ$-divergences including the test of Berk and Jones as well as Tukey's higher criticism test. The optimality of this family is already known for the heterogeneous normal mixture model. We now present a technique to transfer this optimality to more general models. For illustration we apply our results to dense signal and sparse signal models including the exponential-$χ^2$ mixture model and general exponential families a…
We analyze the information geometric structure of time reversibility for parametric families of irreducible transition kernels of Markov chains. We define and characterize reversible exponential families of Markov kernels, and show that irreducible and reversible Markov kernels form both a mixture family and, perhaps surprisingly, an exponential family in the set of all stochastic kernels. We propose a parametrization of the entire manifold of reversible kernels, and inspect reversible geodesics. We define information projections onto the reversible manifold, and derive closed-form expressions…
Motivated by problems in data clustering, we establish general conditions under which families of nonparametric mixture models are identifiable, by introducing a novel framework involving clustering overfitted \emph{parametric} (i.e. misspecified) mixture models. These identifiability conditions generalize existing conditions in the literature, and are flexible enough to include for example mixtures of Gaussian mixtures. In contrast to the recent literature on estimating nonparametric mixtures, we allow for general nonparametric mixture components, and instead impose regularity assumptions on …
We study the behavior of exponential random graphs in both the sparse and the dense regime. We show that exponential random graphs are approximate mixtures of graphs with independent edges whose probability matrices are critical points of an associated functional, thereby satisfying a certain matrix equation. In the dense regime, every solution to this equation is close to a block matrix, concluding that the exponential random graph behaves roughly like a mixture of stochastic block models. We also show existence and uniqueness of solutions to this equation for several families of exponential …
