Expressing Estimators of Expected Quality Adjusted Survival as Functions of Nelson-Aalen Estimators

Quality adjusted survival has been increasingly advocated in clinical trials to be assessed as a synthesis of survival and quality of life. We investigate nonparametric estimation of its expectation for a general multistate process with incomplete follow-up data. Upon establishing a representation of expected quality adjusted survival through marginal distributions of a set of defined events, we propose two estimators for expected quality adjusted survival. Expressed as functions of Nelson-Aalen estimators, the two estimators are strongly consistent and asymptotically normal. We derive their asymptotic variances and propose sample-based variance estimates, along with evaluation of asymptotic relative efficiency. Monte Carlo studies show that these estimation procedures perform well for practical sample sizes. We illustrate the methods using data from a national, multicenter AIDS clinical trial.     

High dimensional asymptotics for the naive Hotelling T2 statistic in pattern recognition

Abstract

This paper examines the high dimensional asymptotics of the naive Hotelling T² statistic. Naive Bayes has been utilized in high dimensional pattern recognition as a method to avoid singularities in the estimated covariance matrix. The naive Hotelling T² statistic, which is equivalent to the estimator of the naive canonical correlation, is a statistically important quantity in naive Bayes and its high dimensional behavior has been studied under several conditions. In this paper, asymptotic normality of the naive Hotelling T² statistic under a high dimension low sample size setting is developed using the central limit theorem of a martingale difference sequence.     

A note on stable limit theory for the OLSE with non usual rates and the heteroskedasticity robust Wald test

We are occupied with an example concerning the limit theory of the ordinary least squares estimator (OLSE) when the innovation process of the regression has the form of a martingale transform the iid part of which lies in the domain of attraction of an α-stable distribution, the scaling sequence has a potentially diverging truncated α-moment, and the regressor process has a potentially divergent truncated second moment. We obtain matrix rates that reflect the stability parameter as well as the slow variations present in the aforementioned sequences, and stable limits. We also derive asymptotic exactness, consistency, and local asymptotic unbiasedness under appropriate local alternatives for a heteroskedasticity robust Wald test based on subsampling. The results could be useful for inference on the factor loadings in an instance of the APT model.     

ESTIMATION OF SPATIAL ARMA MODELS

Spatial ARMA models are considered using the nonsymmetric half plane ordering on a lattice of data. A method is given for the estimation of the orders and the coefficients of such models under an identifiability condition and the condition that the beat linear predictor is the best predictor in the mean square sense. Under these conditions, the strong consistency of the estimators ia established. The usual methods for ARMA modelling in Time Series Analysis require estimation of the innovations. The method of this paper introduces an inveree model complementary to the original model so that the estimation of the innovations is avoided. This leads to a substantial reduction in the computational complexity in the two-dimensional case.     

Testing recapture model MT for departure from homogeneity

This paper presents a method of testing for animal heterogeneity when the capture effort is not constant from occasion to occasion. The basic set of statistics used is a lower triangular matrix whose (I,k)th element is the proportion of those animals caught on the next (i+1)the occasion who have been caught k times until now. Both theoretical and simulation evidence is presented that the new procedure is superior to methods previously suggested. The method may breakdown for certain kinds of behavioural response.     

A Simplified Method of Calculating an Overall Goodness-of-Fit Test for the Cox Proportional Hazards Model

Gronnesby and Borgan (1996) propose an overall goodness-of-fit test for the Cox proportional hazards model. The basis of their test is a grouping of subjects by their estimated risk score. We show that the Gronnesby and Borgan test is algebraically identical to one obtained from adding group indicator variables to the model and testing the hypothesis the coefficients of the group indicator variables are zero via the score test. Thus showing that the test can be calculated using existing software. We demonstrate that the table of observed and estimated expected number of events within each group of the risk score is a useful adjunct to the test to help identify potential problems in fit.     

Optimal investment,consumption, and life insurance in an incomplete market

ABSTRACT

In this article, we solve an optimal insurance-consumption-investment problem for a wage earner in an incomplete market, where the stock price has a mean-reverting drift. By using the martingale method, we analyze this problem and derive the optimal strategies. Explicit solutions are found for both power and logarithmic utilities.     

Non-parametric Estimation of a Survival Function with Two-stage Design Studies

Abstract.  The two-stage design is popular in epidemiology studies and clinical trials due to its cost effectiveness. Typically, the first stage sample contains cheaper and possibly biased information, while the second stage validation sample consists of a subset of subjects with accurate and complete information. In this paper, we study estimation of a survival function with right-censored survival data from a two-stage design. A non-parametric estimator is derived by combining data from both stages. We also study its large sample properties and derive pointwise and simultaneous confidence intervals for the survival function. The proposed estimator effectively reduces the variance and finite-sample bias of the Kaplan–Meier estimator solely based on the second stage validation sample. Finally, we apply our method to a real data set from a medical device postmarketing surveillance study.     

Goodness-of-fit Tests for Semi-Markov and Markov Survival Models with One Intermediate State

Survival data with one intermediate state are described by semi-Markov and Markov models for counting processes whose intensities are defined in terms of two stopping times T ₁< T ₂. Problems of goodness-of-fit for these models are studied. The test statistics are proposed by comparing Nelson–Aalen estimators for data stratified according to T ₁. Asymptotic distributions of these statistics are established in terms of the weak convergence of some random fields. Asymptotic consistency of these test statistics is also established. Simulation studies are included to indicate their numerical performance.