Nonparametric bounds for the survivor function under general dependent truncation

Truncation occurs in cohort studies with complex sampling schemes. When truncation is ignored or incorrectly assumed to be independent of the event time in the observable region, bias can result. We derive completely nonparametric bounds for the survivor function under truncation and censoring; these extend prior nonparametric bounds derived in the absence of truncation. We also define a hazard ratio function that links the unobservable region in which event time is less than truncation time, to the observable region in which event time is greater than truncation time, under dependent truncation. When this function can be bounded, and the probability of truncation is known approximately, it yields narrower bounds than the purely nonparametric bounds. Importantly, our approach targets the true marginal survivor function over its entire support, and is not restricted to the observable region, unlike alternative estimators. We evaluate the methods in simulations and in clinical applications.     

Ergodicity and truncation bounds for inhomogeneous birth and death processes with additional transitions from and to origin

ABSTRACT

In this paper, we present the extension of the analysis of time-dependent limiting characteristics the class of continuous-time birth and death processes defined on non-negative integers with special transitions from and to the origin. From the origin transitions can occur to any state. But being in any other state, besides ordinary transitions to neighboring states, a transition to the origin can occur. All possible transition intensities are assumed to be non-random functions of time and may depend on the state of the process. We improve previously known ergodicity and truncation bounds for this class of processes that were known only for the case when transitions from the origin decay exponentially (other intensities must have unique uniform upper bound). We show how the bounds can be obtained if the decay rate is slower than exponential. Numerical results are given in the queueing theory context.     

Computational Algorithms for Moments of Accumulated Markov and Semi-Markov Rewards

Power moments for accumulated rewards defined on Markov and semi-Markov chains are studied. A model with mixed time-space termination of reward accumulation is considered for inhomogeneous in time rewards and Markov chains. Characterization of power moments as minimal solutions of recurrence system of linear equations, sufficient conditions for finiteness of these moments and upper bounds for them, expressed in terms of so-called test functions, are given. Backward recurrence algorithms for funding of power moments of accumulated rewards and various time-space truncation approximations reducing dimension of the corresponding recurrence relations are described.     

IMPROVING THE NUMERICAL TECHNIQUE FOR COMPUTING THE ACCUMULATED DISTRIBUTION OF A QUADRATIC FORM IN NORMAL VARIABLES

This paper is concerned with the technique of numerically evaluating the cumulative distribution function of a quadratic form in normal variables. The efficiency of two new truncation bounds and all existing truncation bounds are investigated. We also find that the suggestion in the literature for further splitting truncation errors might reduce computational efficiency, and the optimum splitting rate could be different in different situations. A practical solution is provided. The paper also discusses a modified secant algorithm for finding the critical value of the distribution at any given significance level.     

IMPROVING THE NUMERICAL TECHNIQUE FOR COMPUTING THE ACCUMULATED DISTRIBUTION OF A QUADRATIC FORM IN NORMAL VARIABLES

ABSTRACT

This paper is concerned with the technique of numerically evaluating the cumulative distribution function of a quadratic form in normal variables. The efficiency of two new truncation bounds and all existing truncation bounds are investigated. We also find that the suggestion in the literature for further splitting truncation errors might reduce computational efficiency, and the optimum splitting rate could be different in different situations. A practical solution is provided. The paper also discusses a modified secant algorithm for finding the critical value of the distribution at any given significance level.     

Computation of the central and noncentral f distributions

Recursion relations suitable for rapid computation are derived for the cumulative distribution of F′ = (X/m)/(Y/n) where X is χ²(λ, m) and Y is independently χ²(n). When n is even no complicated function evaluations are needed. For n odd, a special doubly noncentral t distribution is needed to start the computation. Series representations for this t distribution are given with rigorous bounds on truncation errors. Proper recursion techniques for numerical evaluation of the special functions are given.     

Truncation invariant dependence structures

In this paper, a special class of m-dimensional distribution functions which can be uniquely determined in terms of their 2-dimensional marginals is studied. The members of the class can be characterized as having truncation invariant dependence structure. The representation given in this paper provides a physical meaning to the multivariate Cook-Johnson distribution, and introduces a systematic way of generating higher dimensional distributions by using rich 2-dimensional distributions provided that the 2-dimensional marginals are compatible. A class of 3-dimensional multivariate normal distribution has been generated and bounds in terms of lower dimensional marginals are provided.     

Information theoretic results for circular distributions

The broad class of generalized von Mises (GvM) circular distributions has certain optimal properties with respect to information theoretic quantities. It is shown that, under constraints on the trigonometric moments, and using the Kullback–Leibler information as the measure, the closest circular distribution to any other is of the GvM form. The lower bounds for the Kullback–Leibler information in this situation are also provided. The same problem is also considered using a modified version of the Kullback–Leibler information. Finally, series expansions are given for the entropy and the normalizing constants of the GvM distribution.     

Inequalities for Expected Current Record Statistics

In this article, we obtain sharp distribution-free bounds for the expected value of the gap between the current records and record values as well as upper sharp bounds for the spacings between any two upper current records. We also present two-sided bounds on the errors in approximating the means of current records by inverse hazard functions.     

A polar coordinate transformation for estimating bivariate survival functions with randomly censored and truncated data

This paper proposes a new estimator for bivariate distribution functions under random truncation and random censoring. The new method is based on a polar coordinate transformation, which enables us to transform a bivariate survival function to a univariate survival function. A consistent estimator for the transformed univariate function is proposed. Then the univariate estimator is transformed back to a bivariate estimator. The estimator converges weakly to a zero-mean Gaussian process with an easily estimated covariance function. Consistent truncation probability estimate is also provided. Numerical studies show that the distribution estimator and truncation probability estimator perform remarkably well.     

PROBABILITY APPROXIMATIONS FOR DIVISIBLE DISCRETE DISTRIBUTIONS

If an integer-valued random variable can be represented as a sum of independent random variables, then powerful tools exist to derive approximations to its distribution. We apply this idea to examples in some of which it is not clear how to give a physical interpretation to the independent sum-mands. We consider bounds on the accuracy of single term approximations, Edgeworth expansions and saddlepoint approximations for both individual probabilities and cumulative probabilities.     

Comparison of pivotals for confidence bounds and intervals for the mean of a stationary time series

We construct new pivotals to obtain confidence bounds and confidence intervals for the mean of a stationary process. These follow the approach based on estimating functions. The new pivotals are compared with the standard pivotal based on studentization. We study the first four cumulants of each of these pivotals and explain why the pivotals based on the estimating function approach result in better coverage probabilities. Some simulation results comparing these pivotals have been reported.     

Berry-Esseen-type bounds for the kernel estimator of conditional distribution and conditional quantiles

Berry-Esseen bounds of order O(n^−1/2) have been obtained for several classes of statistics. In this paper, the rates of convergence in central limit theorem for conditional empirical functions and conditional sample quantiles based on kernel estimators are studied for both conditional and unconditional distributions.     

Testing for additivity and joint effects in multivariate nonparametric regression using Fourier and wavelet methods

We consider the problem of testing for additivity and joint effects in multivariate nonparametric regression when the data are modelled as observations of an unknown response function observed on a d-dimensional (d 2) lattice and contaminated with additive Gaussian noise. We propose tests for additivity and joint effects, appropriate for both homogeneous and inhomogeneous response functions, using the particular structure of the data expanded in tensor product Fourier or wavelet bases studied recently by Amato and Antoniadis (2001) and Amato, Antoniadis and De Feis (2002). The corresponding tests are constructed by applying the adaptive Neyman truncation and wavelet thresholding procedures of Fan (1996), for testing a high-dimensional Gaussian mean, to the resulting empirical Fourier and wavelet coefficients. As a consequence, asymptotic normality of the proposed test statistics under the null hypothesis and lower bounds of the corresponding powers under a specific alternative are derived. We use several simulated examples to illustrate the performance of the proposed tests, and we make comparisons with other tests available in the literature.     

A simple formula based on quantiles for the moments of beta generalized distributions

In this article, we derive explicit expansions for the moments of beta generalized distributions from power series expansions for the quantile functions of the baseline distributions. We apply our formula to the beta normal, beta Student t, beta gamma and beta beta generalized distributions. We propose a simple way to express the quantile function of any beta generalized distribution as a power series expansion with known coefficients.     

On the validity of the perturbation method in asymptotic theory

The perturbation methods and the Taylor expansions are very often used to obtain test statistics approximations in multivariate analysis (Specially in Principal Component and Canonical Analyses). These approximations are then used to obtain formal Edgeworth expransions of the distribution functions of the statistics. BHATTACHARYA and GHOSH 1978 have justified these practices under suitable assumptions. In this paper a non classical perturbation problem is solved in order to obtain almost surely expansions of test statistics     

Bounds on the covariate-time transformation for competing-risks survival analysis

A fundamental problem with the latent-time framework in competing risks is the lack of identifiability of the joint distribution. Given observed covariates along with assumptions as to the form of their effect, then identifiability may obtain. However it is difficult to check any assumptions about form since a more general model may lose identifiability. This paper considers a general framework for modelling the effect of covariates, with the single assumption that the copula dependency structure of the latent times is invariant to the covariates. This framework consists of a set of functions: the covariate-time transformations. The main result produces bounds on these functions, which are derived solely from the crude incidence functions. These bounds are a useful model checking tool when considering the covariate-time transformation resulting from any particular set of further assumptions. An example is given where the widely-used assumption of independent competing risks is checked.     

Percentile Related Bounds and Estimator for the Mean of the Negative Binomial Distribution

Upper and lower bounds are obtained for the mean of the negative binomial distribution. These bounds are simple functions of a percentile determined by the shape parameter. The result is then used to obtain a robust estimate of the mean when the shape parameter is known.     

Shortening the distance between Edgeworth and Berry–Esseen in the classical case

We obtain near optimal Berry–Esseen bounds for standardized sums of independent identically distributed random variables. This is achieved by distinguishing the lattice and the non-lattice cases, as one-term Edgeworth expansions do. The main tool is an easy inequality involving the usual second modulus of continuity, in substitution of Esseen's smoothing inequality. An illustrative example concerning the exponential distribution is also considered.     

Conditional Quantile Estimation for Truncated and Associated Data

Abstract

In survival or reliability studies, it is common to have data which are not only incomplete but weakly dependent too. Random truncation and censoring are two common forms of such data when they are neither independent nor strongly mixing but rather associated. The focus of this paper is on estimating conditional distribution and conditional quantile functions for randomly left truncated data satisfying association condition. We aim at deriving strong uniform consistency rates and asymptotic normality for the estimators and thereby, extend to association case some results stated under iid and α-mixing hypotheses. The performance of the quantile function estimator is evaluated on simulated data sets.