Order restricted inference for comparing the cumulative incidence of a competing risk over several populations

There is a substantial literature on testing for the equality of the cumulative incidence functions associated with one specific cause in a competing risks setting across several populations against specific or all alternatives. In this paper we propose an asymptotically distribution-free test when the alternative is that the incidence functions are linearly ordered, but not equal. The motivation stems from the fact that in many examples such a linear ordering seems reasonable intuitively and is borne out generally from empirical observations. These tests are more powerful when the ordering is justified. We also provide estimators of the incidence functions under this ordering constraint, derive their asymptotic properties for statistical inference purposes, and show improvements over the unrestricted estimators when the order restriction holds.Comment: Published in at http://dx.doi.org/10.1214/193940307000000040 the IMS Collections (http://www.imstat.org/publications/imscollections.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

A unified approach to testing for and against a set of linear inequality constraints in the product multinomial setting

AbstractA problem that is frequently encountered in statistics concerns testing for equality of multiple probability vectors corresponding to independent multinomials against an alternative they are not equal. In applications where an assumption of some type of stochastic ordering is reasonable, it is desirable to test for equality against this more restrictive alternative. Similar problems have been considered heretofore using the likelihood ratio approach. This paper aims to generalize the existing results and provide a unified technique for testing for and against a set of linear inequality constraints placed upon on any r(r≥1) probability vectors corresponding to r independent multinomials. The paper shows how to compute the maximum likelihood estimates under all hypotheses of interest and obtains the limiting distributions of the likelihood ratio test statistics. These limiting distributions are of chi bar square type and the expression of the weighting values is given. To illustrate our theoretical results, we use a real life data set to test against second-order stochastic ordering

A linear-quadratic distributional identity

In this paper we extend the result of Bateman (1949) to obtain the joint conditional characteristic function of a sum of squares of n noncentral squares of independent normal variates and s1 of their linear combinations given that these variates lie in a subspace of dimension n - s2.Characteristic function Chi-bar square Subspace

Restricted estimation of the cumulative incidence functions corresponding to competing risks

Open accessIn the competing risks problem, an important role is played by the cumulative incidence function (CIF), whose value at time t is the probability of failure by time t from a particular type of failure in the presence of other risks. In some cases there are reasons to believe that the CIFs due to various types of failure are linearly ordered. El Barmi et al. [3] studied the estimation and inference procedures under this ordering when there are only two causes of failure. In this paper we extend the results to the case of k CIFs, where k > 3. Although the analyses are more challenging, we show that most of the results in the 2-sample case carry over to this fc-sample case.Peer reviewe

A unified approach to testing for and against a set of linear inequality constraints in the product multinomial setting

A problem that is frequently encountered in statistics concerns testing for equality of multiple probability vectors corresponding to independent multinomials against an alternative they are not equal. In applications where an assumption of some type of stochastic ordering is reasonable, it is desirable to test for equality against this more restrictive alternative. Similar problems have been considered heretofore using the likelihood ratio approach. This paper aims to generalize the existing results and provide a unified technique for testing for and against a set of linear inequality constraints placed upon on any probability vectors corresponding to r independent multinomials. The paper shows how to compute the maximum likelihood estimates under all hypotheses of interest and obtains the limiting distributions of the likelihood ratio test statistics. These limiting distributions are of chi bar square type and the expression of the weighting values is given. To illustrate our theoretical results, we use a real life data set to test against second-order stochastic ordering.Chi bar square Inequality constraints Lagrange multipliers Likelihood ratio Orthant probabilities Stochastic ordering

Transformation-based density estimation for weighted distributions

In this paper we consider the estimation of a density f on the basis of random sample from a weighted distribution G with density g given by g(x) = w(x)f(x)/ õw, where w(u)>0 for all u and õw = â« w(u)f(u)du < âÂÂ. A special case of this situation is that of length-biased sampling, where w(x) = x. In this paper we examine a simple transformation-based approach to estimating the density f. The approach is motivated by the form of the nonparametric estimator of f in the same context and under a monotonicity constraint. Since the method does not depend on the specific density estimate used (only the transformation), it can be used to construct both simple density estimates (histograms or frequency polygons) and more complex methods with favorable properties (e.g., local or penalized likelihood estimates). Monte Carlo simulations indicate that transformation-based density estimation can outperform the kernel-based estimator of Jones (1991) depending on the weight function w, and leads to much better estimation of monotone densities than the nonparametric maximum likelihood estimator.Statistics Working Papers Serie