For a response variable $Y$, and a $d$ dimensional vector of covariates $\mathbf{X}$, the first projective direction, $\mathbf{\vartheta}$, is defined as the direction that accounts for the most variability in $Y$. The asymptotic distribution of an estimator of a trimmed version of $\mathbf{\vartheta}$ has been characterized only under the assumption of the single index model (SIM). This paper proposes the use of a flexible trimming function in the objective function, which…
The classical F test for the hypothesis of no main fixed effects in the two-way crossed mixed effects design is derived under model assumptions that include normality, variance homogeneity, and symmetry. The symmetry assumption specifies that the random main and interaction effects are independent. While it is known that the F test is robust against violations of the normality assumption and/or the homoscedasticity assumption in the balanced case, the effects of the symmetry…
Summary Consider a pair of random variables, both subject to random right censoring. New estimators for the bivariate and marginal distributions of these variables are proposed. The estimators of the marginal distributions are not the marginals of the corresponding estimator of the bivariate distribution. Both estimators require estimation of the conditional distribution when the conditioning variable is subject to censoring. Such a method of estimation is proposed. The weak…
Consider a heteroscedastic regression model $Y = m(X) + \sigma(X)\varepsilon$, where the functions $m$ and $\sigma$ are “smooth,” and $\varepsilon$ is independent of $X$. The response variable $Y$ is subject to random censoring, but it is assumed that there exists a region of the covariate $X$ where the censoring of $Y$ is “light.” Under this condition, it is shown that the assumed nonparametric regression model can be used to transfer tail information from regions of light…
Nonparametric hypotheses of no main effects, no simple effects, and no interaction effects are considered in the context of factorial designs with censored observations. To test these hypotheses, the asymptotic distribution of quadratic forms of rank statistics is derived using the methodology of martingales for counting processes. The weights used reduce to the usual logistic scores with uncensored data but are different from the weights commonly used for testing the…
Factorial designs are studied with independent observations, fixed number of levels, and possibly unequal number of observations per factor level combination. In this context, the nonparametric null hypotheses introduced by Akritas and Arnold are considered. New rank statistics are derived for testing the nonparametric hypotheses of no main effects, no interaction, and no factor effects in unbalanced crossed classifications. The formulation of all results includes tied…
The Theil-Sen estimator of the slope parameter in simple linear regression is extended to data with both the response and the covariate subject to censoring. Based on inverting a suitable version of Kendall's τ statistic, this estimator requires weak assumptions and is simple to compute, and a simple estimate of its asymptotic variance is obtained. A second extension of the Theil-Sen estimator, based on a direct estimation of the median of pairwise slopes, is given. These…
We consider the problem of estimating the bivariate distribution of the random vector $(X, Y)$ when $Y$ may be subject to random censoring. The censoring variable $C$ is allowed to depend on $X$ but it is assumed that $Y$ and $C$ are conditionally independent given $X = x$. The estimate of the bivariate distribution is obtained by averaging estimates of the conditional distribution of $Y$ given $X = x$ over a range of values of $x$. The weak convergence of the centered…
We introduce nonparametric versions for many of the hypotheses tested in analysis of variance and repeated measures models, such as the hypotheses of no main effects, no interaction effects, and no factor effects. These natural extensions of the nonparametric hypothesis of equality of the k distributions in the k sample problem have appealing practical interpretations. We concentrate on multivariate repeated measures designs and obtain simple rank statistics for testing…
The use of the rank transform for testing problems in some two-factor designs is considered. The rank transform procedure consists of replacing the observations by their ranks in the combined sample and performing one of the standard analysis of variance (ANOVA) procedures on these ranks. The asymptotic version of the rank transform is introduced, and its usefulness as a means for understanding the nature of the rank transform is examined. It is demonstrated that the…
An interval censoring model is carefully defined and the range of its applicability is illustrated. A class of rank tests for the two- and $k$-sample problems is proposed. The statistic is based on the exact ranks of the observed responses and resembles the rank statistic in the usual uncensored case. This statistic is shown to be asymptotically equivalent to the efficient scores test statistic under an assumed parametric model. An alternative rank statistic, based on the…
Chi-squared tests are proposed for testing goodness of fit when the data may or may not be subject to random censoring. Both the simple and compositive null hypotheses are considered. For the simple null hypothesis the resulting test statistic in the uncensored case differs from Pearson's statistic only in that the expected frequency for cell Aj is taken to be , where is the empirical distribution function, instead of n ∫ A j d F o ; this results in k df instead of the usual…
In the two-sample problem under random censorship we consider the uncensored observations and the censored ones as two separate groups. For each group a suitable rank statistic is obtained, and these two are then combined to a final one. This idea, which was first employed in Akritas (1983), is shown here to produce tests that (a) closely resemble the ordinary rank tests for the uncensored case and do not require the calculation of the Kaplan-Meier estimator, (b) are…
In the two-sample problem under random censorship we consider the uncensored observations and the censored ones as two separate groups. For each group a suitable rank statistic is obtained, and these two are then combined to a final one. This idea, which was first employed in Akritas (1983), is shown here to produce tests that (a) closely resemble the ordinary rank tests for the uncensored case and do not require the calculation of the Kaplan-Meier estimator, (b) are…
A class of empirical processes associated with $V$-statistics ($V$-empirical process) under random censoring, and a class of nonparametric estimators based on the corresponding quantile process are defined. The $V$-empirical process is the censored data analogue of the $U$-empirical process considered by Silverman (1976, 1983). The class of estimators is the analogue of the class of generalized $L$-statistics introduced by Serfling (1984) and it includes the results of…
We establish contiguity of certain families of probability measures indexed by $T$, as $T \rightarrow \infty$, for classes of stochastic processes with stationary, independent increments whose sample paths are discontinuous. Many important consequences pertaining to properties of tests and estimates then apply. A new expression for the Radon-Nikodym derivative of these processes is obtained.
For a response variable Y, and a d dimensional vector of covariates X, the first projective direction, ϑ, is defined as the direction that accounts for the most variability in Y. The asymptotic distribution of an estimator of a trimmed version of ϑ has been characterized only under the assumption of the single index model (SIM). This paper proposes the use of a flexible trimming function in the objective function, which…
Consider a heteroscedastic regression model Y=m(X)+σ(X)ε, where the functions m and σ are “smooth,” and ε is independent of X. The response variable Y is subject to random censoring, but it is assumed that there exists a region of the covariate X where the censoring of Y is “light.” Under this condition, it is shown that the assumed nonparametric regression model can be used to transfer tail information from regions of light…
We consider the problem of estimating the bivariate distribution of the random vector (X,Y) when Y may be subject to random censoring. The censoring variable C is allowed to depend on X but it is assumed that Y and C are conditionally independent given X=x. The estimate of the bivariate distribution is obtained by averaging estimates of the conditional distribution of Y given X=x over a range of values of x. The weak convergence of the centered…
An interval censoring model is carefully defined and the range of its applicability is illustrated. A class of rank tests for the two- and k-sample problems is proposed. The statistic is based on the exact ranks of the observed responses and resembles the rank statistic in the usual uncensored case. This statistic is shown to be asymptotically equivalent to the efficient scores test statistic under an assumed parametric model. An alternative rank statistic, based on the…
Empirical Processes Associated with V-Statistics and a Class of Estimators Under Random Censoring
A class of empirical processes associated with V-statistics (V-empirical process) under random censoring, and a class of nonparametric estimators based on the corresponding quantile process are defined. The V-empirical process is the censored data analogue of the U-empirical process considered by Silverman (1976, 1983). The class of estimators is the analogue of the class of generalized L-statistics introduced by Serfling (1984) and it includes the results of…
We establish contiguity of certain families of probability measures indexed by T, as T→∞, for classes of stochastic processes with stationary, independent increments whose sample paths are discontinuous. Many important consequences pertaining to properties of tests and estimates then apply. A new expression for the Radon-Nikodym derivative of these processes is obtained.