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We consider marked empirical processes indexed by a randomly projected functional covariate to construct goodness-of-fit tests for the functional linear model with scalar response. The test statistics are built from continuous functionals over the projected process, resulting in computationally efficient tests that exhibit root-n convergence rates and circumvent the curse of dimensionality. The weak convergence of the empirical process is obtained conditionally on a random direction, whilst the almost surely equivalence between the testing for significance expressed on the original and on the projected functional covariate is proved. The computation of the test in practice involves calibration by wild bootstrap resampling and the combination of several p-values, arising from different projections, by means of the false discovery rate method. The finite sample properties of the tests are illustrated in a simulation study for a variety of linear models, underlying processes, and alternatives. The software provided implements the tests and allows the replication of simulations and data applications., Paper: 23 pages, 4 figures, 1 table. Supplementary material: 17 pages, 4 figures, 3 tables

From a multiple testing viewpoint, Kolmogorov–Smirnov (KS)-type tests are union-intersection tests which can be redefined in terms of local levels. The local level perspective offers a new viewpoint on ranges of sensitivity of KS-type tests and the design of new tests. We study the finite and asymptotic local level behavior of weighted KS tests which are either tail, intermediate or central sensitive. Furthermore, we provide new tests with approximately equal local levels and prove that the asymptotics of such tests with sample sizes $m$ and $n$ coincides with the asymptotics of one-sample higher criticism tests with sample size $\min (m,n)$. We compare the overall power of various tests and introduce local powers that are in line with local levels. Finally, suitably parameterized local level shape functions can be used to design new tests. We illustrate how to combine tests with different sensitivity in terms of local levels.

When fitting, by least squares, a linear model (with an intercept term) with $p$ parameters to $n$ data points, the asymptotic behavior of the residual empirical process is shown to be the same as in the single sample problem provided $p^3 \log^2 (p) /n \to 0$ for any error density having finite variance and a bounded first derivative. No further conditions are imposed on the sequence of design matrices. The result is extended to more general estimates with the property that the average error and average squared error in the fitted values are on the same order as for least squares.

The power of Kolmogorov-Smirnov tests can be increased by transforming the empirical process into a new process that converges to a Wiener process under the null hypothesis and by choosing the transformation in such a way that some families of local alternatives become as noticeable as possible.

This paper establishes the asymptotic uniform linearity of M- and R-scores in a family of nonlinear time series and regression models. It also gives an asymptotic expansion of the standardized sequential residual empirical process in these models. These results are, in turn, used to obtain the asymptotic normality of certain classes of M-, R- and minimum distance estimators of the underlying parameters. The classes of estimators considered include analogs of Hodges-Lehmann, Huber and LAD (least absolute deviation) estimators. Some applications to the change point and testing of the goodness-of-fit problems in threshold and amplitude-dependent exponential autoregression models are also given. The paper thus offers a unified functional approach to some aspects of robust inference for a large class of nonlinear time series models.

It has been over 60 years since Kolmogorov introduced a distribution-free omnibus test for the simple null hypothesis that a distribution function coincides with a given distribution function. Doob subsequently observed that Kolmogorov's approach could be simplified by transforming the empirical process to an empirical process based on uniform random variables. Recent use of more sophisticated transformations has led to the construction of asymptotically distribution-free omnibus tests when unknown parameters are present. The purpose of the present paper is to use the transformation approach to construct an asymptotically distribution-free omnibus test for independence of a survival time from a covariate. The test statistic is obtained from a certain test statistic process (indexed by time and covariate), which is shown to converge in distribution to a Brownian sheet. A simulation study is carried out to investigate the finite sample properties of the proposed test and an application to data from the British Medical Research Council's 4th myelomatosis trial is given.

Dufour gives a conjecture concerning a characterization of the exponential distribution based on type 2 right censored samples. This conjecture, if true, generalizes the characterization based on complete samples of Seshadri, Csorgo and Stephens (1969) and Dufour, Maag and van Eeden (1984). In this paper it is shown that Dufour's conjecture is true if the number of censored observations is no larger than $(1/3)n - 1$, where $n$ is the sample size. The result has implications for testing fit of censored data to the exponential distribution.

How often does the empirical distribution pass over an acceptance band around the hypothetical distribution function? Sheffer polynomials are applied to derive the exact distribution of this goodness-of-fit distribution and its two-sample analogue. This method leads to new proofs and extensions of Takacs's (1971) results.

Let $(\phi_\nu)^\infty_{\nu=0}$ be a complete orthonormal system of functions on a interval $\lbrack a, b\rbrack$ and let $w$ be a function defined on $\mathbb{R}$ with support $\lbrack a, b\rbrack$ and strictly positive on $(a, b)$. Let $(x_j)^\infty_{j=1}$ be i.i.d.rv's absolutely continuous with density function $f$ with respect to Lebesgue measure $\mu$ on $\mathbb{R}$. Let $f_n$ be the estimate of $f$ defined for $x\in (a, b)$ by $f_n(x) = \Sigma^m_{\nu=0}a_\nu(m)\hat{d}_\nu\phi_\nu(x)/w(x)$, where $a_\nu(m), (m = 0, 1, \cdots, \nu = 0, 1, \cdots, m)$ is a sequence in $\mathbb{R}$ and $\hat{d}_\nu = n^{-1}\Sigma^n_{j=1}\phi_\nu(X_j)w(X_j)$. In this work the asymptotic distributions of the functionals $T_n = n(m + 1)^{-1} \int^b_a(f_n - Ef_n)^2w^2 dx$ and $T^\ast_n = n(m + 1)^{-1} \int^b_a(f_n - f)^2w^2 dx$ are found. These results are used to construct tests of goodness-of-fit analogous to those proposed by Bickel and Rosenblatt. The basic idea in obtaining the results consists in finding the asymptotic distribution of $T_n(T^\ast_n)$ with $f_n$ replaced by a conveniently chosen Gaussian process and showing that the two functionals converge to the same law. For this the normalized and centered sample distribution function is approximated by an appropriate Brownian motion process by using a Skorohod-like imbedding due to Brillinger and Breiman.

Let $X_1, \cdots, X_n$ be a random sample from a full-rank multivariate normal distribution $N(\mu, \Sigma)$. The two cases (i) $\mu$ unknown and $\Sigma = \sigma^2\Sigma_0, \Sigma_0$ known, and (ii) $\mu$ and $\Sigma$ completely unknown are considered here. Transformations are given that transform the observation vectors to a (smaller) set of i.i.d. uniform rv's. These transformations can be used to construct goodness-of-fit tests for these multivariate normal distributions. Two examples are given to illustrate the application of these tests to numerical problems.

Two Pearson-type goodness-of-fit test statistics for parametric families are considered for randomly right-censored data. Asymptotic distribution theory for the test statistics is based on the result that the product-limit process with MLE for nuisance parameters converges weakly to a Gaussian process. The Chernoff-Lehmann (1954) result extends to a generalized Pearson statistic. A modified Pearson statistic is shown to have a limiting chi-square null distribution.