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In many settings, multiple data collections and analyses on the same topic are summarised separately through statistical estimators of parameters and variances, and yet there are scientific...

Linear mixed models (LMMs) are widely used for regression analysis of data that are assumed to be clustered or correlated. Assessing model fit is important for valid inference but to date no confirmatory tests are available to assess the adequacy of the fixed effects part of LMMs against general alternatives. We therefore propose a class of goodness-of-fit tests for the mean structure of LMMs. Our test statistic is a quadratic form of the difference between observed values and the values expected under the estimated model in cells defined by a partition of the covariate space. We show that this test statistic has an asymptotic chi-squared distribution when model parameters are estimated by maximum likelihood or by least squares and method of moments, and study its power under local alternatives both analytically and in simulations. Data on repeated measurements of thyroglobulin from individuals exposed to the accident at the Chernobyl power plant in 1986 are used to illustrate the proposed test.

After a brief historical survey of parametric survival models, from actuarial, biomedical, demographical and engineering sources, this paper discusses the persistent reasons why parametric models still play an important role in exploratory statistical research. The phase-type models are advanced as a flexible family of latent-class models with interpretable components. These models are now supported by computational statistical methods that make numerical calculation of likelihoods and statistical estimation of parameters feasible in theory for quite complicated settings. However, consideration of Fisher Information and likelihood-ratio type tests to discriminate between model families indicates that only the simplest phase-type model topologies can be stably estimated in practice, even on rather large datasets. An example of a parametric model with features of mixtures, multiple stages or 'hits', and a trapping-state is given to illustrate simple computational tools in R, both on simulated data and on a large SEER 1992-2002 breast-cancer dataset.

Necessary and sufficient analytical conditions are given for homogeneous multiple Wiener-Itô integral processes (MWIs) to be mixing, and sufficient conditions are given for mixing of general square-integrable Gaussian-subordinated processes. It is shown that every finite or infinite sum Y of MWIs (i.e. every real square-integrable stationary polynomial form in the variables of an underlying weakly mixing Gaussian process) is mixing if the process defined separately by each homogeneous-order term is mixing, and that this condition is necessary for a large class of Gaussian-subordinated processes. Moreover, for homogeneous MWIs Y1, for sums of MWIs of order ≤ 3, and for a large class of square-integrable infinite sums Y1, of MWIs, mixing holds if and only if Y2 has correlation-function decaying to zero for large lags. Several examples of the criteria for mixing are given, including a second-order homogeneous MWI, i.e. a degree two polynomial form, orthogonal to all linear forms, which has auto-correlations tending to zero for large lags but is not mixing.

Consistency issues related to autocorrelation estimation for Gaussian processes with mixed spectra are clarified. The sample autocorrelation is known not to be consistent when the spectrum contains spectral atoms. This fact is verified by computing explicitly its mean-square limit in terms of the random measure assigned to the atoms of the process. The alternative estimator constructed from the zero-crossing rate is likewise not consistent in general. However, it is consistent for a spectrum supported at a single frequency, or a spectrum for which the ‘signal’ and ‘noise’ have the exact same first-order autocorrelation. This is proved, using recent results from multiple Wiener-Ito integral expansions for level-crossing counts, by direct computation of the spectrum of the zero-crossing indicator process when the underlying process has a mixed spectrum. In general, regardless of the spectral type, the asymptotic zero-crossing rate admits values between the lowest and highest positive frequencies with probability one. For band-limited processes, this fact provides an easy way to assess the precision of functions of the zero-crossing rate.

This paper applies the stochastic calculus of multiple Wiener-Ito integral expansions to express the number of crossings of the mean level by a stationary (discrete- or continuous-time) Gaussian process within a fixed time interval [0,T]. The resulting expansions involve a class of hypergeometric functions, for which recursion and differential relations and some asymptotic properties are derived. The representation obtained for level-crossing counts is applied to prove a central limit theorem of Cuzick (1976) for level crossings in continuous time, using a general central limit theorem of Chambers and Slud (1989a) for processes expressed via multiple Wiener-Ito integral expansions in terms of a stationary Gaussian process. Analogous results are given also for discrete-time processes. This approach proves that the limiting variance is strictly positive, without additional assumptions needed by Cuzick.

Let $\mathbf{X} = (X_t, t \geq 0)$ be a stationary Gaussian process with zero mean, continuous spectral distribution and twice-differentiable correlation function. An explicit representation is given for the number $N_\psi(T)$ of crossings of a $C^1$ curve $\psi$ by $\mathbf{X}$ on the bounded interval $\lbrack 0, T\rbrack$, in a multiple Wiener-Ito integral expansion. This continues work of the author in which the result was given for $\psi \equiv 0$. The representation is applied to prove new central and noncentral limit theorems for numbers of crossings of constant levels, and some consequences for asymptotic variances are given in mixed-spectrum settings.

The asymptotic distribution theory of sequentially computed modified-Wilcoxon scores is developed for two-sample survival data with random staggered entry and random loss to follow-up. The asymptotic covariance indicates generally dependent modified-Wilcoxon increments, contradicting (the authors' reading of) Jones and Whitehead (1979). A repeated significance testing procedure is presented for testing the equality of two survival distributions based on the asymptotic theory. The early stopping properties of this procedure are illustrated by a prostate cancer example.

Let X=(X t ,t∈ℝ) be a stationary Gaussian process on (Ω, ℱ,P), letH(X) be the Hilbert space of variables inL 2 (Ω,P) which are measurable with respect toX, and let (U s ,s∈ℝ) be the associated family of time-shift operators. We sayY∈H(X) (withE(Y)=0) satisfies the functional central limit theorem or FCLT [respectively, the central limit theorem of CLT if in [respectively, ], where $$Y_T (t) \equiv {{\int\limits_0^{Tt} {U_s \circ Yds} } \mathord{\left/ {\vphantom {{\int\limits_0^{Tt} {U_s \circ Yds} } {\left\{ {Var\left( {\int\limits_0^T {U_s \circ Yds} } \right)} \right\}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}} \right. \kern-\nulldelimiterspace} {\left\{ {Var\left( {\int\limits_0^T {U_s \circ Yds} } \right)} \right\}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}$$ andW(•) is a standard Wiener process on [0,1]. This paper provides some general sufficient conditions onX andY ensuring thatY satisfies the CLT or FCLT. Examples ofY are given which satisfy the CLT but not the FCLT. This work extends CLT's of Maruyama (1976) and Breuer and Major (1983).

A new class of reliability point-process models for dependent components is introduced. The dependence is expressed through a regression, following a form suggested by Cox (1972) for survival data analysis involving the current life-length of the components. After formulating the current-life process as a Markov process with stationary transitions and stating some general results on asymptotic behavior, we describe the stationary distributions in some bivariate examples. Finally, we discuss statistical inference for the new models, exhibiting and justifying full- and partial-likelihood methods for their analysis.

In two-sample semiparametric survival models other than the Cox proportional-hazards regression model, it is shown that partial-likelihood inference of structural parameters in the presence of fully nonpararnetric nuisance-hazards typically has relative efficiency zero compared with fuii-Iikelihood infer -ence. The practical Interpretation of efficiencies in the pres-ence of infinite-dimensional nuisance-parameters is discussed, with reference to two important examples, namely a recent sur-vival regression-model of Clayton and Cuzick and a class of additive excess-risk models. Under the excess-risk models, a formula is derived for the large-sample information [which here is the same as the limiting Fisher information when the nuisance-parameter dimension gets large] for estimating the parameter of difference between two samples, as the nuisance function becomes fully nonpararnetric.

and let H(X) = L’(f), u(X), P) denote the space of square-integrable functionals of X. Say that Y t H(X) with EY =O satisfies the Central Limit Theorem (CLT) if A family of martingales (Z,(t), t 2 0) is exhibited for which 2, (“c) = Z,, and martingale tech- niques and results are used to provide suficient conditions on X and Y for the CLT. These conditions are then shown to be necessary for slightly more restrictive central limit behavior of Y. AMS Subject Ch\jfication.s: hOF05, 60G10, 60644 multiple Wiener-It8 integral 3 stochastic integral with respect to square-integrable martingale * predictable variance process * martingale Functional Central Limit Theorem * Band- weighted Central Limit Theorem

It is shown that the level-crossings process of zeroes and ones corresponding to a stationary but not independent Gaussian sequence can never be exactly ( m -step) Markov, although its correlation-sequence can agree exactly with that of a Markov sequence.

If a strictly stationary process $\{Z_k\}$ has residuals $Z_{k+1} - \sum^k_{j=1} a_{k,j}Z_j$ independent of $(Z_1, \cdots, Z_k)$ for all $k \geq m$, it is shown that the process is Gaussian or degenerate or $m$-step Markovian. Generalized (nonlinear) autoregressive stationary processes are defined and partially characterized.