A probabilistic model of aging is considered. It is based on the assumption that a random resource, a stochastic process of aging (wear) and the corresponding anti-aging process are embedded at birth. A death occurs when the accumulated wear exceeds the initial random resource. It is assumed that the anti-aging process decreases wear in each increment. The impact of environment (lifestyle) is also taken into account. The corresponding relations for the observed and the conditional hazard rate (force of mortality) are obtained. Similar to some demographic models, the deceleration of mortality phenomenon is explained via the concept of frailty. Simple examples are considered.

The weaknesses of established model selection procedures based on hypothesis testing and similar criteria are discussed and an alternative based on synthetic (composite) estimation is proposed. It is developed for the problem of prediction in ordinary regression and its properties are explored by simulations for the simple regression. Extensions to a general setting are described and an example with multiple regression is analysed. Arguments are presented against using a selected model for any inferences.

As already mentioned in the Preface to Volume I, and then detailed in its Section 1.5, in Volume II of this monograph constructional aspects of suitable designs are to be discussed, particularly with regard to some statistical concepts essential for planning of experiments. In the present chapter various methods of constructing designs will be described, along with some combinatorial and statistical properties of block designs, mainly those related to the efficiency factors of the design for estimating the corresponding basic contrasts in the intra-block analysis, as defined in Section 3.4. In general, the classification of block designs adopted here is based on the new terminology proposed in Section 4.4 (see, in particular, Definition 4.4.2). It may be helpful to recall it shortly.

The Reed-Frost epidemic model is a simple stochastic process with parameter q that describes the spread of an infectious disease among a closed population. Given data on the final outcome of an epidemic, it is possible to perform Bayesian inference for q using a simple Gibbs sampler algorithm. In this paper it is illustrated that by choosing latent variables appropriately, certain monotonicity properties hold which facilitate the use of a perfect simulation algorithm. The methods are applied to real data.

Let {X_t;t∈ℤ be a strictly stationary nonlinear process of the formX_t=ε_t+∑ _r=1^∞W_rt, whereW_rt can be written as a functiong_r(ε_t−1,...ε_t-r-q), {ε_t;t∈ℤ is a sequence of independent and identically distributed (i.i.d.) random variables withE|ε₁|^g < ∞ for some γ>0 andq≥0 is fixed integer. Under certain mild regularity conditions ofg_r and {ε_t} we then show thatX₁ has a density functionf and that the standard kernel type estimator $$\hat f_n (x)$$ baded on a realization {X₁,...,X_n} from {X_t} is, asymptotically, normal and converges a.s. tof(x) asn→∞.

In this chapter, we consider second-order properties of block bootstrap estimators for estimating the sampling distribution of a statistic of interest. The basic tool for studying second-order properties of block bootstrap distribution function estimators is based on the theory of Edgeworth expansions.

The jackknife variance estimator and the infinitesimal jackknife variance estimator are shown to be asymptotically equivalent if the functional of interest is a smooth function of the mean or a trimmed L-statistic with Hölder continuous weight function.

We discuss in this paper the assessment of local influence in univariate elliptical linear regression models. This class includes all symmetric continuous distributions, such as normal, Student-t, Pearson VII, exponential power and logistic, among others. We derive the appropriate matrices for assessing the local influence on the parameter estimates and on predictions by considering as influence measures the likelihood displacement and a distance based on the Pearson residual. Two examples with real data are given for illustration.

Various methods of constructing confidence sets are introduced in this chapter, along with studies of properties of confidence sets. Throughout this chapter X = (X₁, …,X_n) denotes a sample from a population P ∈ P; θ = θ(P) denotes a functional from P to Θ ⊂ R^k for a fixed integer k; and C(X) denotes a confidence set for θ, a set in ß_Θ (the class of Borel sets on Θ) depending only on X. We adopt the basic concepts of confidence sets introduced in §2.4.3. In particular, inf_P∈PP(θ ∈ C(X)) is the confidence coefficient of C(X) and, if the confidence coefficient of C(X) is ≥ 1- α for fixed α ∈ (0, 1), then we say that C(X) has significance level 1-α or C(X) is a level 1 - α confidence set.