Fitting parametric frailty and mixture models under biased sampling

Biased sampling from an underlying distribution with p.d.f. f(t), t>0, implies that observations follow the weighted distribution with p.d.f. f ^w (t)=w(t)f(t)/E[w(T)] for a known weight function w. In particular, the function w(t)=t ^α has important applications, including length-biased sampling (α=1) and area-biased sampling (α=2). We first consider here the maximum likelihood estimation of the parameters of a distribution f(t) under biased sampling from a censored population in a proportional hazards frailty model where a baseline distribution (e.g. Weibull) is mixed with a continuous frailty distribution (e.g. Gamma). A right-censored observation contributes a term proportional to w(t)S(t) to the likelihood; this is not the same as S ^w (t), so the problem of fitting the model does not simply reduce to fitting the weighted distribution. We present results on the distribution of frailty in the weighted distribution and develop an EM algorithm for estimating the parameters of the model in the important Weibull–Gamma case. We also give results for the case where f(t) is a finite mixture distribution. Results are presented for uncensored data and for Type I right censoring. Simulation results are presented, and the methods are illustrated on a set of lifetime data.     

ON TRANSITION PROBABILITIES OF SKIP-FREE MARKOV CHAINS

Consider an ergodic Markov chain X(t) in continuous time with an infinitesimal matrix Q = (q_ij) defined on a finite state space {0, 1,…, N}. In this note, we prove that if X(t) is skip-free positive (negative, respectively), i.e., q_ij, = 0 for j > i+ 1 (i > j+ 1), then the transition probability p_ij(t) = Pr[X(t)=j | X(0) =i] can be represented as a linear combination of p_0N(t) (p^(m)_(N0)(t)), 0 ≤ m ≤N, where f^(m)(t) denotes the mth derivative of a function f(t) with f⁽⁰⁾(t) =f(t). If X(t) is a birth-death process, then p_ij(t) is represented as a linear combination of p_0N^(m)(t), 0 ≤m≤N - |i-j|.     

Model fitting and inference under latent equilibrium processes

This paper presents a methodology for model fitting and inference in the context of Bayesian models of the type f(Y | X,θ)f(X|θ)f(θ), where Y is the (set of) observed data, θ is a set of model parameters and X is an unobserved (latent) stationary stochastic process induced by the first order transition model f(X ^(t+1)|X ^(t),θ), where X ^(t) denotes the state of the process at time (or generation) t. The crucial feature of the above type of model is that, given θ, the transition model f(X ^(t+1)|X ^(t),θ) is known but the distribution of the stochastic process in equilibrium, that is f(X|θ), is, except in very special cases, intractable, hence unknown. A further point to note is that the data Y has been assumed to be observed when the underlying process is in equilibrium. In other words, the data is not collected dynamically over time. We refer to such specification as a latent equilibrium process (LEP) model. It is motivated by problems in population genetics (though other applications are discussed), where it is of interest to learn about parameters such as mutation and migration rates and population sizes, given a sample of allele frequencies at one or more loci. In such problems it is natural to assume that the distribution of the observed allele frequencies depends on the true (unobserved) population allele frequencies, whereas the distribution of the true allele frequencies is only indirectly specified through a transition model. As a hierarchical specification, it is natural to fit the LEP within a Bayesian framework. Fitting such models is usually done via Markov chain Monte Carlo (MCMC). However, we demonstrate that, in the case of LEP models, implementation of MCMC is far from straightforward. The main contribution of this paper is to provide a methodology to implement MCMC for LEP models. We demonstrate our approach in population genetics problems with both simulated and real data sets. The resultant model fitting is computationally intensive and thus, we also discuss parallel implementation of the procedure in special cases.     

On the weak convergence of kernel density estimators in Lp spaces

Since its introduction, the pointwise asymptotic properties of the kernel estimator f?_n of a probability density function f on ?^d, as well as the asymptotic behaviour of its integrated errors, have been studied in great detail. Its weak convergence in functional spaces, however, is a more difficult problem. In this paper, we show that if f_n(x)=(f?_n(x)) and (r_n) is any nonrandom sequence of positive real numbers such that r_n/√n→0 then if r_n(f?_n?f_n) converges to a Borel measurable weak limit in a weighted L^p space on ?^d, with 1≤p<∞, the limit must be 0. We also provide simple conditions for proving or disproving the existence of this Borel measurable weak limit.     

On kernel estimators of density ratio

Let f(x) and g(x) denote two probability density functions and g(x)≠0. There are two ways to estimate the density ratio f(x)/g(x). One is to estimate f(x) and g(x) first and then the ratio, the other is to estimate f(x)/g(x) directly. In this paper, we derive asymptotic mean square errors and central limit theorems for both estimators.     

PLUG-IN ESTIMATION OF GENERAL LEVEL SETS

Given an unknown function (e.g. a probability density, a regression function, …) f and a constant c, the problem of estimating the level set L(c) ={f≥c} is considered. This problem is tackled in a very general framework, which allows f to be defined on a metric space different from . Such a degree of generality is motivated by practical considerations and, in fact, an example with astronomical data is analyzed where the domain of f is the unit sphere. A plug‐in approach is followed; that is, L(c) is estimated by L_n(c) ={f_n≥c} , where f_n is an estimator of f. Two results are obtained concerning consistency and convergence rates, with respect to the Hausdorff metric, of the boundaries ?L_n(c) towards ?L(c) . Also, the consistency of L_n(c) to L(c) is shown, under mild conditions, with respect to the L₁ distance. Special attention is paid to the particular case of spherical data.     

On a Class of Distributions Defined by the Relationship Between Their Density and Distribution Functions

Knowledge concerning the family of univariate continuous distributions with density function f and distribution function F defined through the relation f(x) = F ^α(x)(1 ? F(x))^β, α, β ? , is reviewed and modestly extended. Symmetry, modality, tail behavior, order statistics, shape properties based on the mode, L-moments, and—for the first time—transformations between members of the family are the general properties considered. Fully tractable special cases include all the complementary beta distributions (including uniform, power law and cosine distributions), the logistic, exponential and Pareto distributions, the Student t distribution on 2 degrees of freedom and, newly, the distribution corresponding to α = β = 5/2. The logistic distribution is central to some of the developments of the article.     

The T-type estimate of a class of partially non linear models

Abstract

This paper considers a partially non linear model E(Y|X, z, t) = f(X, β) + z^Tg(t) and gives its T-type estimate, which is a weighted quasi-likelihood estimate using sieve method and can be obtained by EM algorithm. The influence functions and asymptotic properties of T-type estimate (consistency and asymptotic normality) are discussed, and convergence rate of both parametric and non parametric components are obtained. Simulation results show the shape of influence functions and prove that the T-type estimate performs quite well. The proposed estimate is also applied to a data set and compared with the least square estimate and least absolute deviation estimate.     

APPROXIMATION OF EXCEEDANCE PROCESSES IN LARGE POPULATIONS

We consider a population of n individuals. Each of these individuals generates a discrete time branching stochastic process. We study the number of ancestors S(n,t) whose offspring at time t exceeds level θ(t), where θ(t) is some positive valued function. It is proved that S(n,t) may be approximated as t → ∞ and n → ∞ by some stochastic processes with independent increments.

     

Estimation for semi-Markov models with partial observations via self-consistency equations

Nonparametric estimators of the survival function S(t) = P(T ≥ t) for a partially observed time variable T have been defined by several methods, in particular, by integral self-consistency equations. The author establishes explicit expressions of the estimators in an additive form and extend this approach to several cases: a left-truncated and right-censored variable and the left-censored or left-truncated sojourn times of a right-censored semi-Markov process. These estimators are always identical to the product-limit estimators if hazard functions may be defined.     

Tests of regression coefficients under ridge regression models

This paper investigates two “non-exact” t-type tests, t( k₂) and t(k₂), of the individual coefficients of a linear regression model, based on two ordinary ridge estimators. The reported results are built on a simulation study covering 84 different models. For models with large standard errors, the ridge-based t-tests have correct levels with considerable gain in powers over those of the least squares t-test, t(0). For models with small standard errors, t(k₁) is found to be liberal and is not safe to use while, t(k₂) is found to slightly exceed the nominal level in few cases. When tie two ridge tests art: not winners, the results indicate that they don't loose much against t(0).     

A Fast Wavelet Algorithm for Analyzing One-Dimensional Signal Processing and Asymptotic Distribution of Wavelet Coefficients With Numerical Example and Simulation

In this article, we consider a sample point (t _j , s _j ) including a value s _j  = f(t _j ) at height s _j and abscissa (time or location) t _j . We apply wavelet decomposition by using shifts and dilations of the basic Häar transform and obtain an algorithm to analyze a signal or function f. We use this algorithm in practical to approximating function by numerical example. Some relationships between wavelets coefficients and asymptotic distribution of wavelet coefficients are investigated. At the end, we illustrate the results on simulated data by using MATLAB and R software.     

An (s,k, S) fluid inventory model with exponential leadtimes and order cancellations

We consider a stochastic fluid inventory model based on a (s, k, S) policy. The content level W = {W(t): t ≥ 0} increases or decreases according to a fluid-flow rate modulated by an n-state continuous time Markov chain (CTMC). W starts at W(0) = S; whenever W(t) drops to level s, an order is placed to take the inventory back to level S, which the supplier will carry out after an exponential leadtime. However, if during the leadtime the content level reaches k, the order is suppressed. We obtain explicit formulas for the expected discounted costs. The derivations are based on the optional sampling theorem (OST) to the multidimensional martingale and on fluid flow techniques.     

A lower bound for the transition density function of a stochastic differential equation

Let W_t be a one-dimensional Brownian motion on the probability space (Ω,F,P), and let dx_t = a(x_t)dt + b(x_t)dw_t, b²(x) > 0, be a one-dimensional Ito stochastic differential equation. For a(x) = a₀ + a₁x + … + a_nxⁿ on a bounded interval we obtain a lower bound for p(t,x,y), the transition density function of the homogeneous Markov process x_t, depending directly on the coefficients a₀,a₁, …, a_n, and b(x).     

Berry–Esseen type bounds in heteroscedastic errors-in-variables model

ABSTRACT

Consider the heteroscedastic partially linear errors-in-variables (EV) model y_i = x_iβ + g(t_i) + ε_i, ξ_i = x_i + μ_i (1 ? i ? n), where ε_i = σ_ie_i are random errors with mean zero, σ²_i = f(u_i), (x_i, t_i, u_i) are non random design points, x_i are observed with measurement errors μ_i. When f( · ) is known, we derive the Berry–Esseen type bounds for estimators of β and g( · ) under {e_i,?1 ? i ? n} is a sequence of stationary α-mixing random variables, when f( · ) is unknown, the Berry–Esseen type bounds for estimators of β, g( · ), and f( · ) are discussed under independent errors.     

Relation of modified power series distributions to lagrangian probability distributions

The class of Lagrangian probability distributions ‘LPD’, given by the expansion of a probability generating function f‘t’ under the transformation u = t/g‘t’ where g ‘t’ is also a p.g.f., has been substantially widened by removing the restriction that the defining functions g ‘t’ and f‘t’ be probability generating functions. The class of modified power series distributions defined by Gupta ‘1974’ has been shown to be a sub-class of the wider class of LPDs     

FIRST-EXIT TIMES FOR POISSON SHOT NOISE

For a shot-noise process X(t) with Poisson arrival times and exponentially diminishing shocks of i.i.d. sizes, we consider the first time T _b at which a given level b > 0 is exceeded. An integral equation for the joint density of T _b and X(T _b ) is derived and, for the case of exponential jumps, solved explicitly in terms of Laplace transforms (LTs). In the general case we determine the ordinary LT of the function ? P(T _b > t) in terms of certain LTs derived from the distribution function H(x; t) = P(X(t) ≤ x), considered as a function of both variables x and t. Moreover, for G(t, u) = P(T _b > t, X(t) < u), that is the joint distribution function of sup_{0 ≤ s ≤ t} X(s) and X(t), an integro-differential equation is presented, whose unique solution is G(t, u).     

Density estimation for samples satisfying a certain absolute regularity condition

Let {ξ_i} be an absolutely regular sequence of identically distributed random variables having common density function f(x). Let H_k(x,y) (k=1, 2,…) be a sequence of Borel-measurable functions and f_n(x)=n^?1(H_n(x,ξ₁)+…+H_n(x,ξ_n)) the empirical density function. In this paper, the asymptotic property of the probability P(sup_x|f_n(x)?f(x)|>ε) (n→∞) is studied.     

Admissible treatment rules for a risk-averse planner with experimental data on an innovation

Consider a planner choosing treatments for observationally identical persons who vary in their response to treatment. There are two treatments with binary outcomes. One is a status quo with known population success rate. The other is an innovation for which the data are the outcomes of an experiment. Karlin and Rubin [1956. The theory of decision procedures for distributions with monotone likelihood ratio. Ann. Math. Statist. 27, 272–299] assumed that the objective is to maximize the population success rate and showed that the admissible rules are the KR-monotone   rules. These assign everyone to the status quo if the number of experimental successes is below a specified threshold and everyone to the innovation if experimental success exceeds the threshold. We assume that the objective is to maximize a concave-monotone function f(·)$f(·)$ of the success rate and show that admissibility depends on the curvature of f(·)$f(·)$. Let a fractional monotone   rule be one where the fraction of persons assigned to the innovation weakly increases with the number of experimental successes. We show that the class of fractional monotone rules is complete if f(·)$f(·)$ is concave and strictly monotone. Define an M-step monotone rule   to be a fractional monotone rule with an interior fractional treatment assignment for no more than M$M$ consecutive values of the number of experimental successes. The M$M$-step monotone rules form a complete class if f(·)$f(·)$ is differentiable and has sufficiently weak curvature. Bayes rules and the minimax-regret rule depend on the curvature of the welfare function.