Estimation in Semiparametric Marginal Shared Gamma Frailty Models

The semiparametric marginal shared frailty models in survival analysis have the non–parametric hazard functions multiplied by a random frailty in each cluster, and the survival times conditional on frailties are assumed to be independent. In addition, the marginal hazard functions have the same form as in the usual Cox proportional hazard models. In this paper, an approach based on maximum likelihood and expectation–maximization is applied to semiparametric marginal shared gamma frailty models, where the frailties are assumed to be gamma distributed with mean 1 and variance θ. The estimates of the fixed–effect parameters and their standard errors obtained using this approach are compared in terms of both bias and efficiency with those obtained using the extended marginal approach. Similarly, the standard errors of our frailty variance estimates are found to compare favourably with those obtained using other methods. The asymptotic distribution of the frailty variance estimates is shown to be a 50–50 mixture of a point mass at zero and a truncated normal random variable on the positive axis for θ₀ = 0. Simulations demonstrate that, for θ₀ < 0, it is approximately an x −(100 − x )%, 0 ≤ x ≤ 50, mixture between a point mass at zero and a truncated normal random variable on the positive axis for small samples and small values of θ₀; otherwise, it is approximately normal.     

Correctings tests for trend in proportions for skewness

We develop the score test for the hypothesis that a parameter of a Markov sequence is constant over time, against the alternatives that it varies over time, i.e., θ_t = θ + U_t; t = 1,2,…, where {U_t; t = 1,2,...} is a sequence of independently and identically distributed random variables with mean zero and variance σ^z _u and θ is a fixed constant. The asymptotic null distribution of the test statistic is proved to be normal. We illustrate our procedure by examples and a real life data analysis.     

The interpretation of maximum-likelihood estimation

Maximum-likelihood estimation is interpreted as a procedure for generating approximate pivotal quantities, that is, functions u(X;θ) of the data X and parameter θ that have distributions not involving θ. Further, these pivotals should be efficient in the sense of reproducing approximately the likelihood function of θ based on X, and they should be approximately linear in θ. To this end the effect of replacing θ by a parameter ϕ = ϕ(θ) is examined. The relationship of maximum-likelihood estimation interpreted in this way to conditional inference is discussed. Examples illustrating this use of maximum-likelihood estimation on small samples are given.     

Non-parametric estimation of an acceleration parameter

Let X₁,… X_m be a random sample of m failure times under normal conditions with the underlying distribution F(x) and Y₁,…,Y_n a random sample of n failure times under accelerated condititons with underlying distribution G(x);G(x)=1?[1?F(x)]^θ with θ being the unknown parameter under study.Define:U_ij=1 otherwise.The joint distribution of _ijdoes not involve the distribution F and thus can be used to estimate the acceleration parameter θ.The second approach for estimating θ is to use the ranks of the Y-observations in the combined X- and Y-samples.In this paper we establish that the rank of the Y-observations in the pooled sample form a sufficient statistic for the information contained in the U_ii 's about the parameter θ and that there does not exist an unbiassed estimator for the parameter θ.We also construct several estimators and confidence interavals for the parameter θ.     

THE q-BETA-GEOMETRIC DISTRIBUTION AS A MODEL FOR FECUNDABILITY

The number of sterile couples in a retrospective study of the number of cycles to conception is necessarily zero; this is not so for a prospective study. The paper puts forward a modification of Weinberg and Gladen's beta geometric model for cycles to conception that is suitable for both types of investigation. The probability that a couple achieves conception at the xth cycle, but not earlier, is assumed to take the form R_x = (1 ? ρ)/(1 ? m ^x?1 ρ/u), instead of μ/(1 ? θ + θx). The set of parameter restraints (0 < m < 1, 0< ρ < 1, 1 < u) is appropriate for retrospective data, whilst the alternative set of restraints (1 < m, 1 < ρ, 0 < u < 1) is appropriate for prospective data. The decrease in R_x over time can be interpreted not only as a time effect, but also as a heterogeneity effect by replacing Weinberg and Gladen's beta mixture of geometric distributions by a q-beta mixture.     

Confidence Intervals for Nonparametric Regression Functions with Missing Data

Suppose that we have a nonparametric regression model Y = m(X) + ε with X ∈ R^p, where X is a random design variable and is observed completely, and Y is the response variable and some Y-values are missing at random. Based on the “complete” data sets for Y after nonaprametric regression imputation and inverse probability weighted imputation, two estimators of the regression function m(x₀) for fixed x₀ ∈ R^p are proposed. Asymptotic normality of two estimators is established, which is used to construct normal approximation-based confidence intervals for m(x₀). We also construct an empirical likelihood (EL) statistic for m(x₀) with limiting distribution of χ²₁, which is used to construct an EL confidence interval for m(x₀).     

THE LIMITING FORM OF THE NON-CENTRAL WISHART DISTRIBUTION1,2

Let S (p×p) have a Wishart distribution -with v degrees of freedom and non-centrality matrix θ= [θ_jK] (p×p). Define θ₀= min {| θ_jk |}, let θ₀→∞, and suppose that | θ_jK | = 0(θ_o). Then the limiting form of the standardized non-central distribution, as θ while n? remains fixed, is a multivariate Gaussian distribution. This result in turn is used to obtain known asymptotic properties of multivariate chi-square and Rayleigh distributions under somewhat weaker conditions.     

Minimax estimation of a lower-bounded scale parameter of a gamma distribution for scale-invariant squared-error loss

Let X have a gamma distribution with known shape parameter θr;aL and unknown scale parameter θ. Suppose it is known that θ ≥ a for some known a > 0. An admissible minimax estimator for scale-invariant squared-error loss is presented. This estimator is the pointwise limit of a sequence of Bayes estimators. Further, the class of truncated linear estimators C = {θ_ρ|θ_ρ(x) = max(a, ρ), ρ > 0} is studied. It is shown that each θ_ρ is inadmissible and that exactly one of them is minimax. Finally, it is shown that Katz's [Ann. Math. Statist., 32, 136–142 (1961)] estimator of θ is not minimax for our loss function. Some further properties of and comparisons among these estimators are also presented.     

Optimality of the posterior predictive p-value based on the posterior. Odds

Thompson (1997) considered a wide definition of p-value and found the Baves p-value for testing a ooint null hypothesis H₀: θ= θ₀ versus H₁: θ ≠ θ₀. In this paper, the general case of testing H₀: θ ∈ ?₀ versus H₁: θ ∈ ?^c ₀ is studied. A generalization of the concept of p-value is given, and it is proved that the posterior predictive p-value based on the posterior odds is (asymptotically) a Bayes p-value. Finally, it is suggested that this posterior predictive p-value could be used as a reference p-value     

NON-EXISTENCE OF AN EXACT β-CONFIDENCE INTERVAL FOR A SCALE PARAMETER*

Abstract There are given k (≥22) independent distributions with c.d.f.'s F(x;θ_j) indexed by a scale parameter θ_j, j = 1,…, k. Let θ_[i] (i = 1,…, k) denote the ith smallest one of θ₁,…, θ_k. In this paper we wish to show that, under some regularity conditions, there does not exist an exact β-level (0≤β1) confidence interval for the ith smallest scale parameter θ_i based on k independent samples. Since the log transformation method may not yield the desired results for the scale parameter problem, we will treat the scale parameter case directly without transformation. Application is considered for normal variances. Two conservative one-sided confidence intervals for the ith smallest normal variance and the percentage points needed to actually apply the intervals are provided.     

Two Applications of Statistics to Traffic Models

Two statistical applications for estimation and prediction of flows in traffic networks are presented. In the first, the number of route users are assumed to be independent α-shifted gamma Γ(θ, λ₀) random variables denoted H(α, θ, λ₀), with common λ₀. As a consequence, the link, OD (origin-destination) and node flows are also H(α, θ, λ₀) variables. We assume that the main source of information is plate scanning, which permits us to identify, totally or partially, the vehicle route, OD and link flows by scanning their corresponding plate numbers at an adequately selected subset of links. A Bayesian approach using conjugate families is proposed that allows us to estimate different traffic flows. In the second application, a stochastic demand dynamic traffic model to predict some traffic variables and their time evolution in real networks is presented. The Bayesian network model considers that the variables are generalized Beta variables such that when marginally transformed to standard normal become multivariate normal. The model is able to provide a point estimate, a confidence interval or the density of the variable being predicted. Finally, the models are illustrated by their application to the Nguyen Dupuis network and the Vermont-State example. The resulting traffic predictions seem to be promising for real traffic networks and can be done in real time.     

MINIMUM MSE ESTIMATION BY LINEAR COMBINATIONS OF ORDER STATISTICS1

In this paper we assume that in a random sample of size ndrawn from a population having the pdf f(x; θ) the smallest r₁ observations and the largest r₂ observations are censored (r₁0, r₂0). We consider the problem of estimating θ on the basis of the middle n-r₁-r₂ observations when either f(x;θ)=θ^-1f(x/θ) or f(x;θ) = (aθ)¹f(x-θ)/aθ) where f(·) is a known pdf, a (<0) is known and θ (>0) is unknown. The minimum mean square error (MSE) linear estimator of θ proposed in this paper is a “shrinkage” of the minimum variance linear unbiased estimator of θ. We obtain explicit expressions of these estimators and their mean square errors when (i) f(·) is the uniform pdf defined on an interval of length one and (ii) f(·) is the standard exponential pdf, i.e., f(x) = exp(–x), x0. Various special cases of censoring from the left (right) and no censoring are considered.     

A Bayesian inference of P(λ1<λ2) for two Poisson parameters

The statistical inference drawn from the difference between two independent Poisson parameters is often discussed in the medical literature. However, such discussions are usually based on the frequentist viewpoint rather than the Bayesian viewpoint. Here, we propose an index θ=P(λ_{1, post}<λ_{2, post}), where λ_{1, post} and λ_{2, post} denote Poisson parameters following posterior density. We provide an exact and an approximate expression for calculating θ using the conjugate gamma prior and compare the probabilities obtained using the approximate and the exact expressions. Moreover, we also show a relation between θ and the p-value. We also highlight the significance of θ by applying it to the result of actual clinical trials. Our findings suggest that θ may provide useful information in a clinical trial.     

A family of admissible minimax estimators of the mean of a multivariate,normal distribution

Let X has a p-dimensional normal distribution with mean vector θ and identity covariance matrix I. In a compound decision problem consisting of squared-error estimation of θ, Strawderman (1971) placed a Beta (α, 1) prior distribution on a normal class of priors to produce a family of Bayes minimax estimators. We propose an incomplete Gamma(α, β) prior distribution on the same normal class of priors to produce a larger family of Bayes minimax estimators. We present the results of a Monte Carlo study to demonstrate the reduced risk of our estimators in comparison with the Strawderman estimators when θ is away from the zero vector.     

Local asymptotic normality of intermediate and central order statistics

Consider n independent random variables Z_i,…, Z_n on R with common distribution function F, whose upper tail belongs to a parametric family F(t) = F_θ(t),t ≥ x₀, where θ ∈ ? ? R ^d. A necessary and sufficient condition for the family F_θ, θ ∈ ?, is established such that the k-th largest order statistic Z_n?k+1:n alone constitutes the central sequence yielding local asymptotic normality ( LAN ) of the loglikelihood ratio of the vector (Z_n?i+1:n)¹ _i=kof the k largest order statistics. This is achieved for k = k(n)→_n→∞∞ with k/n→_n→∞ 0.

In the case of vectors of central order statistics ( Z_r:n, Z_r+1:n,…, Z_s:n ), with r/n and s/n both converging to q ∈ ( 0,1 ), it turns out that under fairly general conditions any order statistic Z_m:n with r ≤ m ≤s builds the central sequence in a pertaining LAN expansion.These results lead to asymptotically optimal tests and estimators of the underlying parameter, which depend on single order statistics only     

Asymptotic properties of Bayes risk for one-sided tests

Consider a given sequence {T_n} of estimators for a real-valued parameter θ. This paper studies asymptotic properties of restricted Bayes tests of the following form: reject H₀:θ ≤ θ₀ in favour of the alternative θ > θ₀ if T_n ≤ C_n, where the critical point C_n is determined to minimize among all tests of this form the expected probability of error with respect to the prior distribution. Such tests may or may not be fully Bayes tests, and so are called T_n-Bayes. Under fairly broad conditions it is shown that and the T_n-Bayes risk where a_n is the order of the standard error of T_n, - is the prior density, and μ is the median of F, the limit distribution of (T_n – θ)/a_nb(θ). Several examples are given.     

On the bounded regret of empirical bayes estimators

Let (θ₁,x₁),…,(θ_n,x_n) be independent and identically distributed random vectors with E(xθ) = θ and Var(x|θ) = a + bθ + cθ². Let t_i be the linear Bayes estimator of θ_i and θ~_i be the linear empirical Bayes estimator of θ_i as proposed in Robbins (1983). When Ex and Var x are unknown to the statistician. The regret of using θ~_i instead of t_i because of ignorance of the mean and the variance is r_i = E(θ_i ? θ_i)² ?E(t_i ?θ_i)². Under appropriate conditions cumulative regret R_n = r₁+…r_n is shown to have a finite limit even when n tends to infinity. The limit can be explicitly computed in terms of a,b,c and the first four moments of x.     

Empirical Likelihood in a Semi-Parametric Model for Missing Response Data

Let Y be a response and, given covariate X,Y has a conditional density f(y | x, θ), where θ is a unknown p-dimensional vector of parameters and the marginal distribution of X is unknown. When responses are missing at random, with auxiliary information and imputation, we define an adjusted empirical log-likelihood ratio for the mean of Y and obtain its asymptotic distribution. A simulation study is conducted to compare the adjusted empirical log-likelihood and the normal approximation method in terms of coverage accuracies.