SOME PROPERTIES OF PROFILE BOOTSTRAP CONFIDENCE INTERVALS

We consider the problem of finding an equi-tailed confidence interval, with coverage probability (1-α), for a scalar parameter θ₀ in the presence of a (possibly infinite dimensional) nuisance parameter ψ₀. It is supposed that the value taken by θ₀ does not restrict the value that ψ₀ may take and vice-versa. Given a sensible estimate ψ_n of ψ₀, profile bootstrap confidence interval for θ₀ is defined to be the exact equi-tailed confidence interval with coverage probability (1-α) assuming that ψ₀=ψ_n. We compare the properties of the profile bootstrap confidence interval and the ordinary bootstrap confidence interval when they are based on studentised and unstudentised quantities. Under mild regularity conditions the profile bootstrap confidence interval is always a subset of the set of allowable values of θ₀ and is transformation-respecting when based on either an unstudentised quantity or a studentised quantity satisfying certain restrictions. As a confidence interval for the autoregressive parameter of an AR(1) process, the profile bootstrap confidence interval has important advantages over the ordinary bootstrap confidence interval based on a studentised quantity.     

EXACT P‐VALUES FOR DISCRETE MODELS OBTAINED BY ESTIMATION AND MAXIMIZATION

In constructing exact tests from discrete data, one must deal with the possible dependence of the P‐value on nuisance parameter(s) ψ as well as the discreteness of the sample space. A classical but heavy‐handed approach is to maximize over ψ. We prove what has previously been understood informally, namely that maximization produces the unique and smallest possible P‐value subject to the ordering induced by the underlying test statistic and test validity. On the other hand, allowing for the worst case will be more attractive when the P‐value is less dependent on ψ. We investigate the extent to which estimating ψ under the null reduces this dependence. An approach somewhere between full maximization and estimation is partial maximization, with appropriate penalty, as introduced by Berger & Boos (1994, P values maximized over a confidence set for the nuisance parameter. J. Amer. Statist. Assoc. 89 , 1012–1016). It is argued that estimation followed by maximization is an attractive, but computationally more demanding, alternative to partial maximization. We illustrate the ideas on a range of low‐dimensional but important examples for which the alternative methods can be investigated completely numerically.     

Exponential inequalities for the Robbins–Monro’s algorithm under mixing condition

Abstract

In this work, we establish exponential inequalities for the Robbins–Monro’s algorithm with ψ-mixing variables, and we give a result on the almost complete convergence rate.     

Tables of cumulative distribution functions and percentiles of the standardized stable random variables

It is shown that Zolotarev's (1964) integral representation of the cumulative distribution function (c.d.f.) of stable random variables and the IMSL subroutine DCADRE (for numerical integration ) provide a natural and practically simple method for finding the values of c.d.f., the percentiles and the density function of such random variables. For symmetric stable random variables (r.v.'s ) Z_∝, values of P_∝(z) … P(0<Z_∝<z) for z … 0(.02)4.08 and ∝=.1(.2)1.9, as well as percentiles of these r.v.'s for ∝=.5(.1)2 and the percentage points .6, .7(.05).85(.025).9(.01).96(.005).995, are presented. For asymmetric stable r.v.'s we present values of their c.d.f.'s for z … 0(.1)4, ß= ?1(.25)1 and ∝=.1(.2)1.9. These result sare compared with related results of others which were obtained by using different procedure and standardization.     

Bayesian analysis of growth curves with AR(1) dependence

In this paper we consider Bayesian analysis of the generalized growth curve model when the covariance matrix Σ = σ²C where C = (ϱ^{i − j}), σ² > 0 and −1 < ϱ < 1 are unknown. We consider both parameter estimation and prediction of future values. Results are illustrated with real and simulated data.     

Shrinkage estimation of P(Y < X) in the exponential distribution mixing with exponential distribution

ABSTRACT

The problem of estimation of R = P(Y < X) have been used in the paper. Let X has exponential distribution mixing with exponential distribution with parameters β and θ and Y independently of X has exponential distribution with parameter λ. By using a prior guess or estimate R₀, different shrinkage estimators of R are derived. Then the performance of the estimators are discussed. Finally, we compare these results with Baklizei and Dayyeh (2003) approaches.     

The Distribution of Sums of Certain I.I.D. Pareto Variates

ABSTRACT

Though the Pareto distribution is important to actuaries and economists, an exact expression for the distribution of the sum of n i.i.d. Pareto variates has been difficult to obtain in general. This article considers Pareto random variables with common probability density function (pdf) f(x) = (α/β) (1 + x/β)^α+1 for x > 0, where α = 1,2,… and β > 0 is a scale parameter. To date, explicit expressions are known only for a few special cases: (i) α = 1 and n = 1,2,3; (ii) 0 < α < 1 and n = 1,2,…; and (iii) 1 < α < 2 and n = 1,2,…. New expressions are provided for the more general case where β > 0, and α and n are positive integers. Laplace transforms and generalized exponential integrals are used to derive these expressions, which involve integrals of real valued functions on the positive real line. An important attribute of these expressions is that the integrands involved are non oscillating.     

On the distribution of a test for exponentiality based on progressively type-II right censored spacings

A test for exponentiality based on progressively Type-II right censored spacings has been proposed recently by Balakrishnan et al. (2002). They derived the asymptotic null distribution of the test statistic. In this work, we utilize the algorithm of Huffer and Lin (2001) to evaluate the exact null probabilities and the exact critical values of this test statistic.     

On computer generation of gamma random variables by rejection and composition procedures 2

The paper presents various algorithms for generating gamma random variables, by combining rejection and composition procedures. Two efficient algorithms are given for the case when the parameter of the gamma distribution is 0<v<l. For the case vl, several algorithms are given but they (except one), work reliably only for small values of v. Results of some computer tests together with FORTRAN subroutines are also presented.     

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.     

Additional approximations to the distribution of a structural coefficient estimator for the case of two included endogenous variables

Four new approximations t o the exact distribution of the two-stage l e a s t squares estimator of astructuralcoefficient for

the case of two included endogeneous variables are introduced and compared with the others in the literatur e . Two of the new approximations are based on the Pearson distribution and are found to be adequate throughout the parameter space. A normal approximation using exact moments and an approximation based on the saddlepoint method (Holly and Phillips,1979) are found to be

poor for a wide range of parameter values.     

A Nonrandomized,Nonconservative Version of the Fisher Exact Test

The Fisher exact test has been unjustly dismissed by some as ‘only conditional,’ whereas it is unconditionally the uniform most powerful test among all unbiased tests, tests of size α and with power greater than its nominal level of significance α. The problem with this truly optimal test is that it requires randomization at the critical value(s) to be of size α. Obviously, in practice, one does not want to conclude that ‘with probability x the we have a statistical significant result.’ Usually, the hypothesis is rejected only if the test statistic's outcome is more extreme than the critical value, reducing the actual size considerably.

The randomized unconditional Fisher exact is constructed (using Neyman–structure arguments) by deriving a conditional randomized test randomizing at critical values c(t) by probabilities γ(t), that both depend on the total number of successes T (the complete-sufficient statistic for the nuisance parameter—the common success probability) conditioned upon.

In this paper, the Fisher exact is approximated by deriving nonrandomized conditional tests with critical region including the critical value only if γ (t) > γ₀, for a fixed threshold value γ₀, such that the size of the unconditional modified test is for all value of the nuisance parameter—the common success probability—smaller, but as close as possible to α. It will be seen that this greatly improves the size of the test as compared with the conservative nonrandomized Fisher exact test.

Size, power, and p value comparison with the (virtual) randomized Fisher exact test, and the conservative nonrandomized Fisher exact, Pearson's chi-square test, with the more competitive mid-p value, the McDonald's modification, and Boschloo's modifications are performed under the assumption of two binomial samples.     

ROBUST ESTIMATION UNDER TYPE II CENSORING

The properties of robust M-estimators with type II censored failure time data are considered. The optimal members within two classes of ψ-functions are characterized. The first optimality result is the censored data analogue of the optimality result described in Hampel et al. (1986); the estimators corresponding to the optimal members within this class are referred to as the optimal robust estimators. The second result pertains to a restricted class of ψ-functions which is the analogue of the class of ψ-functions considered in James (1986) for randomly censored data; the estimators corresponding to the optimal members within this restricted class are referred to as the optimal James-type estimators. We examine the usefulness of the two classes of ψ-functions and find that the breakdown point and efficiency of the optimal James-type estimators compare favourably with those of the corresponding optimal robust estimators. From the computational point of view, the optimal James-type ψ-functions are readily obtainable from the optimal ψ-functions in the uncensored case. The ψ-functions for the optimal robust estimators require a separate algorithm which is provided. A data set illustrates the optimal robust estimators for the parameters of the extreme value distribution.     

Simultaneous estimation of the multivariate normal mean under balanced loss function

This paper considers simultaneous estimation of multivariate normal mean vector using Zellner's(1994) balanced loss function which is defined as follows:

where 0 < w < 1 and for i = 1,…,p and j = 1,…,n, X_ij is distributed as normal with mean θi and variance 1. It is shown that the sample mean, X, is admissible when p <3. For p ≥3, we obtain that James-Stein type estimator which has uniformly smaller risk than that of sample mean X.     

Minimax Estimation of the Bounded Parameter of Some Discrete Distributions Under LINEX Loss Function

For a class of discrete distributions, including Poisson(θ), Generalized Poisson(θ), Borel(m, θ), etc., we consider minimax estimation of the parameter θ under the assumption it lies in a bounded interval of the form [0, m] and a LINEX loss function. Explicit conditions for the minimax estimator to be Bayes with respect to a boundary supported prior are given. Also for Bernoulli(θ)-distribution, which is not in the mentioned class of discrete distributions, we give conditions for which the Bayes estimator of θ ∈ [0, m], m < 1 with respect to a boundary supported prior is minimax under LINEX loss function. Numerical values are given for the largest values of m for which the corresponding Bayes estimators of θ are minimax.     

Global Partial Likelihood for Nonparametric Proportional Hazards Models

As an alternative to the local partial likelihood method of Tibshirani and Hastie and Fan, Gijbels, and King, a global partial likelihood method is proposed to estimate the covariate effect in a nonparametric proportional hazards model, λ(t|x) = exp{ψ(x)}λ(0)(t). The estimator, ψ?(x), reduces to the Cox partial likelihood estimator if the covariate is discrete. The estimator is shown to be consistent and semiparametrically efficient for linear functionals of ψ(x). Moreover, Breslow-type estimation of the cumulative baseline hazard function, using the proposed estimator ψ?(x), is proved to be efficient. The asymptotic bias and variance are derived under regularity conditions. Computation of the estimator involves an iterative but simple algorithm. Extensive simulation studies provide evidence supporting the theory. The method is illustrated with the Stanford heart transplant data set. The proposed global approach is also extended to a partially linear proportional hazards model and found to provide efficient estimation of the slope parameter. This article has the supplementary materials online.     

Bivariate exponential-type distributions with linear regression

Let Xbe a bivariate exponential-type random vector (BIDLIKAR, PATIL (1968)), than it is proved:

1. If P(X ≥0) = 1 is valid, then Xhas linear regression to both directions if and only if Xpossesses a symmetric Γ-distribution.

2. Xpossesses linear regression to both directions with constant regression coefficients (independent of the parameter vector ? of the exponential-type distribution (BIDLIKAR, PATIL (1968)) if and only if Xis normal distributed.     

Estimation of the mean of the exponential distribution with an outlying observation

Given a life testing experiment consisting of n items, n-1 of which have the expected life λ while one could have an expected life λ/α with 0 < α < 1 the problem is. to find a mean square error (MSE) minimizing estimation function. The standard estimators for the homogeneous case (α = 1) overestimate the expected life and their MSE tend to infinity when a tends to 0.

Looking at the estimation problem as an insurance (see Anscombe (1960)) two different “testimators” are compared with respect to their MSE, Numerical results show that an estimation function based on the “Epstein-statistic” x_(n)/[xbar] is the best one.