Estimation of the Quantile Function of an IFRA Distribution

Let F and G be lifetime distributions and consider the problem of estimating F ⁻¹ when it is known that G ⁻¹ F is star-shaped. Estimators of F ⁻¹ are considered here which are shown to be uniformly strongly consistent. The case of censored data is also presented. Asymptotic confidence intervals and bands for F ⁻¹ are provided. The result are applicable, for example, to the estimation of quantile functions of k -out-of- n systems in reliability. The special case of an IFRA distribution follows immediately from the more general case presented here     

A Remark on the Zhang Omnibus Test for Normality

Zhang (1999) proposed a novel test statistic Q for testing normality based on the ratio of two unbiased standard deviation estimators, q₁ and q₂, for the true population standard deviation σ. Mingoti & Neves (2003) discussed some properties of q₁ and q₂ and showed that the variance of q₁ increases as the true population variance increases. In this paper, we show that the distribution of q₁ is not normal. As a result, normality percentage points for Q are not appropriate. In this paper, percentage points of Q are obtained using simulations. Monte Carlo simulations are provided to evaluate the performance of the new method and Zhang's method.     

Computation of the Generalized F Distribution

Exact expressions for the cumulative distribution function of a random variable of the form ( α ₁ X ₁+ α ₂ X ₂)/ Y are given where X ₁, X ₂ and Y are independent chi-squared random variables. The expressions are applied to the detection of joint outliers and Hotelling's mis-specified T ² distribution.     

Goodness-of-fit Tests for Semi-Markov and Markov Survival Models with One Intermediate State

Survival data with one intermediate state are described by semi-Markov and Markov models for counting processes whose intensities are defined in terms of two stopping times T ₁< T ₂. Problems of goodness-of-fit for these models are studied. The test statistics are proposed by comparing Nelson–Aalen estimators for data stratified according to T ₁. Asymptotic distributions of these statistics are established in terms of the weak convergence of some random fields. Asymptotic consistency of these test statistics is also established. Simulation studies are included to indicate their numerical performance.     

Testing equality of proportions with incomplete correlated data

Let (ψ_i,φ_i) be independent, identically distributed pairs of zero-one random variables with (possible) dependence of ψ_i and φ_i within the pair. For n pairs, both variables are observed, but for m₁ additional pairs only ψ_i is observed and for m₂ others φ_i is observed. If π_1· = P{ψ_i = 1} and π_·1=P{φ_i, the problem is to test π_1·=π_·1. Maximum likelihood estimates of π_1· and π_·1 are obtained via the EM algorithm. A test statistic is developed whose null distribution is asymptotically chi-square with one degree of freedom (as n and either m₁ or m₂ tend to infinity). If m₁ = m₂ = 0 the statistic reduces to that of McNemar's test; if n = 0, it is equivalent to the statistic for testing equality of two independent proportions. This test is compared with other tests by means of Pitman efficiency. Examples are presented.     

Approximations of distributions for some standardized partial sums in sequential analysis

In sequential analysis it is often necessary to determine the distributions of √t Y _t and/or √a Y _t , where t is a stopping time of the form t = inf{ n ≥ 1 : n+S_n+ξ_n> a }, Y _n is the sample mean of n independent and identically distributed random variables (iidrvs) Y_i with mean zero and variance one, S_n is the partial sum of iidrvs X_i with mean zero and a positive finite variance, and { ξ_n } is a sequence of random variables that converges in distribution to a random variable ξ as n →∞ and ξ_n is independent of ( X_n+1, Y_n+1), (X_n+2, Y_n+2), . . . for all n ≥ 1. Anscombe's (1952) central limit theorem asserts that both √t Y _t and √a Y _t are asymptotically normal for large a , but a normal approximation is not accurate enough for many applications. Refined approximations are available only for a few special cases of the general setting above and are often very complex. This paper provides some simple Edgeworth approximations that are numerically satisfactory for the problems it considers.     

Balanced e-optimal designs of resolution III for the 2×3 series

We consider the problem of constructing designs which are E-optimal in the class of all balanced resolution III designs for the 2^m×3ⁿ series. The inverse of the information matrix for general resolution III balanced 2^m×3ⁿ designs is obtained. Optimal designs are constructed for the cases (m,n)=(3, 1), (4, 1), (2, 2) and (3, 2) for various numbers of runs in the practical range.     

Exact tests of binomial p=1/2 when one binomial event possibly has different probability

The data are n independent random binomial events, each resulting in success or failure. The event outcomes are believed to be trials from a binomial distribution with success probability p, and tests of p=1/2 are desired. However, there is the possibility that some unidentified event has a success probability different from the common value p for the other n?1 events. Then, tests of whether this common p equals 1/2 are desired. Fortunately, two-sided tests can be obtained that simultaneously are applicable for both situations. That is, the significance level for a test is same when one event has a different probability as when all events have the same probability. These tests are the usual equal-tail tests for p=1/2 (based on n independent trials from a binomial distribution).     

An Efficient Simulation Method for the Computation of a Class of Conditional Expectations

Suppose that the random vector X and the random variable Y are jointly continuous. Also suppose that an observation x of X can be easily simulated and that the probability density function of Y conditional on X = x is known. The paper presents an efficient simulation-based algorithm for estimating E{ g ( X , Y ) | h ( X , Y ) = r } where g and h are real-valued functions. This algorithm is applicable to time series problems in which X = ( X ₁, . . . , X _n−1) and Y = X_n where { x_t } is a discrete time stochastic process for which ( X₁ , . . . , X_n ) is a continuous random vector. A numerical example from time series analysis illustrates the algorithim, for prediction for an ARCH(1) process.     

Some Bounds on the Distribution of Certain Quadratic Forms in Normal Random Variables

The paper derives bounds on the distribution of the quadratic forms Z = y ^H( X Γ X ^H)⁻¹ y and W = y ^H(σ² I + X Γ X ^H)⁻¹ y , where the elements of the M × 1 vector y and the M × N matrix X are independent identically distributed (i.i.d.) complex zero mean Normal variables, Γ is some N × N diagonal matrix with positive diagonal elements, I , is the identity, σ² is a constant and ^H denotes the Hermitian transpose. The bounds are convenient for numerical work and appear to be tight for small values of M . This work has applications in digital mobile radio for a specific channel where M antennas are used to receive a signal with N interferers. Some of these applications in radio communication systems are discussed.     

Lattice paths combinatorics applied to transient queue length distribution of C2/M/1 queues and busy period analysis of bulk queues C2/M/1

This paper aims at deriving explicit transient queue length distribution for GI/M/1 system and busy period analysis of bulk queue GI^b/M/1 through lattice paths (LPs) combinatorics. The general interarrival time distribution is approximated by two-phase Cox distribution, C₂, that has Markovian property, enabling us to represent the processes by two-dimensional LPs. As distributions C₂ cover a wide class of distributions that have rational Laplace–Stieltjes transforms (LSTs) with square coefficient of variation lying in , the results obtained are applicable to a large class of real life situations. Some numerical results for the C₂^b/M/1 model are also given.     

A NOTE ON THE MULTIVARIATE LINEAR MODEL WITH CONSTRAINTS ON THE DEPENDENT VECTOR

It is shown that the least squares estimators of B and Σ in the multivariate linear model {E Y _i= X ₁ B , D ( Y _i) =Σ, 1 ≤ i ≤ n , Y ₁ Y _n uncorrelated} subject to the constraints Y _i M = X _i N are just the usual least squares estimators = ( X'X )-¹ X'Y and ΣC = 1/n( Y-X )( Y-X ) in the unconstrained model where Σ has full rank. Tests of hypotheses concerning B are discussed for situations in which each Y _i has a multivariate normal distribution, and examples of the applicability of the model reviewed.     

Computing the conditional stationary distribution in Markov chains of level-dependent M/G/1-type

This paper considers the computation of the conditional stationary distribution in Markov chains of level-dependent M/G/1-type, given that the level is not greater than a predefined threshold. This problem has been studied recently and a computational algorithm is proposed under the assumption that matrices representing downward jumps are nonsingular. We first show that this assumption can be eliminated in a general setting of Markov chains of level-dependent G/G/1-type. Next we develop a computational algorithm for the conditional stationary distribution in Markov chains of level-dependent M/G/1-type, by modifying the above-mentioned algorithm slightly. In principle, our algorithm is applicable to any Markov chain of level-dependent M/G/1-type, if the Markov chain is irreducible and positive-recurrent. Furthermore, as an input to the algorithm, we can set an error bound for the computed conditional distribution, which is a notable feature of our algorithm. Some numerical examples are also provided.     

Estimating smooth monotone functions

Many situations call for a smooth strictly monotone function f of arbitrary flexibility. The family of functions defined by the differential equation D  ² f  = w Df , where w is an unconstrained coefficient function comprises the strictly monotone twice differentiable functions. The solution to this equation is f = C ₀ + C ₁  D ⁻¹{exp( D ⁻¹ w )}, where C ₀ and C ₁ are arbitrary constants and D ⁻¹ is the partial integration operator. A basis for expanding w is suggested that permits explicit integration in the expression of f . In fitting data, it is also useful to regularize f by penalizing the integral of w ² since this is a measure of the relative curvature in f . Applications are discussed to monotone nonparametric regression, to the transformation of the dependent variable in non-linear regression and to density estimation.     

Multiple comparisons with a control in repeated measures incomplete block designs using R-estimators

New multiple comparison with a control (MCC) procedures are developed in repeated measures incomplete block design settings based on R-estimates. It is assumed that the errors within each subject are exchangeable random variables. The R-estimators of the treatment effects are obtained by minimizing a sum of Jaeckel (1972)-type dispersion functions. Based on the R-estimators, Dunnett-type multiple comparison procedures are developed for comparing test-treatments with a control-treatment. Under exchangeable errors, it is demonstrated that for Cox-type designs, the new procedures are more efficient than the existing nonparametric procedures. The new MCC procedures are applied to a data set in a clinical trial which consists of patients with reversible obstructive pulmonary disease.     

Bayesian analysis of Er/M/1 and Er/M/c queues

Bayesian inference and prediction tasks for Er/M/1 and Er/M/c queues are undertaken. Equilibrium probabilities of the queue size and waiting time distributions are estimated using conditional Monte-Carlo simulation methods. We illustrate that some standard queueing measures do not exist when independent priors are used for the arrival and service rates of a G/M/1 queue.     

ON THE CONVERGENCE OF EMPIRICAL PROCESSES OF MIXING VARIABLES1

Conditions are given for the weak convergence of (t—t²)L_N(a_N^-1( t )) to a Gaussian process where v<1/2, a _N is a cdf and L _N is the normalized weighted empirical cumulative distribution function (cdf) for an α-mixing sample of random variables in R which may be non-stationary with discontinuous marginals.     

Estimation and testing stationarity for double-autoregressive models

Summary.  The paper considers the double-autoregressive model y _t  =  φ y _{t −1}+ ɛ _t with ɛ _t  =     . Consistency and asymptotic normality of the estimated parameters are proved under the condition E  ln | φ  +√ α η _t |<0, which includes the cases with | φ |=1 or | φ |>1 as well as     . It is well known that all kinds of estimators of φ in these cases are not normal when ɛ _t are independent and identically distributed. Our result is novel and surprising. Two tests are proposed for testing stationarity of the model and their asymptotic distributions are shown to be a function of bivariate Brownian motions. Critical values of the tests are tabulated and some simulation results are reported. An application to the US 90-day treasury bill rate series is given.     

The flower intersection problem for Kirkman triple systems

The flower at a point x in a Steiner triple system is the set of all triples containing x. Denote by I_R^*[r] the set of all integers k such that there exists a pair of KTS(2r+1) having k+r triples in common, r of them being the triples of a common flower. In this article we determine the set I_R^*[r] for any positive integer r≡1 (mod 3) (only nine cases are left undecided for r=7,13,16,19), and establish that I_R^*[r]=J[r] for r≡1 (mod 3) and r22 where J[r]={0,1,…,2r(r−1)/3−6,2r(r−1)/3−4,2r(r−1)/3}.     

One-eighth- and one-sixteenth-fraction quaternary code designs with high resolution

The development of a general methodology for the construction of good two-level nonregular designs has received significant attention over the last 10 years. Recent works by Phoa and Xu (2009) and Zhang et al. (2011) indicate that quaternary code (QC) designs are very promising in this regard. This paper explores a systematic construction for 1/8th and 1/16th fraction QC designs with high resolution for any number of factors. The 1/8th fraction QC designs often have larger resolution than regular designs of the same size. A majority of the 1/16th fraction QC designs also have larger resolution than comparable two-level regular designs.