THE EXACT BIAS OF THE LOG-PERIODOGRAM REGRESSION ESTIMATOR

The paper makes two contributions. First, we provide a formula for the exact distribution of the periodogram evaluated at any arbitrary frequency, when the sample is taken from any zero-mean stationary Gaussian process. The inadequacy of the asymptotic distribution is demonstrated through an example in which the observations are generated by a fractional Gaussian noise process. The results are then applied in deriving the exact bias of the log-periodogram regression estimator (Geweke and Porter-Hudak (1983), Robinson (1995)). The formula is computable. Practical bounds on this bias are developed and their arithmetic mean is shown to be accurate and useful.     

Generalized portmanteau statistics and tests of randomness

This paper considers the problem of testing the randomness of Gaussian and non–Gaussian time series. A general class of parametric portmanteau statistics, which include the Box–Pierce and the Ljung–Box statistics, is introduced. Using the exact first and second moments of the sample autocorrelations when the observations are i.i.d. normal with unknown mean, the exact expected value of any portmanteau statistics is obtained for this case. Two new portmanteau statistics, which exploit the exact moments of the sample autocorrelations, are studied. For the nonparametric case, a rank portmanteau statistic is introduced. The latter has the same distribution for any series of exchangeable random variables and uses the exact moments of the rank autocorrelations. We show that its asymptotic distribution is chi–squate. Simulation results indicate that the new portmanteau statistics are better approximated by the chi–square asymptotic distribution than the Ljung–Box statistics. Several analytical results presented in the paper were derived by usig a symbolic manipulation program.     

Final outcome probabilities for SIR epidemic model

Abstract

We consider an SIR stochastic epidemic model in which new infections occur at rate f(x, y), where x and y are, respectively, the number of susceptibles and infectives at the time of infection and f is a positive sequence of real functions. A simple explicit formula for the final size distribution is obtained. Some efficient recursive methods are proved for the exact calculation of this distribution. In addition, we give a Gaussian approximation for the final distribution using a diffusion process approximation.     

Skew Gaussian Process for Nonlinear Regression

In this article, we extend the Gaussian process for regression model by assuming a skew Gaussian process prior on the input function and a skew Gaussian white noise on the error term. Under these assumptions, the predictive density of the output function at a new fixed input is obtained in a closed form. Also, we study the Gaussian process predictor when the errors depart from the Gaussianity to the skew Gaussian white noise. The bias is derived in a closed form and is studied for some special cases. We conduct a simulation study to compare the empirical distribution function of the Gaussian process predictor under Gaussian white noise and skew Gaussian white noise.     

Information loss on the mean for spatial processes when some values are missing

The loss of information on the mean due to the presence of missing values is discussed for a Gaussian univariate process on a rectangular lattice. The exact as well as the approximate formulae for this loss are given for general conditional autoregressive (CAR) and simultaneous autoregressive (SAR) processes. The formulae are evaluated for some low order CAR and SAR processes. The approximate formula is shown to give a good insight into how the loss varies over the different configurations of missing sites.     

Mean success run length

In this paper we consider mean of success run lengths appearing in a sequence of binary trials. We derive the exact and limiting distributions of mean success run length for i.i.d. Bernoulli trials. The exact distribution of the corresponding random variable is also derived for a sequence of Markov-dependent Bernoulli trials. In addition, a combinatorial formula for the distribution of any success run statistic defined on Markov-dependent trials is presented.     

Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling

The normal inverse Gaussian distribution is defined as a variance-mean mixture of a normal distribution with the inverse Gaussian as the mixing distribution. The distribution determines an homogeneous Lévy process, and this process is representable through subordination of Brownian motion by the inverse Gaussian process. The canonical, Lévy type, decomposition of the process is determined. As a preparation for developments in the latter part of the paper the connection of the normal inverse Gaussian distribution to the classes of generalized hyperbolic and inverse Gaussian distributions is briefly reviewed. Then a discussion is begun of the potential of the normal inverse Gaussian distribution and Lévy process for modelling and analysing statistical data, with particular reference to extensive sets of observations from turbulence and from finance. These areas of application imply a need for extending the inverse Gaussian Lévy process so as to accommodate certain, frequently observed, temporal dependence structures. Some extensions, of the stochastic volatility type, are constructed via an observation-driven approach to state space modelling. At the end of the paper generalizations to multivariate settings are indicated.     

On the failure rate estimation of the inverse gaussian distribution

New estimators of the inverse Gaussian failure rate are proposed based on the maximum likelihood predictive densities derived by Yang (1999). These estimators are compared, via Monte Carlo simulation, with the usual maximum likelihood estimators of the failure rate and found to be superior in terms of bias and mean squared error. Sensitivity of the estimators against the departure from the inverse Gaussian distribution is studied.     

Small-sample results for maximum-likelihood estimation in branching processes

In this article, small sample properties of the maximum-likelihood estimator (m.l.e.) for the offspring distribution (pk) and its mean m are considered in the context of the simple branching process. A representation theorem is given for the m.l.e. of (Pk) from which the m.l.e. of m is obtained. The case where p0 + p1 + p2 = 1 is studied in detail: numerical results are given for the exact bias of these estimators as a function of the age of the process; a curve fitting analysis expresses the bias of m? as a function of the mean and the variance of the offspring distribution and finally an “approximate m.l.e.” for (pk) is given.     

Sample size determinations when two binomial proportions are very small

Sample size determination for testing the hypothesis of equality of proportions with a specified type I and type I1 error probabilitiesis of ten based on normal approximation to the binomial distribution. When the proportionsinvolved are very small, the exact distribution of the test statistic may not follow the assumed distribution. Consequently, the sample size determined by the test statistic may not result in the sespecifiederror probabilities. In this paper the author proposes a square root formula and compares it with several existing sample size approximation methods. It is found that with small proportion (p≦.01) the squar eroot formula provides the closest approximation to the exact sample sizes which attain a specified type I and type II error probabilities. Thes quare root formula is simple inform and has the advantage that equal differencesare equally detectable.     

Explicit vector expression of exact score for time series models in state space form

Koopman and Shephard (1992) [1] and Segal and Weinstein (1989) [4] propose a formula for calculating the exact score vector for a general form of linear Gaussian state space models. However, for applying their method, one needs to calculate the derivatives of functions with respect to vectors and matrices, which can be intractable in many practical cases. Koopman and Shephard (1992) [1] derive its explicit expression only for a particular case. In this note, we complement Koopman and Shephard (1992) [1] and Segal and Weinstein [4] by deriving an explicit vector expression of the exact score vector for the general form of linear Gaussian state space models.     

Dependence bias in conventional p-charts and its correction with an approximate lot quality distribution

In this study, the variance of mean fraction nonconformance under the exact lot quality distribution with first order dependence is examined. The extent of the validity of binomial assumption with respect to the exact distribution in constructing the p-charts for dependent production processes is investigated and it is shown that neglect of dependence for small lot sizes may cause bias in the width of the area between the control limits. An approximate lot quality distribution with great versatility is proposed and is used for correcting the bias due to dependence in p-charts.     

Efficient bias-Robust M-Estimators of location in Kolmogorov normal neighbourhoods

We consider minimax-bias M-estimation of a location parameter in a Kolmogorov neighbourhood K() of a normal distribution. The maximum asymptotic bias of M-estimators for the Kolmogorov normal neighbourhood is derived, and its relation with the gross-error sensitivity of the estimator at the nominal model (the Gaussian case) is found. In addition, efficient bias-robust M-estimators T_i are constructed. Numerical results are also obtained to show the percentage of increase in maximum asymptotic bias and the efficiency we can achieve for some well-known -functions.     

Statistical theory of shape under elliptical models via QR decomposition

The statistical shape theory via QR decomposition and based on Gaussian and isotropic models is extended in this paper to the families of non-isotropic elliptical distributions. The new shape distributions are easily computable and then the inference procedure can be studied with the resulting exact densities. An application in Biology is studied under two Kotz models, the best distribution (non-Gaussian) is selected by using a modified Bayesian information criterion (BIC)*.     

Exact Distributional Results for Random Resistance Trees

With a view to the study of, for instance, arterial trees, this paper presents some exact distributional results on finite trees with (reciprocal) inverse Gaussian and gamma resistances. In particular, it is shown that under the specified model the conditional distribution of the minimal sufficient statistic given the total resistance of the tree is a convolution of gamma distributions and two-dimensional reciprocal inverse Gaussian distributions.     

Modeling dropouts by conditional distribution, a copula-based approach

In this paper the concept of copulas is implemented into the methodology for solving the imputation problem in correlated incomplete data. We use the Gaussian copula as alternative to the joint distribution for modeling the conditional distribution, conditioned by the observed values of measurements. The general formula for imputation and its application for compound symmetry correlation structure are given.     

Formulas for stationary planar fibre processes III – intersections with fibre systems

In this paper we investigate the distribution of the periodogram, respectively, the periodogram matrix for stationary random sequences. These .distributions are consid¬ered in the case of a fixed frequency as well as in the case of a finite number of frequencies for Gaussian sequences and for sequences of independent random variables. The exact distribution is obtained in the case of a fixed frequence for one-dimensional GAUSsian sequences. Asymptotic expansions, respectively, the rate of convergence to the asymptotic distribution are given in the case mentioned above     

A method for estimating parameters and quantiles of the three-parameter inverse Gaussian distribution based on statistics invariant to unknown location

The inverse Gaussian (IG) distribution, also known as the Wald distribution, is a long-tailed positively skewed distribution and a well-known lifetime distribution. In this paper, we propose an efficient method of estimation for the parameters and quantiles of the three-parameter IG distribution, which is based on statistics invariant to unknown location. Through a Monte Carlo simulation study, we then show that the proposed method performs well compared with other prominent methods in terms of bias and variance. Finally, we present two illustrative examples.     

Estimation of Parameters of a Mixed Failure Time Distribution Which is a Mixture of Degenerate (Degenerate at Zero) and I. G. Distribution

The problem of estimation of parameters of a mixture of degenerate (at zero) and exponential distribution is considered by Jayade and Prasad (1990). The sampling scheme proposed in it is extended in this paper to a mixture of degenerate and Inverse Gaussian distribution. The Inverse Gaussian distribution is very relevant for studying reliability and life-testing problems. The inverse Gaussian being the first passage time distribution for Wiener process makes it particularly appropriate for failure or reaction time data analysis.     

Is There a Jump in the Transition?

This article develops a statistical test for the presence of a jump in an otherwise smooth transition process. In this testing, the null model is a threshold regression and the alternative model is a smooth transition model. We propose a quasi-Gaussian likelihood ratio statistic and provide its asymptotic distribution, which is defined as the maximum of a two parameter Gaussian process with a nonzero bias term. Asymptotic critical values can be tabulated and depend on the transition function employed. A simulation method to compute empirical critical values is also developed. Finite-sample performance of the test is assessed via Monte Carlo simulations. The test is applied to investigate the dynamics of racial segregation within cities across the United States.