A statistical test for the hypothesis of Gaussian random function

A Gaussian random function is a functional version of the normal distribution. This paper proposes a statistical hypothesis test to test whether or not a random function is a Gaussian random function. A parameter that is equal to 0 under Gaussian random function is considered, and its unbiased estimator is given. The asymptotic distribution of the estimator is studied, which is used for constructing a test statistic and discussing its asymptotic power. The performance of the proposed test is investigated through several numerical simulations. An illustrative example is also presented.     

Characterizations of exponentiality within the HNBUE class and related tests

A harmonic new better than used in expectation (HNBUE) variable is a random variable which is dominated by an exponential distribution in the convex stochastic order. We use a recently obtained condition on stochastic equality under convex domination to derive characterizations of the exponential distribution and bounds for HNBUE variables based on the mean values of the order statistics of the variable. We apply the results to generate discrepancy measures to test if a random variable is exponential against the alternative that is HNBUE, but not exponential.     

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.     

Parameter estimation for generalized random coefficient autoregressive processes

A generalized random coefficient autoregressive (GRCA) process is introduced in which the random coefficients are permitted to be correlated with the error process. The ordinary random coefficient autoregressive process, the Markovian bilinear model and its generalization, and the random coefficient exponential autoregressive process, among others, are seen to be special cases of the GRCA process. Conditional least squares, and weighted least-squares estimators of the mean of the random coefficient vector are derived and their limit distributions are studied. Estimators of the variance-covariance parameters are also discussed. A simulation study is presented which shows that the weighted least-squares estimator dominates the unweighted least-squares estimator.     

A class of neutral to the right priors induced by superposition of beta processes

A random distribution function on the positive real line which belongs to the class of neutral to the right priors is defined. It corresponds to the superposition of independent beta processes at the cumulative hazard level. The definition is constructive and starts with a discrete time process with random probability masses obtained from suitably defined products of independent beta random variables. The continuous time version is derived as the corresponding infinitesimal weak limit and is described in terms of completely random measures. It takes the interpretation of the survival distribution resulting from independent competing failure times. We discuss prior specification and illustrate posterior inference on a real data example.     

Some simulation results for the linear expenditure system with random marginal budget shares

The estimation of the linear expenditure system (LES) with aggregate time series data is criticized since it requires the assumption of homogeneous consumers. A cross-section random coefficient version of the LES is proposed and maximum likelihood estimators are derived. Some Monte Carlo results are described which imply that the random coefficient approach does a reasonably good job of parameter estimation in certain circumstances.     

A characterization of probability distributions similar to the exponential

A concept of the lack-of-memory property at a given time point c > 0 is introduced. It is equivalent to the concept of the almost-lack-of-memory (ALM) property of the random variables. A representation theorem is given for the cumulative distribution function of such random variables as well as for corresponding decompositions in terms of independent random variables. It is shown that a periodic failure rate for a random variable is equivalent to the ALM property. In addition some properties of the service time of an unreliable server are observed.     

On the strong convergence for weighted sums of random variables

A strong convergence result is obtained for weighted sums of identically distributed negatively associated random variables which have a suitable moment condition. This result improves the result of Cai (Metrika 68:323–331, 2008).     

Deriving the Probability Density for Sums of Uniform Random Variables

A simple procedure for deriving the probability density function (pdf) for sums of uniformly distributed random variables is offered. This method is suited to introductory courses in probability and mathematical statistics. In our experience, deriving and working with the pdf for sums of random variables facilitates an understanding of the convergence properties of the density of such sums and motivates consideration of other algebraic manipulation for random variables.     

Convergence of the moment generating function of standardized quantiles

A sequence of independent, identically distributed random variables is considered. Given a simple local condition on the distribution of these random variables, we give necessary and sufficient conditions on the tails of the distribution for the moment generating function of a standardized quantile of the first n observations to converge to the moment generating function of an appropriate normal distribution as n →infinity;. This result is actually a special case of a more general result which can also be used to show convergence in distribution and convergence of moments of standardized quantiles.     

Estimation and tests of hypotheses for the initial mean and covariance in the kalman filter model

Kalman filtering techniques are widely used by engineers to recursively estimate random signal parameters which are essentially coefficients in a large-scale time series regression model. These Bayesian estimators depend on the values assumed for the mean and covariance parameters associated with the initial state of the random signal. This paper considers a likelihood approach to estimation and tests of hypotheses involving the critical initial means and covariances. A computationally simple convergent iterative algorithm is used to generate estimators which depend only on standard Kalman filter outputs at each successive stage. Conditions are given under which the maximum likelihood estimators are consistent and asymptotically normal. The procedure is illustrated using a typical large-scale data set involving 10-dimensional signal vectors.     

Bayesian Prediction of the Median of the Burr Distribution with Fixed and Random Sample Sizes

sLingappaiah (1986) was the first to introduce the idea of Bayesian prediction in life testing when the size of the future sample is a random variable. In this paper we discuss the Bayesion prediction of the sample median when the parent distribution is a generalized Burr distribution (GBD), the old sample is censored type II and the size of the future sample is a random variable. A numerical illustration is provided.     

Stability of the roots of random algebraic polynomials

In this paper we consider the behavior of the roots of random algebraic polynomials. A code was developed which generates a sample of random algebraic polynomials, calculates the roots of each sample polynomial, and then calculates the averages of the roots. Finally, the roots of the deterministic algebraic polynomial whose coefficients are the averages of the sample coefficients are calculated. These data are then tabulated and graphically displayed. The relationship between the averages of the roots of the sample polynomials and the roots of the average polynomial is discussed.     

On the Evaluation of the Kolmogorov Statistic

At least two computer program packages, SPSS and STRATA, use simulated Bernoulli trials to draw (without replacement) a random sample of records from a finite population of records. Therefore, the size of the sample is a random variable. Two estimators of a population total under this sampling procedure are compared with the usual estimator under simple random sampling. Conditions under which the Bernoulli sampling estimators have almost the same mean squared error as the simple random-sample estimator are illustrated.     

On the distribution of the maximum of bivariate normal random variables with general means and variances

A great amount of effort has been devoted to achieving exact expressions for moments of order statistics of independent normal random variables, as well as the dependent case with the same correlation coefficients, means and variances. It does not seem as if there are handy formulae for the order statistics of even the simple bivariate normal random variables when the means and variances are allowed to be different. In this paper we give an explicit formula for the Lanl ace-Stielties Transform of the maximum of bivariate normal random variables by which we obtain formulae for the first two moments in the standard way.     

A Bayesian approach for analysing longitudinal nominal outcomes using random coefficients transitional generalized logit model: an application to the labour force survey data

A random-effects transition model is proposed to model the economic activity status of household members. This model is introduced to take into account two kinds of correlations; one due to the longitudinal nature of the study, which will be considered using a transition parameter, and the other due to the existing correlation between responses of members of the same household which is taken into account by introducing random coefficients into the model. The results are presented based on the homogeneous (all parameters are not changed by time) and non-homogeneous Markov models with random coefficients. A Bayesian approach via the Gibbs sampling is used to perform parameter estimation. Results of using random-effects transition model are compared, using deviance information criterion, with those of three other models which exclude random effects and/or transition effects. It is shown that the full model gains more precision due to the consideration of all aspects of the process which generated the data. To illustrate the utility of the proposed model, a longitudinal data set which is extracted from the Iranian Labour Force Survey is analysed to explore the simultaneous effect of some covariates on the current economic activity as a nominal response. Also, some sensitivity analyses are performed to assess the robustness of the posterior estimation of the transition parameters to the perturbations of the prior parameters.     

A Model of Aging and a Shape of the Observed Force of Mortality

A probabilistic model of aging is considered. It is based on the assumption that a random resource, a stochastic process of aging (wear) and the corresponding anti-aging process are embedded at birth. A death occurs when the accumulated wear exceeds the initial random resource. It is assumed that the anti-aging process decreases wear in each increment. The impact of environment (lifestyle) is also taken into account. The corresponding relations for the observed and the conditional hazard rate (force of mortality) are obtained. Similar to some demographic models, the deceleration of mortality phenomenon is explained via the concept of frailty. Simple examples are considered.     

Family of Estimators of Population Mean Using Two Auxiliary Variables in Stratified Random Sampling

A general family of estimators, which use the information of two auxiliary variables in the stratified random sampling, is proposed to estimate the population mean of the variable under study. Under stratified random sampling without replacement scheme, the expressions of bias and mean square error (MSE) up to the first- and second-order approximations are derived. The family of estimators in its optimum case is discussed. Also, an empirical study is carried out to show the properties of the proposed estimators.     

Impact of the non-distinctness and non-ignorability on the inference by multiple imputation in multivariate multilevel data: a simulation assessment

Multiple imputation (MI) is an increasingly popular method for analysing incomplete multivariate data sets. One of the most crucial assumptions of this method relates to mechanism leading to missing data. Distinctness is typically assumed, which indicates a complete independence of mechanisms underlying missingness and data generation. In addition, missing at random or missing completely at random is assumed, which explicitly states under which conditions missingness is independent of observed data. Despite common use of MI under these assumptions, plausibility and sensitivity to these fundamental assumptions have not been well-investigated. In this work, we investigate the impact of non-distinctness and non-ignorability. In particular, non-ignorability is due to unobservable cluster-specific effects (e.g. random-effects). Through a comprehensive simulation study, we show that MI inferences suggest that nonignoriability due to non-distinctness do not immediately imply dismal performance while non-ignorability due to missing not at random leads to quite subpar performance.     

Random stopping sets in a sequential analysis of random measures and fields

A random stopping set is defined and some of its properties are proved. Namely, we prove the theorem on the absolute continuity of measures on a σ-algebra F_τ connected with a random stopping set τ, which can be applied to the sequential analysis of random measures and fields.