Test for model selection using Cramér–von Mises distance in a fixed design regression setting

In this paper a test for model selection is proposed which extends the usual goodness-of-fit test in several ways. It is assumed that the underlying distribution H depends on a covariate value in a fixed design setting. Secondly, instead of one parametric class we consider two competing classes one of which may contain the underlying distribution. The test allows to select one of two equally treated model classes which fits the underlying distribution better. To define the distance of distributions various measures are available. Here the Cramér-von Mises has been chosen. The null hypothesis that both parametric classes have the same distance to the underlying distribution H can be checked by means of a test statistic, the asymptotic properties of which are shown under a set of suitable conditions. The performance of the test is demonstrated by Monte Carlo simulations. Finally, the procedure is applied to a data set from an endurance test on electric motors.     

Cramér-Rao lower bounds for estimators based on censored data

The Cramér-Rao lower bounds for the variances of unbiased estimators based on censored data are given. Useful techniques of evaluation are then derived for these lower bounds. Examples are given to illustrate these techniques. Small-sample comparisons are made between the resulting lower bounds, the variances of the best linear unbiased estimators, and the variances of unbiased esti-mators which are based on the maximum likelihood estimators.     

Durbin–Watson and Generalized Durbin–Watson Tests for Autocorrelations and Randomness

The Durbin–Watson (DW) test for lag 1 autocorrelation has been generalized (DWG) to test for autocorrelations at higher lags. This includes the Wallis test for lag 4 autocorrelation. These tests are also applicable to test for the important hypothesis of randomness. It is found that for small sample sizes a normal distribution or a scaled beta distribution by matching the first two moments approximates well the null distribution of the DW and DWG statistics. The approximations seem to be adequate even when the samples are from nonnormal distributions. These approximations require the first two moments of these statistics. The expressions of these moments are derived.     

Marshall–Olkin q-Weibull distribution and max–min processes

In this paper, we introduce a new probability model known as Marshall–Olkin q-Weibull distribution. Various properties of the distribution and hazard rate functions are considered. The distribution is applied to model a biostatistical data. The corresponding time series models are developed to illustrate its application in times series modeling. We also develop different types of autoregressive processes with minification structure and max–min structure which can be applied to a rich variety of contexts in real life. Sample path properties are examined and generalization to higher orders are also made. The model is applied to a time series data on daily discharge of Neyyar river in Kerala, India.     

On the pade tablé and its relationship to the r and sarrays and arm a modeling

The Box-Jenkins method is a popular and important technique for modeling and forecasting of time series. Unfortunately the problem of determining the appropriate ARMA forecasting model (or indeed if an ARMA model holds) is a major drawback to the use of the Box-Jenkins methodology. Gray et al. (1978) and Woodward and Gray (1979) have proposed methods of estimating p and qin ARMA modeling based on the R and Sarrays that circumvent some of these modeling difficulties.

In this paper we generalize the R and S arrays by showing a relationship to Padé approximunts and then show that these arrays have a much wider application than in just determining model order. Particular non-ARMA models can be identified as well. This includes certain processes that consist of deterministic functions plus ARMA noise, indeed we believe that the combined R and S arrays are the best overall tool so fur developed for the identification of general 2nd order (not just stationary) time scries models.     

Bias–variance and breadth–depth tradeoffs in respondent-driven sampling

Respondent-driven sampling (RDS) is a link-tracing network sampling strategy for collecting data from hard-to-reach populations, such as injection drug users or individuals at high risk of being infected with HIV. The mechanism is to find initial participants (seeds), and give each of them a fixed number of coupons allowing them to recruit people they know from the population of interest, with a mutual financial incentive. The new participants are again given coupons and the process repeats. Currently, the standard RDS estimator used in practice is known as the Volz–Heckathorn (VH) estimator. It relies on strong assumptions about the underlying social network and the RDS process. Via simulation, we study the relative performance of the plain mean and VH estimators when assumptions of the latter are not satisfied, under different network types (including homophily and rich-get-richer networks), participant referral patterns, and varying number of coupons. The analysis demonstrates that the plain mean outperforms the VH estimator in many but not all of the simulated settings, including homophily networks. Also, we highlight the implications of multiple recruitment and varying referral patterns on the depth of RDS process. We develop interactive visualizations of the findings and RDS process to further build insight into the various factors contributing to the performance of current RDS estimation techniques.     

The modified Fréchet distribution and its properties

The Fréchet distribution is an absolutely continuous model which has wide applicability in extreme value theory. In this paper, we propose a new three-parameter model, so-called the modified Fréchet distribution, to extend the Fréchet distribution. By using the Lambert function, we obtain some properties of the new distribution. We provide a simulation study to illustrate the performance of the maximum likelihood estimates. The flexibility of the introduced distribution is illustrated by means of a real data set. We use some goodness-of-fit statistics to verify the adequacy of the proposed model. We prove empirically that it is appropriate for lifetime applications.     

Karhunen–Loève expansions of Lévy processes

We derive the basis functions and joint distribution of the stochastic coefficients of the Karhunen–Loève expansion of a square-integrable Lévy process. Further, we demonstrate a method for simulating the coefficients via a shot-noise representation.     

Lévy Processes,Phase-Type Distributions,and Martingales

Lévy processes are defined as processes with stationary independent increments and have become increasingly popular as models in queueing, finance, etc.; apart from Brownian motion and compound Poisson processes, some popular examples are stable processes, variance gamma processes, CGMY Lévy processes (tempered stable processes), NIG (normal inverse Gaussian) Lévy processes, and hyperbolic Lévy processes. We consider here a dense class of Lévy processes, compound Poisson processes with phase-type jumps in both directions and an added Brownian component. Within this class, we survey how to explicitly compute a number of quantities that are traditionally studied in the area of Lévy processes, in particular two-sided exit probabilities and associated Laplace transforms, the closely related scale function, one-sided exit probabilities and associated Laplace transforms coming up in queueing problems, and similar quantities for a Lévy process with reflection in 0. The solutions are in terms of roots to polynomials, and the basic equations are derived by purely probabilistic arguments using martingale optional stopping; a particularly useful martingale is the so-called Kella-Whitt martingale. Also, the relation to fluid models with a Brownian component is discussed.     

Age–period–cohort decompositions using principal components and partial least squares

Age–period–cohort decomposition requires an identification assumption because there is a linear relationship between age, survey period, and birth cohort (age+cohort=period). This paper proposes new decomposition methods based on factor models such as principal components model and partial least squares model. Although factor models have been applied to overcome the problem of many observed variables with possible co-linearity, they are applied to overcome the perfect co-linearity among age, period, and cohort dummy variables. Since any unobserved factor in the factor model is represented as a linear combination of the observed variables, the parameter estimates for age, period, and cohort effects are automatically obtained after the application of these factor models. Simulation results suggest that in almost all cases, the performance of the proposed method is better than that of a conventional econometric method. Empirical examples are also provided.     

A modified Conway–Maxwell–Poisson type binomial distribution and its applications

This article proposes a generalized binomial distribution, which is derived from the finite capacity queueing system with state-dependent service and arrival rates. This distribution is also generated from the conditional Conway–Maxwell–Poisson (CMP) distribution given a sum of two CMP variables. In this article, we consider the properties of the probability mass function, indices of dispersion, skewness and kurtosis, and give applications of the proposed distribution. The estimation method and simulation study are also considered.     

The uniform local asymptotics for a Lévy process and its overshoot and undershoot

ABSTRACT

In this article, we obtain the uniform local asymptotics for a Lévy process with a heavy-tailed Lévy measure and for the overshoot and undershoot of the Lévy process. As applications, we get the uniform asymptotics of the finite-time ruin probability and the local ruin probability for the Lévy risk model with a heavy-tailed Lévy measure. By the above results, we find that in the compound Poisson model perturbed by a Brownian motion, the effect of the Brownian component on the asymptotics of the finite-time ruin probability and the local ruin probability washes out.     

Marshall–Olkin bivariate Weibull distributions and processes

In this paper we introduce a new probability model known as type 2 Marshall–Olkin bivariate Weibull distribution as an extension of type 1 Marshall–Olkin bivariate Weibull distribution of Marshall–Olkin (J Am Stat Assoc 62:30–44, 1967). Various properties of the new distribution are considered. Bivariate minification processes with the two types of Weibull distributions as marginals are constructed and their properties are considered. It is shown that the processes are strictly stationary. The unknown parameters of the type 1 process are estimated and their properties are discussed. Some numerical results of the estimates are also given.     

Testing for co-directional interactions using union–intersection and intersection–union methods

When interaction terms exist in a two-factor, factorial experiment, the consideration and analysis of main effects are often restricted to those situations where the interaction between factors is not significant. Hinkelman and Kempthorne [4 Hinkelmann, K. and Kempthorne, O. 1994. Design and Analysis of Experiments. Volume 1: Introduction to Experimental Design, New York: Wiley.  [Google Scholar]] softened that stance somewhat and advocate testing main effects when the interaction is deemed co-directional but not anti-directional. A test for the main effects in that situation may be pragmatic to the practitioner and appealing to researchers in other disciplines. Intersection–union and union–intersection methods are examined for assessing the directional nature of significant interactions so that the main effects in a two-factor factorial may be evaluated. The tests suggested are conceptually straightforward and practical and maintain the nominal Type-I error rate. Examples are provided to illustrate the methods.