A Test for bivariate exponentiality against bhnbue alternative

A test statistic for testing bivariate expotentiality against the alternative representing the property ‘bivariate harmonic new better than used in expectation’ (Bhnbue) is developed.     

Multiple comparisons with ‘best’ for multivariate normal populations

Comparisons of multivariate normal populations are made using a mul-tivariate approach (instead of reducing the problem to a univariate one). A rather negative finding is that, for comparisons with the ‘best’ of each variate, repeated univariate comparisons appear to be almost as efficient as multivariate comparisons, at least for the bivariate case and, under certain circumstances, for higher dimensional cases. Investigations are done on comparisons with the ‘MAX-best’ population (that one having the largest maximum of the marginal means), the ‘MIN-best’ (having the largest minimum) and the ‘O-best’ (being closest to largest in all marginal means). Detailed results are given for the bivariate normal with extensions indicated for the multivariate.     

Two group inspection-based control charts for dispersion matrix

In this article, we propose the two control charts, i.e. the ‘VMAX Group Runs’ (VMAX-GR) and ‘VMAX Modified Group Runs’ (VMAX-MGR) control charts based on the bivariate normal processes, for monitoring the covariance matrix. The proposed charts give the faster detection of a process change and have better diagnostic feature. It is verified that the VMAX-GR and the VMAX-MGR charts give a significant reduction in the out-of-control ‘Average Run Length’ (ARL) in the zero state, as well as in the steady state, as compared to the synthetic control chart based on the VMAX statistic and the generalized variance |S| chart.     

Asymptotics of bivariate penalised splines

We study the class of bivariate penalised splines that use tensor product splines and a smoothness penalty. Similar to Claeskens, G., Krivobokova, T., and Opsomer, J.D. [(2009), ‘Asymptotic Properties of Penalised Spline Estimators’, Biometrika, 96(3), 529–544] for the univariate penalised splines, we show that, depending on the number of knots and penalty, the global asymptotic convergence rate of bivariate penalised splines is either similar to that of tensor product regression splines or to that of thin plate splines. In each scenario, the bivariate penalised splines are found rate optimal in the sense of Stone, C.J. [(12, 1982), ‘Optimal Global Rates of Convergence for Nonparametric Regression’, The Annals of Statistics, 10(4), 1040–1053] for a corresponding class of functions with appropriate smoothness. For the scenario where a small number of knots is used, we obtain expressions for the local asymptotic bias and variance and derive the point-wise and uniform asymptotic normality. The theoretical results are applicable to tensor product regression splines.     

Joint models for mixed categorical outcomes: a study of HIV risk perception and disease status in Mozambique

Two types of bivariate models for categorical response variables are introduced to deal with special categories such as ‘unsure’ or ‘unknown’ in combination with other ordinal categories, while taking additional hierarchical data structures into account. The latter is achieved by the use of different covariance structures for a trivariate random effect. The models are applied to data from the INSIDA survey, where interest goes to the effect of covariates on the association between HIV risk perception (quadrinomial with an ‘unknown risk’ category) and HIV infection status (binary). The final model combines continuation-ratio with cumulative link logits for the risk perception, together with partly correlated and partly shared trivariate random effects for the household level. The results indicate that only age has a significant effect on the association between HIV risk perception and infection status. The proposed models may be useful in various fields of application such as social and biomedical sciences, epidemiology and public health.     

Globally intensity-reweighted estimators for K- and pair correlation functions

We introduce new estimators of the inhomogeneous K-function and the pair correlation function of a spatial point process as well as the cross K-function and the cross pair correlation function of a bivariate spatial point process under the assumption of second-order intensity-reweighted stationarity. These estimators rely on a ‘global’ normalisation factor which depends on an aggregation of the intensity function, while the existing estimators depend ‘locally’ on the intensity function at the individual observed points. The advantages of our new global estimators over the existing local estimators are demonstrated by theoretical considerations and a simulation study.     

On the distribution of quotient of random variables conditioned to the positive quadrant

Abstract

Motivated by Caginalp and Caginalp [Physica A—Statistical Mechanics and Its Applications, 499, 2018, 457–471], we derive the exact distribution of X/Y conditioned on X?>?0, Y?>?0 for more than ten classes of distributions, including the bivariate t, bivariate Cauchy, bivariate Lomax, Arnold and Strauss’ bivariate exponential, Balakrishna and Shiji’s bivariate exponential, Mohsin et al.’s bivariate exponential, Morgenstern type bivariate exponential, bivariate gamma exponential and bivariate alpha skew normal distributions. The results can be useful in finance and other areas.     

A GQL estimation approach for analysing non-stationary over-dispersed BINAR(1) time series

This paper proposes a generalized quasi-likelihood (GQL) function for estimating the vector of regression and over-dispersion effects for the respective series in the bivariate integer-valued autoregressive process of order 1 (BINAR(1)) with Negative Binomial (NB) marginals. The auto-covariance function in the proposed GQL is computed using some ‘robust’ working structures. As for the BINAR(1) process, the inter-relation between the series is induced mainly by the correlated NB innovations that are subject to different levels of over-dispersion. The performance of the GQL approach is tested via some Monte-Carlo simulations under different combination of over-dispersion together with low and high serial- and cross-correlation parameters. The model is also applied to analyse a real-life series of day and night accidents in Mauritius.     

Estimation of the bivariate distribution function for censored gap times

In many medical studies, patients may experience several events during follow-up. The times between consecutive events (gap times) are often of interest and lead to problems that have received much attention recently. In this work, we consider the estimation of the bivariate distribution function for censored gap times. Some related problems such as the estimation of the marginal distribution of the second gap time and the conditional distribution are also discussed. In this article, we introduce a nonparametric estimator of the bivariate distribution function based on Bayes’ theorem and Kaplan–Meier survival function and explore the behavior of the four estimators through simulations. Real data illustration is included.     

An alternative circular smoothing method to nonparametric estimation of periodic functions

This article provides alternative circular smoothing methods in nonparametric estimation of periodic functions. By treating the data as ‘circular’, we solve the “boundary issue” in the nonparametric estimation treating the data as ‘linear’. By redefining the distance metric and signed distance, we modify many estimators used in the situations involving periodic patterns. In the perspective of ‘nonparametric estimation of periodic functions’, we present the examples in nonparametric estimation of (1) a periodic function, (2) multiple periodic functions, (3) an evolving function, (4) a periodically varying-coefficient model and (5) a generalized linear model with periodically varying coefficient. In the perspective of ‘circular statistics’, we provide alternative approaches to calculate the weighted average and evaluate the ‘linear/circular–linear/circular’ association and regression. Simulation studies and an empirical study of electricity price index have been conducted to illustrate and compare our methods with other methods in the literature.     

Nonparametric analysis of recurrent gap time data

Abstract

Recurrent event data are frequently encountered in longitudinal studies. In many applications, the times between successive recurrent events (gap times) are often of interest and lead to problems that have received much attention recently. In this article, using the approach of inverse probability-of-censoring weights (IPCW), we propose nonparametric estimators for the estimation of the bivariate distribution and survival functions for gap times of recurrent event data. We also consider the estimation of Kendall’s tau for two gap times by expressing it as an integral functional of the bivariate survival function. The asymptotic properties of the proposed estimators are established. Simulation studies are conducted to investigate their finite sample performance.     

Comparisons of some bivariate regression models

The bivariate negative binomial regression (BNBR) and the bivariate Poisson log-normal regression (BPLR) models have been used to describe count data that are over-dispersed. In this paper, a new bivariate generalized Poisson regression (BGPR) model is defined. An advantage of the new regression model over the BNBR and BPLR models is that the BGPR can be used to model bivariate count data with either over-dispersion or under-dispersion. In this paper, we carry out a simulation study to compare the three regression models when the true data-generating process exhibits over-dispersion. In the simulation experiment, we observe that the bivariate generalized Poisson regression model performs better than the bivariate negative binomial regression model and the BPLR model.     

Household budget-share distributions and welfare implications: an application of multivariate distributional statistics

In this paper the consequences of considering the household ‘food share’ distribution as a welfare measure, in isolation from the joint distribution of itemized budget shares, is examined through the unconditional and conditional distribution of ‘food share’ both parametrically and nonparametrically. The parametric framework uses Dirichlet and Beta distributions, while the nonparametric framework uses kernel smoothing methods. The analysis, in a three commodity setup (‘food’, ‘durables’, ‘others’), based on household level rural data for West Bengal, India, for the year 2009–2010 shows significant underrepresentation of households by the conventional unconditional ‘food share’ distribution in the higher range of food budget shares that correspond to the lower end of the income profile. This may have serious consequences for welfare measurement.     

A note on characterizations based on truncated expectations

Many characterization results of the bivariate exponential distribution and the bivariate geometric distribution have been proved in the literature. Recently Nair and Nair (1988b, Ann. Inst. Statist. Math. 40 (2), 267–271) obtained a characterization result of the Gumbel bivariate exponential distribution and a bivariate geometric distribution based on truncated moments. In this note, we extend the results of Nair and Nair (1988b) to obtain a general result, characterizing these two bivariate distributions based on the truncated expectation of a function h, satisfying some mild conditions.     

More Efficient Approximation of Multiple Integrals using Steady State Ranked Simulated Sampling

This article extends the concept of using the steady state ranked simulated sampling approach (SRSIS) by Al-Saleh and Samawi (2000) for improving Monte Carlo methods for single integration problem to multiple integration problems. We demonstrate that this approach provides unbiased estimators and substantially improves the performance of some Monte Carlo methods for bivariate integral approximations, which can be extended to multiple integrals’ approximations. This results in a significant reduction in costs and time required to attain a certain level of accuracy. In order to compare the performance of our method with the Samawi and Al-Saleh (2007) method, we use the same two illustrations for the bivariate case.     

Some bivariate families of lagrangian probability distributions

The bivariate Lagrange expansion, given by Poincare (1986), has been explained and slightly modified which gives bivariate Lagrangian probability models. A generalized bivariate Lagrangian Poisson distribution with six parameters has been obtained and studied. Also, the bivariate Lagrangian binomial, bivariate Lagrangian negative binomial and bivariate Lagrangian logarithmic series distribution have been obtained.     

A comparison of the independent-samples t-test and the paired-samples t-test when the observations are nonnegatively correlated pairs

An account of the behavior of the independent-samples t-test when applied to homoschedastic bivariate normal data is presented, and a comparison is made with the paired-samples t-test. Since the significance level is not violated when applying the independent-samples t-test to data which consist of positively correlated pairs and since the estimate of the variance is based on a larger number of ‘degrees of freedom’, the results suggest that when the sample size is small, one should not worry much about the possible existence of weak positive correlation. One may do better, powerwise, to ignore such correlation and use the independent-samples t-test, as though the samples were independent.     

On certain alternative mean square error estimators in complex survey sampling

Rao (J. Indian Statist. Assoc. 17 (1979) 125) has given a ‘necessary form’ for an unbiased mean square error (MSE) estimator to be ‘uniformly non-negative’. The MSE is of a homogeneous linear estimator ‘subject to a specified constraint’, for a survey population total of a real variable of interest. We present a corresponding theorem when the ‘constraint’ is relaxed. Certain results are added presenting formulae for estimators of MSEs when the variate-values for the sampled individuals are not ascertainable. Though not ascertainable, they are supposed to be suitably estimated either by (1) randomized response techniques covering sensitive issues or by (2) further sampling in ‘subsequent’ stages in specific ways when the initial sampling units are composed of a number of sub-units. Using live numerical data, practical uses of the proposed alternative MSE estimators are demonstrated.     

On bivariate transformation of scale distributions

ABSTRACT

Elsewhere, I have promoted (univariate continuous) “transformation of scale” (ToS) distributions having densities of the form 2g(Π^?1(x)) where g is a symmetric distribution and Π is a transformation function with a special property. Here, I develop bivariate (readily multivariate) ToS distributions. Univariate ToS distributions have a transformation of random variable relationship with Azzalini-type skew-symmetric distributions; the bivariate ToS distribution here arises from marginal variable transformation of a particular form of bivariate skew-symmetric distribution. Examples are given, as are basic properties—unimodality, a covariance property, random variate generation—and connections with a bivariate inverse Gaussian distribution are pointed out.     

FACTORIAL ANALYSIS OF RECIDIVIST DATA

This paper presents a method of fitting factorial models to recidivism data consisting of the (possibly censored) time to ‘fail’ of individuals, in order to test for differences between groups. Here ‘failure’ means rearrest, reconviction or reincarceration, etc. A proportion P of the sample is assumed to be ‘susceptible’ to failure, i.e. to fail eventually, while the remaining 1-P are ‘immune’, and never fail. Thus failure may be described in two ways: by the probability P that an individual ever fails again (‘probability of recidivism’), and by the rate of failure Λ for the susceptibles. Related analyses have been proposed previously: this paper argues that a factorial approach, as opposed to regression approaches advocated previously, offers simplified analysis and interpretation of these kinds of data. The methods proposed, which are also applicable in medical statistics and reliability analyses, are demonstrated on data sets in which the factors are Parole Type (released to freedom or on parole), Age group (≤ 20 years, 20–40 years, > 40 years), and Marital Status. The outcome (failure) is a return to prison following first or second release.