On Box–Muller Transformation and Simulation of Normal Record Data

Record data are commonly encountered in many fields such as sports, geography, finance, and reliability. In this article, we use the well-known Box–Muller transformation to develop an efficient method of simulating record data from the normal distribution. Another method based on exponential records is also discussed. Then, the performance of these algorithms is compared with some standard simulation methods.     

Simulation of the typical Poisson–Voronoi–Cox–Voronoi cell

We consider stationary Poisson–Voronoi tessellations (PVT) in the Euclidean plane and study the properties of Voronoi tessellations induced by linear Poisson processes on the edges of the PVT. We are especially interested in simulation algorithms for the typical cell. Two different simulation algorithms are introduced. The first algorithm directly simulates the typical cell, whereas the second algorithm simulates cells from which distributional properties of the typical cell can be obtained. This second algorithm can also be used for simulating the typical cell of other Cox–Voronoi tessellations. The implementation of both algorithms is tested for their correctness using random software tests. Then different cell characteristics are studied by simulation and compared with the typical cell of PVT and Cox–Voronoi tessellations based on linear Poisson processes on the lines of Poisson line processes. Our results can be applied, for example, in the analysis of telecommunication networks and vesicle paths on cytoskeletal networks.     

D–optimality of a class of saturated main–effect plans and allied results

This paper applies the theory of unimodular matrices to prove that all saturated main effect plans of an s₁ × s₂ factorial are equivalent from the point of view of D–optimality and are hence all D–optimal. The A– and E–optimal plans in this context have also been derived. An application in sequential experimentation has been considered     

Kullback–Leibler information of a censored variable and its applications

In this paper, we study the Kullback–Leibler (KL) information of a censored variable, which we will simply call it censored KL information. The censored KL information is shown to have the necessary monotonicity property in addition to inherent properties of nonnegativity and characterization. We also present a representation of the censored KL information in terms of the relative risk and study its relation with the Fisher information in censored data. Finally, we evaluate the estimated censored KL information as a goodness-of-fit test statistic.     

Three-stage and ‘accelerated’ sequential procedures for the mean of a normal population with known coefficient of variation

Three-stage and ‘accelerated’ sequential procedures are developed for estimating the mean of a normal population when the population coefficient of variation (CV) is known. In spite of the usual estimator, i.e. the sample mean, Searls' (1964 Searls, DT. (1964). The utilization of a known coefficient of variation in the estimation procedure. J. Amer. Statist. Assoc, 50: 1225–1226.  ) estimator is utilized for the estimation purpose. It is established that Searls' estimator dominates the sample mean under the two sampling schemes.     

Small area estimation strategies for large population surveys: a comparison of design and model-based methods

Small area estimation (SAE) concerns with how to reliably estimate population quantities of interest when some areas or domains have very limited samples. This is an important issue in large population surveys, because the geographical areas or groups with only small samples or even no samples are often of interest to researchers and policy-makers. For example, large population health surveys, such as Behavioural Risk Factor Surveillance System and Ohio Mecaid Assessment Survey (OMAS), are regularly conducted for monitoring insurance coverage and healthcare utilization. Classic approaches usually provide accurate estimators at the state level or large geographical region level, but they fail to provide reliable estimators for many rural counties where the samples are sparse. Moreover, a systematic evaluation of the performances of the SAE methods in real-world setting is lacking in the literature. In this paper, we propose a Bayesian hierarchical model with constraints on the parameter space and show that it provides superior estimators for county-level adult uninsured rates in Ohio based on the 2012 OMAS data. Furthermore, we perform extensive simulation studies to compare our methods with a collection of common SAE strategies, including direct estimators, synthetic estimators, composite estimators, and Datta GS, Ghosh M, Steorts R, Maples J.'s [Bayesian benchmarking with applications to small area estimation. Test 2011;20(3):574–588] Bayesian hierarchical model-based estimators. To set a fair basis for comparison, we generate our simulation data with characteristics mimicking the real OMAS data, so that neither model-based nor design-based strategies use the true model specification. The estimators based on our proposed model are shown to outperform other estimators for small areas in both simulation study and real data analysis.     

Stein's two sample test of a general linear hypothesis and a noncentral f–type distribution

Stein's two–sample procedure for a general linear model is studied and derived in terms of matrices in which the error tems are distributed as multivatriate student t–error terms. Tests and confidence regions are constructed in a similar way to classical linear models which involves percentage points of student t and F distributions. The advantages of taking two samples are: the variance of the error terms is known, and the power of tests are size of confidence regions are controllable. A new distribution called noncentral F–type distribution different from the nencentral F is found when considerinf the power of the test of general linear hypothesis.     

A Decomposition of Pearson–Fisher and Dzhaparidze–Nikulin Statistics and Some Ideas for a More Powerful Test Construction

An explicit decomposition on asymptotically independent distributed as chi-squared with one degree of freedom components of the Pearson–Fisher and Dzhaparidze–Nikulin tests is presented. The decomposition is formally the same for both tests and is valid for any partitioning of a sample space. Vector-valued tests, components of which can be not only different scalar tests based on the same sample, but also scalar tests based on components or groups of components of the same statistic are considered. Numerical examples illustrating the idea are presented.     

On the comparison of Horvitz–Thompson and Murthy's sampling strategies for estimating finite population proportions in direct and randomized response surveys

We consider the problem of unbiased estimation of a finite population proportion and compare the relative efficiency of the unequal probability sampling strategies due to Horvitz and Thompson (1952 Horvitz, D.G., Thompson, D.J. (1952). A generalization of sampling without replacement. J Am Stat Assoc. 47:663–685.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and Murthy (1957 Murthy, M.N. (1957). Ordered and unordered estimators in sampling without replacement. Sankhya 18:379–390. [Google Scholar]) under a super-population model. It is shown that the model expected variance is smaller for the Murthy's (1957 Murthy, M.N. (1957). Ordered and unordered estimators in sampling without replacement. Sankhya 18:379–390. [Google Scholar]) strategy both when these two sampling strategies are based on data obtained from (i) a direct survey, and (ii) a randomized response (RR) survey employing some RR technique following a general RR model.     

Likelihood Function and Canonical Correlation Analysis of the Peña–Box Model

The Peña–Box model is a type of dynamic factor model whose factors try to capture the time-effect movements of a multiple time series. The Peña–Box model can be expressed as a vector autoregressive (VAR) model with constraints. This article derives the maximum likelihood estimates and the likelihood ratio test of the VAR model for Gaussian processes. Then a test statistic constructed by canonical correlation coefficients is presented and adjusted for conditional heteroscedasticity. Simulations confirm the validity of adjustments for conditional heteroscedasticity, and show that the proposed statistics perform better than the statistics used in the existing literature.     

Nonparametric estimator for the survival function of quality adjusted lifetime (QAL) in a three-state illness–death model

In this work, we consider nonparametric estimation of QAL distribution in a three-state illness–death model. In our approach, we first write down the expression for the distribution of QAL in terms of the joint distribution of the sojourn times in the three states. The estimate of the QAL distribution is obtained by substituting the estimates of sojourn time distributions in the expression of the QAL distribution. The proposed nonparametric estimate, assuming independence between time to illness and sojourn time in the state of illness, is uniformly consistent. Asymptotic normality has also been established. An estimate of asymptotic variance has been obtained. The performance of the proposed estimator is investigated by simulation. A data set of the Stanford Heart Transplant program has been analyzed for illustration.     

Various measures of dependence of a new asymmetric generalized Farlie–Gumbel–Morgenstern copulas

ABSTRACT

In this paper, we discuss an asymmetric extension of Farlie–Gumbel–Morgenstern copulas studied by several authors and obtain the range of the parameter. We derive an expression for regression function and the properties of these copulas are studied in detail. Also, explicit expressions for various measures of association are obtained. These measures are numerically compared for some particular parametric values of the copulas.     

Measures of biomarker dependence using a copula-based multivariate epsilon–skew–normal family of distributions

In this note we develop a new multivariate copula model based on epsilon–skew–normal marginal densities for the purpose of examining biomarker dependency structures. We illustrate the flexibility and utility of this model via a variety of graphical tools and a data analysis example pertaining to salivary biomarker. The multivariate normal model is a sub-model of the multivariate epsilon–skew–normal distribution.     

Exact distribution of a modified Behrens–Fisher statistic

The exact distribution of a modified Behrens–Fisher statistic is derived. The distribution function is mostly elementary and is simpler than the exact distribution derived by Nel et al. Its practical use (including computationalefficiency and computational convenience) is discussed.     

The asymptotic size and power of the augmented Dickey–Fuller test for a unit root

It is shown that the limiting distribution of the augmented Dickey–Fuller (ADF) test under the null hypothesis of a unit root is valid under a very general set of assumptions that goes far beyond the linear AR(∞) process assumption typically imposed. In essence, all that is required is that the error process driving the random walk possesses a continuous spectral density that is strictly positive. Furthermore, under the same weak assumptions, the limiting distribution of the ADF test is derived under the alternative of stationarity, and a theoretical explanation is given for the well-known empirical fact that the test's power is a decreasing function of the chosen autoregressive order p. The intuitive reason for the reduced power of the ADF test is that, as p tends to infinity, the p regressors become asymptotically collinear.     

A case study of green tree frog population size estimation by repeated capture–mark–recapture method with individual tagging: a parametric bootstrap method vs Jolly–Seber method

This paper deals with estimation of a green tree frog population in an urban setting using repeated capture–mark–recapture (CMR) method over several weeks with an individual tagging system which gives rise to a complicated generalization of the hypergeometric distribution. Based on the maximum likelihood estimation, a parametric bootstrap approach is adopted to obtain interval estimates of the weekly population size which is the main objective of our work. The method is computation-based; and programming intensive to implement the algorithm for re-sampling. This method can be applied to estimate the population size of any species based on repeated CMR method at multiple time points. Further, it has been pointed out that the well-known Jolly–Seber method, which is based on some strong assumptions, produces either unrealistic estimates, or may have situations where its assumptions are not valid for our observed data set.     

Bayesian analysis of hypothesis testing problems for general population: A Kullback–Leibler alternative

We consider a hypothesis problem with directional alternatives. We approach the problem from a Bayesian decision theoretic point of view and consider a situation when one side of the alternatives is more important or more probable than the other. We develop a general Bayesian framework by specifying a mixture prior structure and a loss function related to the Kullback–Leibler divergence. This Bayesian decision method is applied to Normal and Poisson populations. Simulations are performed to compare the performance of the proposed method with that of a method based on a classical z-test and a Bayesian method based on the “0–1” loss.