Exact confidence intervals for randomized response strategies

For surveys with sensitive questions, randomized response sampling strategies are often used to increase the response rate and encourage participants to provide the truth of the question while participants' privacy and confidentiality are protected. The proportion of responding ‘yes’ to the sensitive question is the parameter of interest. Asymptotic confidence intervals for this proportion are calculated from the limiting distribution of the test statistic, and are traditionally used in practice for statistical inference. It is well known that these intervals do not guarantee the coverage probability. For this reason, we apply the exact approach, adjusting the critical value as in [10 J. Frey and A. Pérez, Exact binomial confidence intervals for randomized response, Amer. Statist.66 (2012), pp. 8–15. Available at http://dx.doi.org/10.1080/00031305.2012.663680.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]], to construct the exact confidence interval of the proportion based on the likelihood ratio test and three Wilson-type tests. Two randomized response sampling strategies are studied: the Warner model and the unrelated model. The exact interval based on the likelihood ratio test has shorter average length than others when the probability of the sensitive question is low. Exact Wilson intervals have good performance in other cases. A real example from a survey study is utilized to illustrate the application of these exact intervals.     

EXACT P‐VALUES FOR DISCRETE MODELS OBTAINED BY ESTIMATION AND MAXIMIZATION

In constructing exact tests from discrete data, one must deal with the possible dependence of the P‐value on nuisance parameter(s) ψ as well as the discreteness of the sample space. A classical but heavy‐handed approach is to maximize over ψ. We prove what has previously been understood informally, namely that maximization produces the unique and smallest possible P‐value subject to the ordering induced by the underlying test statistic and test validity. On the other hand, allowing for the worst case will be more attractive when the P‐value is less dependent on ψ. We investigate the extent to which estimating ψ under the null reduces this dependence. An approach somewhere between full maximization and estimation is partial maximization, with appropriate penalty, as introduced by Berger & Boos (1994, P values maximized over a confidence set for the nuisance parameter. J. Amer. Statist. Assoc. 89 , 1012–1016). It is argued that estimation followed by maximization is an attractive, but computationally more demanding, alternative to partial maximization. We illustrate the ideas on a range of low‐dimensional but important examples for which the alternative methods can be investigated completely numerically.     

Modifying the exact test for a binomial proportion and comparisons with other approaches

In this note we provide a simple continuity and tail-corrected approach to the standard exact test for a single binomial proportion commonly used in practice. We redefine the p-value for the two-sided alternative by noting the skewed distribution of the sample proportion under the null hypothesis. We illustrate that for both one and two-sided alternatives the coverage probabilities of the new methodology approaches more closely the desired type I error α and thus recommend these modifications to the applied statistician for consideration.     

Markov chain Monte Carlo exact inference for binomial and multinomial logistic regression models

We develop Metropolis-Hastings algorithms for exact conditional inference, including goodness-of-fit tests, confidence intervals and residual analysis, for binomial and multinomial logistic regression models. We present examples where the exact results, obtained by enumeration, are available for comparison. We also present examples where Monte Carlo methods provide the only feasible approach for exact inference.     

一类Fisher方程的行波解及行波波速

通过引入一种解的形式讨论了双曲型Fisher方程，利用待定系数法得到该方程的新的行波解及行波波速．这个方程被广泛地应用于化学动力学和数学生物学．     

Bayesian analysis of contingency tables: a simulation and graphics-based approach

In this paper we present a simulation and graphics-based model checking and model comparison methodology for the Bayesian analysis of contingency tables. We illustrate the approach by testing the hypotheses of independence and symmetry on complete and incomplete simulated tables.     

Jacobi椭圆函数展开法的应用

通过对Jacobi椭圆函数展开法适用条件——秩的分析，求解了Joseph—Egri方程的精确周期解，并且对椭圆函数展开法进行适当的扩充以求解变系数KdV方程的精确周期解．此方法同样适用于其它具有变系数的非线性演化方程（NLEEs）．     

A note on nonparametric estimation of circular conditional densities

ABSTRACT

The conditional density offers the most informative summary of the relationship between explanatory and response variables. We need to estimate it in place of the simple conditional mean when its shape is not well-behaved. A motivation for estimating conditional densities, specific to the circular setting, lies in the fact that a natural alternative of it, like quantile regression, could be considered problematic because circular quantiles are not rotationally equivariant. We treat conditional density estimation as a local polynomial fitting problem as proposed by Fan et al. [Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika. 1996;83:189–206] in the Euclidean setting, and discuss a class of estimators in the cases when the conditioning variable is either circular or linear. Asymptotic properties for some members of the proposed class are derived. The effectiveness of the methods for finite sample sizes is illustrated by simulation experiments and an example using real data.     

Some unbiased estimators versus mean per unit and ratio estimators in finite population sample surveys

We present some unbiased estimators at the population mean in a finite population sample surveys with simple random sampling design where information on an auxiliary variance x positively correlated with the main variate y is available. Exact variance and unbiased estimate of the variance are computed for any sample size. These estimators are compared for their precision with the mean per unit and the ratio estimators. Modifications of the estimators are suggested to make them more precise than the mean per unit estimator or the ratio estimator regardless of the value of the population correlation coefficient between the variates x and y. Asymptotic distribution of our estimators and confidnece intervals for the population mean are also obtained.     

On the ratio of generalized variances

This paper examines the ratio of the determinants of two independent central Wishart matrices. The ratio is represented as a product of independent beta type-2 random variables and then the exact density is evaluated for the general case. Different representations and particular cases are also discussed.