In this paper, we consider kernel-based estimators in the nonparametric binary regression problem with multidimensional covariates. We propose a local linear type estimator of the response probability function with kernel weighted at each observed covariate. In addition, we discuss the rule of thumb bandwidth selector and the plug-in bandwidth selector. The efficiency of the weighted local linear estimator is determined from results of asymptotic properties and our simulation study.

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Cataract surgery results when a patient decides to undergo lens surgery following a diagnosis of a clinically significant cataract (CSC)νll. Because the presence of a CSC is generally latent and unobserved, a person might not receive cataract surgery even if the person has a CSCνll. This misclassification needs to be adjusted in the statistical analysis of CSC so as to reduce the bias in the parameter estimation. Following Magder and Hughes (1997) and using the cataract surgery data on atomic-bomb survivors at the Radiation Effects Research Foundation, we used this method for estimating the prevalence of CSC in a linear logistic dose response model taking account of the sensitivity and/or specificity of the decision for lens surgery. The estimated sensitivity was 0.385 (95% CI: 0.268, 0.517) and the estimated specificity was perfect. The odds ratio estimate for the radiation dose response changed from 1.39 (95% CI: 1.24, 1.55) to 1.58 (95% CI: 1.26, 1.98) when allowing for the imperfect sensitivity. A large sample simulation study with a continuous covariate was conducted, assuming either imperfect sensitivity or imperfect specificity, to investigate the performance of the method. Results indicated that the parameter estimates are almost correct. We calculated the asymptotic relative efficiency (ARE) for a simple logistic regression slope estimate and showed that the ARE depends only on the values of slope and intercept parameters.

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We derive a Markov basis consisting of moves of degree at most three for a two-state toric homogeneous Markov chain model of arbitrary length without parameters for initial states. Our basis consists of moves of degree three and degree one, which alter the initial frequencies, in addition to moves of degree two and degree one for toric homogeneous Markov chain model with parameters for initial states.

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The structural gradient model is a multivariate statistical model used to extract various interactions of a given data set. In this note, we show that Efron's statistical curvature of the structural gradient model is less than that of a competitive mixture model under a null hypothesis.

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We consider k-sample and change point problems for independent data in a unified way. We propose a test statistic based on the rank statisitcs. The asymptotic distribution under the null hypothesis is shown to be the supremum of the 2-dimensional standard Brownian pillow. Also, the test is shown to be consistent under the alternative that k distribution functions are linearly independent. It is important from practical point of view that our test is not only asymptotically distribution free but also distribution free even for fixed finite sample.

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Consider a sequence of independent observations X₁,...,X_n from a N(θ,cθ) distribution with 0<θ <∞. We assume that θ is unknown, but c(>0) is known. We begin with the problem of testing H₀: θ =θ ₀ against H₁: θ =θ ₁ where θ ₀,θ ₁(θ ₀≠ θ ₁) are specified values of θ. The most powerful (MP) level α test depends upon ∑_i=1ⁿX_i², a complete and sufficient statistic for θ, which has a multiple of a non-central chi-square distribution with its non-centrality parameter involving n and the true parameter value θ under H₀,H₁. We first target type-I and type-II error probabilities α and β respectively, with α >0,β >0,α +β <1. We set out to determine the required exact sample size which will control these error probabilities and provide two useful large-sample approximations for the sample size. The three methods provide nearly the same required sample size whether n is small, moderate or large. We also show how one may derive the minimum variance unbiased estimators (MVUEs) for a number of interesting and useful functionals of θ by combining some previous work from Mukhopadhyay and Cicconetti (2004) and Mukhopadhyay and Bhattacharjee (2010). All methodologies are illustrated with both simulated data and real data.} \keywords{Exact method, large-sample method, minimum variance unbiased estimation, monotone likelihood ratio, most powerful test, non-central chi-square distribution, one-parameter exponential family, required sample size determination, type-I error probability, type-II error probability.

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