In this paper we consider the kernel estimators of a distribution function defined by the stochastic approximation algorithm when the observation are contamined by measurement errors. It is well known that this estimators depends heavily on the choice of a smoothing parameter called the bandwidth. We propose a specific second generation plug-in method of the deconvolution kernel distribution estimators defined by the stochastic approximation algorithm. We show that, using the proposed bandwidth selection and the stepsize which minimize the MISE (Mean Integrated Squared Error), the proposed estimator will be better than the classical one for small sample setting when the error variance is controlled by the noise to signal ratio. We corroborate these theoretical results through simulations and a real dataset.

For a one-sided truncated exponential family of distributions with a truncation parameter and a natural parameter as a nuisance parameter, it is shown by Akahira and Ohyauchi (2016) that the second order asymptotic loss of a bias-adjusted maximum likelihood estimator (MLE) of a truncation parameter for unknown natural parameter relative to a bias-adjusted MLE of a truncation parameter for known natural parameter is obtained. In this paper, in a similar way to Akahira and Ohyauchi (2016), for a two-sided truncated exponential family of distributions with a natural parameter and lower and upper truncation parameters, the stochastic expansions of the bias-adjusted MLE of an upper truncation parameter for known natural and lower truncation parameters, the bias-adjusted MLE of an upper truncation parameter for unknown natural parameter and known lower truncation parameter and the bias-adjusted MLE of an upper truncation parameter for unknown natural and lower truncation parameters are derived, their asymptotic variances are given, and the second order asymptotic losses of the MLEs of an upper truncation parameter for unknown natural parameter and known/unknown lower truncation parameter relative to the MLE of an upper truncation parameter for known natural and lower truncation parameters are also obtained. Further, some examples including an upper-truncated Pareto case are given.

This paper is concerned with the null distribution of the likelihood ratio test statistic −2log Λ for testing the adequacy of a random-effects covariance structure in a parallel profile model. It is known that the null distribution of −2log Λ converges to χ²_f or 0.5χ²_f + 0.5χ²_f+1 when the sample size tends to infinity. In order to extend this result, we derive asymptotic expansions of the null distribution of −2log Λ. The accuracy of the approximations based on the limiting distribution and an asymptotic expansion are compared through numerical experiments.

For a one-sided truncated exponential family of distributions with a truncation parameter γ and a natural parameter θ as a nuisance parameter, the stochastic expansions of the Bayes estimator when θ is known and the Bayes estimator plugging the maximum likelihood estimator (MLE) in θ of when θ is unknown are derived. The second order asymptotic loss of relative to is also obtained through their asymptotic variances. Further, it is shown that and are second order asymptotically equivalent to the bias-adjusted MLEs and when θ is known and when θ is unknown, respectively. Some examples are also given.