A minimum asymptotic mean squared error controller for an IMA(1,1) noise process with a starting offset, and its resetting design

In a discrete-part manufacturing process, the noise is often described by an IMA(1,1) process and the pure unit delay transfer function is used as the feedback controller to adjust it. The optimal controller for this process is the well-known minimum mean square error (MMSE) controller. The starting level of the IMA(1,1) model is assumed to be on target when it starts. Considering such an impractical assumption, we adopt the starting offset. Since the starting offset is not observable, the MMSE controller does not exist. An alternative to the MMSE controller is the minimum asymptotic mean square error controller, which makes the long-run mean square error minimum.Another concern in this article is the un-stability of the controller, which may produce high adjustment costs and/or may exceed the physical bounds of the process adjustment. These practical barriers will prevent the controller to adjust the process properly. To avoid this dilemma, a resetting design is proposed. That is, the resetting procedure in use of the controller is to adjust the process according to the controller when it remains within the reset limit, and to reset the process, otherwise.The total cost for the manufacturing process is affected by the off-target cost, the adjustment cost, and the reset cost. Proper values for the reset limit are selected to minimize the average cost per reset interval (ACR) considering various process parameters and cost parameters. A time non-homogeneous Markov chain approach is used for calculating the ACR. The effect of adopting the starting offset is also studied here.     

Almost sure central limit theorem for the maxima and sums of stationary Gaussian sequences

In this paper, we proved an almost sure central limit theorem for the maxima (after centered at the sample mean) and the partial sums of standardized stationary Gaussian sequences under some conditions related to the convergence rate of covariance functions, which extended the existing results.     

The limit distribution of a modified Shapiro–Wilk statistic for normality to Type II censored data

The Shapiro–Wilk statistic and modified statistics are widely used test statistics for normality. They are based on regression and correlation. The statistics for the complete data can be easily generalized to the censored data. In this paper, the distribution theory for the modified Shapiro–Wilk statistic is investigated when it is generalized to Type II right censored data. As a result, it is shown that the limit distribution of the statistic can be representable as the integral of a Brownian bridge. Also, the power comparison to the other procedure is performed.     

The analysis of tumorigenicity data using a frailty effect

In animal tumorigenicity data, the time of occurrence of the tumor is not observed because the existence of the tumor is looked for only at either the time of death or the time of sacrifice of the animal. Such an incomplete data structure makes it difficult to investigate the impact of treatment on the occurrence of tumors. A three-state model (no tumor–tumor–death) is used to model events that occurred sequentially and to connect them. In this paper, we also employed a frailty effect to model the dependency of death on tumor occurrence. For the inference of parameters, an EM algorithm is considered. The method is applied to a real bladder tumor data set and a simulation study is performed to show the behavior of the proposed estimators.     

Some inferential procedures in randomized repeated measurement design for binary response

We consider a two-treatment two-period crossover design in the presence of possible carryover effects, where the treatment responses are binary. We provide some simple probability models incorporating the possible carryover effects. Asymptotic distributions of the estimates of the parameters under the proposed models are derived. We carry out tests for treatment difference and carryover effects. Finally we use a data set to illustrate the applicability of the proposed procedures.     

Remarks on an integral functional driven by sub-fractional Brownian motion

This paper studies the functionals where is a one-dimension sub-fractional Brownian motion with index H∈(0,1). It shows that there exists a constant p_H∈(1,2) such that p-variation of the process (j=1,2) is equal to 0 if p>p_H, where ?_j, j=1,2, are the local time and weighted local time of S^H, respectively. This extends the classical results for Brownian motion.     

Estimation of the scale parameter of the half-logistic distribution with multiply type II censored sample

In this paper, we consider the maximum likelihood and Bayes estimation of the scale parameter of the half-logistic distribution based on a multiply type II censored sample. However, the maximum likelihood estimator(MLE) and Bayes estimator do not exist in an explicit form for the scale parameter. We consider a simple method of deriving an explicit estimator by approximating the likelihood function and discuss the asymptotic variances of MLE and approximate MLE. Also, an approximation based on the Laplace approximation (Tierney & Kadane, 1986) is used to obtain the Bayes estimator. In order to compare the MLE, approximate MLE and Bayes estimates of the scale parameter, Monte Carlo simulation is used.     

Central limit theorems for LS estimators in the EV regression model with dependent measurements

In this paper, we consider the simple linear errors-in-variables (EV) regression models: η_i=θ+βx_i+ε_i,ξ_i=x_i+δ_i,1≤i≤n, where θ,β,x₁,x₂,… are unknown constants (parameters), (ε₁,δ₁),(ε₂,δ₂),… are errors and ξ_i,η_i,i=1,2,… are observable. The asymptotic normality for the least square (LS) estimators of the unknown parameters β and θ in the model are established under the assumptions that the errors are m-dependent, martingale differences, ?-mixing, ρ-mixing and α-mixing.     

A remark on the Bahadur representation of sample quantiles for negatively associated sequences

Under mild conditions, we investigate further the Bahadur representation of sample quantiles for negatively associated sequences, whose convergence rate is faster than the corresponding one in Ling (2008).     

Convergence properties of conditional medians

Certain convergence theorems, akin to results of Lévy and Lebesgue for conditional expectations, are established for conditional medians. An explicit representation of a conditional median is given.     

Consistency of the least-squares-estimators for linear stochastic difference equations

Assumptions are given for the strong consistency in the stable case and weak consistency in the instable case of the Least-Square-Estimator of the unknown system-parameters of a inhomogeneous linear stochastic difference equation system with constant coefficients.     

Gamma-minimax results for the class of unimodal priors

Olman and Shmundak proved 1985 that in estimating a bounded normal mean under squared error loss the Bayes estimator with respect to the uniform distribution on the parameter interval is gamma-minimax when the parameter interval is sufficiently small and the class of priors consists of all symmetric and unimodal distributions. Recently, one of the authors showed that this result remains valid for quite general families of distributions which satisfy some regularity conditions. In the present paper a generalization to the class of unimodal priors with fixed mode is derived. It is proved that the Bayes estimator with respect to a suitable mixture of two uniform distributions is gamma-minimax for sufficiently small parameter intervals. To that end appropriate characterizations of a saddle point in the corresponding statistical games are established. Some results of a numerical study are presented.     

EDF tests of goodness of fit for transform-both-sides models

Very often in regression analysis, a particular functional form connecting known covariates and unknown parameters is either suggested by previous work or demanded by theoretical considerations so that the deterministic part of the responses has a known form. However, the underlying error structure is often less well understood. In this case, the transform-both-sides (TBS) models are appropriate. In this paper we generalize the usual TBS models and develop tests to assess goodness of fit when fitting TBS or GTBS models. Parameter estimation is discussed, and tests based on the Cramér-von Mises statistic and the Anderson-Darling statistic are presented with a table suitable for finite-sample applications.     

An application of the Bernoulli part to local limit theorems for moving averages on stationary sequences

We consider partial sums S_n of a general class of stationary sequences of integer-valued random variables, and we provide sufficient conditions for S_n to satisfy a local limit theorem. To prove this result, we introduce a concept called the Bernoulli part. The amount of Bernoulli part in S_n determines the extent to which the density of S_n is relatively flat. If in addition S_n satisfies a global central limit theorem, the local limit theorem follows.     

Normal approximation to the hypergeometric distribution in nonstandard cases and a sub-Gaussian Berry–Esseen theorem

In this paper, we consider simple random sampling without replacement from a dichotomous finite population. We investigate accuracy of the Normal approximation to the Hypergeometric probabilities for a wide range of parameter values, including the nonstandard cases where the sampling fraction tends to one and where the proportion of the objects of interest in the population tends to the boundary values, zero and one. We establish a non-uniform Berry–Esseen theorem for the Hypergeometric distribution which shows that in the nonstandard cases, the rate of Normal approximation to the Hypergeometric distribution can be considerably slower than the rate of Normal approximation to the Binomial distribution. We also report results from a moderately large numerical study and provide some guidelines for using the Normal approximation to the Hypergeometric distribution in finite samples.     

Geostochastic calculus

A stochastic calculus for a family of continuous measure-valued Markov processes is developed. Such processes arise naturally in the construction of stochastic models of spatially distributed populations. The stochastic calculus is a tool whereby a class of density-dependent models can be studied in terms of the multiplicative measure diffusion process. In this paper the stochastic integral is introduced in the space-time setting and a Cameron-Martin-Girsanov theorem is established.     

On asymptotic properties of the parameter estimator for a type of SPDE

In this paper, we study a random field _U?(t,x)$U_{?}(t,x)$ governed by some type of stochastic partial differential equations with an unknown parameter θ$\theta$ and a small noise ?$?$. We construct an estimator of θ$\theta$ based on the continuous observation of N   Fourier coefficients of _U?(t,x)$U_{?}(t,x)$, and prove the strong convergence and asymptotic normality of the estimator when the noise ?$?$ tends to zero.     

On moments of the supremum of normed weighted averages

Let {X, X_n; n ≥ 1} be a sequence of real-valued iid random variables, 0 < r < 2 and p > 0. Let D = { A = (a_nk; 1 ≤ k ≤ n, n ≥ 1); a_nk, ? R and sup_{n, k} |a_n,k| < ∞}. Set S_n( A ) = ∑ⁿ_k=1a_{n, k}X_k for A ? D and n ≥ 1. This paper is devoted to determining conditions whereby E{sup_{n ≥ 1}, |S_n( A )|/n^1/r}^p < ∞ or E{sup_{n ≥ 2} |S_n( A )|/2n log n)^1/2}^p < ∞ for every A ? D. This generalizes some earlier results, including those of Burkholder (1962), Choi and Sung (1987), Davis (1971), Gut (1979), Klass (1974), Siegmund (1969) and Teicher (1971).     

Asymptotic properties of a generalized regression-type predictor of a finite population variance in probability sampling

A system of predictors for estimating a finite population variance is defined and shown to be asymptotically design-unbiased (ADU) and asymptotically design-consistent (ADC) under probability sampling. An asymptotic mean squared error (MSE) of a generalized regression-type predictor, generated from the system, is obtained. The suggested predictor attains the minimum expected variance of any design-unbiased estimator when the superpopulation model is correct. The generalized regression-type predictor and the predictor suggested by Mukhopadhyay (1990) are compared.     

Invalidity of average squared error criterion in density estimation

The average squared error has been suggested earlier as an appropriate estimate of the integrated squared error, but an example is given which shows their ratio can tend to infinity. The results of a Monte Carlo study are also presented which suggest the average squared error can seriously underestimate the errors inherent in even the simplest density estimations.