Adaptive sequential confidence intervals for A location Parameter

Let θ + X₁, θ + X₂... be a random sample and X_i's have a common distribution function F, where F has an absolutely continuous density f that is symmetric about origin. For a given confidence coefficient, a fully sequential procedure is proposed for constructing a fixed-width confidence interval for the location parameter θ. The result is based on Stone's (1975) adaptive maximum likelihood estimator. under mild conditions the procedure is shown to be asymptotically consistent and fully efficient.     

Replicated piecewise stopping numbers and sequential analysis

For certain basic problems in sequential analysis,it is shown that a piecewise sequential methodology resting on replicated piecewise stopping numbers provides a natural estimate of the variance of the overall stopping variable without compromising much on the performancecharacteristics of the estimation rules.     

Fixed-width confidence interval of log odds ratio for joint binomial and inverse binomial sampling

This article provides a two-stage procedure to develop a fixed-width confidence interval of log odds ratio in a joint binomial and inverse binomial setting where the stopping rule is obtained by adopting a two-stage procedure on the number of index subjects for the inverse sampling. A purely sequential version of this procedure is also studied. Different asymptotic results associated with the procedures are obtained. The findings are supported by detailed simulation study followed by one data example.     

On Two-Stage Sampling for Fixed-Width Interval Estimation of the Common Variance of Equi-Correlated Normal Distributions

Abstract

The present article presents two-stage procedures for fixed-width interval estimators of the common variance of equi-correlated normal distributions. This study is a continuation of those of Zacks and Ramig (1987 Zacks , S. and Ramig , P. ( 1987 ). Confidence Intervals for the Common Variance of Equicorrelated Normal Random Variables , in Contributions to the Theory and Applications of Statistics, A Volume in Honor of Herbert Soloman , New York : Academic Press . [Google Scholar]) and Ghezzi and Zacks (2005 Ghezzi , D. J. and Zacks , S. ( 2005 ). Inference on the Common Variance of Correlated Normal Random Variables , Communications in Statistics - Theory and Methods 34 : 1517 – 1531 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). We derive formulae for the distributions of total sample sizes and their functionals.     

Multi-stage point estimation of the mean of an inverse Gaussian distribution

In this article, we develop two-stage, three-stage, and accelerated sequential procedures for the point estimation of the mean μ of an inverse Gaussian distribution when the scale parameter λ is unknown. Both minimum risk and bounded risk estimation problems are considered subject to a weighted squared error loss function. We aim at controlling the associated risk functions for all three procedures. Second-order approximations are obtained for the proposed procedures.     

Fixed width interval estimation for the multiple response callibration problem

A multiple response extension is proposed for the univariate calibration problem and a sequential solution is developed. The sequential sampling procedure is a generalization of existing univariate results of Perng and Tong (1974) and is based on the theory developed by Chow and Robbins (1965). It is shown that the proposed multiple response procedure is at least as good as and in many instances better than the univariate method     

On asymptotically Risk-effivient sequential versions of generalized U-statistics

Sequential point estimation of means of generalized U-statistics is considered. Based on a well defined stopping rule, the proposed sequential (estimation) procedure is shown to be asymptotically (first order) risk-efficient. Asymptotic distributions of sequential generalized U-statistics and the allied stopping times are also studied.     

Asymptotic results for Fourier‐PARMA time series

Periodically stationary times series are useful to model physical systems whose mean behavior and covariance structure varies with the season. The Periodic Auto‐Regressive Moving Average (PARMA) process provides a powerful tool for modelling periodically stationary series. Since the process is non‐stationary, the innovations algorithm is useful to obtain parameter estimates. Fitting a PARMA model to high‐resolution data, such as weekly or daily time series, is problematic because of the large number of parameters. To obtain a more parsimonious model, the discrete Fourier transform (DFT) can be used to represent the model parameters. This article proves asymptotic results for the DFT coefficients, which allow identification of the statistically significant frequencies to be included in the PARMA model.     

Minimum risk sequential point estimation of the stress-strength reliability parameter for exponential distribution

Abstract

In this article, using purely and two-stage sequential procedures, the problem of minimum risk point estimation of the reliability parameter (R) under the stress–strength model, in case the loss function is squared error plus sampling cost, is considered when the random stress (X) and the random strength (Y) are independent and both have exponential distributions with different scale parameters. The exact distribution of the total sample size and explicit formulas for the expected value and mean squared error of the maximum likelihood estimator of the reliability parameter under the stress–strength model are provided under the two-stage sequential procedure. Using the law of large numbers and Monte Carlo integration, the exact distribution of the stopping rule under the purely sequential procedure is approximated. Moreover, it is shown that both proposed sequential procedures are finite and for special cases the exact distribution of stopping times has a degenerate distribution at the initial sample size. The performances of the proposed methodologies are investigated with the help of simulations. Finally, using a real data set, the procedures are clearly illustrated.     

A note on the CFR inequality for sequential estimation

Considering the regu1ar j.i.d. case, this note points out that Govindarajalu and Vincze(1989) are mistaken in claiming that their lower bound for the variance of sequential estimators is better than that of Wolfowitz (1847) by showing that the two bounds are identical.     

Subsampling inference for the autocovariances and autocorrelations of long‐memory heavy‐ tailed linear time series

We provide a self‐normalization for the sample autocovariances and autocorrelations of a linear, long‐memory time series with innovations that have either finite fourth moment or are heavy‐tailed with tail index 2 < α < 4. In the asymptotic distribution of the sample autocovariance there are three rates of convergence that depend on the interplay between the memory parameter d and α, and which consequently lead to three different limit distributions; for the sample autocorrelation the limit distribution only depends on d. We introduce a self‐normalized sample autocovariance statistic, which is computable without knowledge of α or d (or their relationship), and which converges to a non‐degenerate distribution. We also treat self‐normalization of the autocorrelations. The sampling distributions can then be approximated non‐parametrically by subsampling, as the corresponding asymptotic distribution is still parameter‐dependent. The subsampling‐based confidence intervals for the process autocovariances and autocorrelations are shown to have satisfactory empirical coverage rates in a simulation study. The impact of subsampling block size on the coverage is assessed. The methodology is further applied to the log‐squared returns of Merck stock.     

Distributions of Sequential and Two-Stage Stopping Times for Fixed-Width Confidence Intervals in Bernoulli Trials: Application in Reliability

Abstract

Albert N. Shiryaev's reminiscence of 50-years old results is astonishing. They look much more modern than their age. Here, some notes on statistics literature peculiarities, a fix of the stationary density figure, and some relations to discrete time numerics are provided.     

Two-stage procedures for the bounded risk point estimation of the parameter and hazard rate in two families of distributions

Two families of distributions are considered that cover a large number of probability distributions useful in investigations involving reliability studies and survival analyses. The problem of bounded risk point estimation of the parameter and hazard rate function of the two families of distribution is handled. Motivated by Mukhopadhyay and Pepe (2006 Mukhopadhyay, N. and Pepe, W. (2006). Exact Bounded Risk Estimation When the Terminal Sample Size and Estimator Are Dependent: The Exponential Case, Sequential Analysis 25: 85–101.[Taylor &; Francis Online] , [Google Scholar]), Roughani and Mahmoudi (2015 Roughani, G. and Mahmoudi, E. (2015). Exact Risk Evaluation of the Two-Stage Estimation of the Gamma Scale Parameter under Bounded Risk Constraint, Sequential Analysis 34: 387–405.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), and Mahmoudi and Lalehzari (2017 Mahmoudi, E. and Lalehzari, R. (2017). Bounded Risk Estimation of the Hazard Rate Function of the Exponential Distribution: Two-Stage Procedure, Sequential Analysis 36: 38–54.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), two-stage procedures are developed based on the maximum likelihood estimator (MLE) as well as uniformly minimum variance unbiased estimator (UMVUE). The estimation problem based on the minimum mean square estimator (MMSE) is also considered. We establish that the MMSE of the parameter and hazard rate provides a smaller risk.     

MULTIVARIATE LOCAL POLYNOMIAL REGRESSION FOR TIME SERIES:UNIFORM STRONG CONSISTENCY AND RATES

Abstract. Local high-order polynomial fitting is employed for the estimation of the multivariate regression function m ( x₁ ,… x_d ) = E {φ( Y_d )φ X ₁= x ₁,…, X_d = x_d }, and of its partial derivatives, for stationary random processes { Y_i , X_i }. The function φ may be selected to yield estimates of the conditional mean, conditional moments and conditional distributions. Uniform strong consistency over compact subsets of R_d , along with rates, are established for the regression function and its partial derivatives for strongly mixing processes.     

A three-stage procedure based on bootstrap critical points

The construction of fixed-width confidence intervals for an unknown population parameter is often based on the normal approximation of an appropriate statistic. In this paper we follow a different approach using bootstrap appro¬ximations, which are known to have a high degree of accuracy. It is expected that such an approximation leads to better results, at least in the second order behaviour.In order to avoid computationally very costly procedures,inherent to a fully sequential method,we implement the bootstrap ideas in a three-stage procedure.Simulation results illustrate the performance of the proposed procedure for small and moderate sample sizes.To support the method we establish some basic asymptotic properties.     

Modified Three-Stage Sampling for Fixed-Width Interval Estimation of the Common Variance of Equicorrelated Normal Distributions

Abstract

In this work, we present a modified three-stage sampling procedure to construct fixed-width confidence intervals of the common variance of equicorrelated normal distributions. We derive the exact distribution of the stopping variable of this sampling scheme and also the exact distribution of the estimator of the common variance at stopping. The modified three-stage sampling substantially reduces the expected sample size compared to that of the two-stage sampling scheme of Haner and Zacks (2013 Haner , D. M. and Zacks , S. ( 2013 ). On Two-Stage Sampling for Fixed-Width Interval Estimation of the Common Variance of Equi-Correlated Normal Distributions , Sequential Analysis 32 : 1 – 13 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The coverage probabilities of the proposed interval estimators are computed exactly and are compared with the coverage probabilities obtained by two-stage sampling. We derive exact formulae for the functionals of the stopping variable and the estimator of the common variance at stopping.