On sequential procedures for the point estimation of the mean of a normal population

The sequential procedures developed by Starr (1966, Ann. Math. Statist., 37, 1173–1185) for estimating the mean of a normal population are further analyzed. Asymptotic properties of the regret and first two moments of the stopping rules are studied and second-order approximations are derived.     

On the estimation of a restricted normal mean

Let X ∼ N(θ,1), where θ ϵ [−m, m], for some m > 0, and consider the problem of estimating θ with quadratic loss. We show that the Bayes estimator δ_m, corresponding to the uniform prior on [−m, m], dominates δ⁰ (x) = x on [−m, m] and it also dominates the MLE over a large part of the parameter interval. We further offer numerical evidence to suggest that δ_m has quite satisfactory risk performance when compared with the minimax estimators proposed by Casella and Strawderman (1981) and the estimators proposed by Bickel (1981).     

Interval estimation of the mean of a normal distribution based on quantized observations

We consider the problem of making statistical inference about the mean of a normal distribution based on a random sample of quantized (digitized) observations. This problem arises, for example, in a measurement process with errors drawn from a normal distribution and with a measurement device or process with a known resolution, such as the resolution of an analog-to-digital converter or another digital instrument. In this paper we investigate the effect of quantization on subsequent statistical inference about the true mean. If the standard deviation of the measurement error is large with respect to the resolution of the indicating measurement device, the effect of quantization (digitization) diminishes and standard statistical inference is still valid. Hence, in this paper we consider situations where the standard deviation of the measurement error is relatively small. By Monte Carlo simulations we compare small sample properties of the interval estimators of the mean based on standard approach (i.e. by ignoring the fact that the measurements have been quantized) with some recently suggested methods, including the interval estimators based on maximum likelihood approach and the fiducial approach. The paper extends the original study by Hannig et al. (2007).     

Multivariate sequential point estimation

For a multivariate normal distribution with unknown mean vector and unknown dispersion matrix, a sequential procedure for estimating the unknown mean vector is suggested. The procedure is shown to be asymptotically “risk efficient” in the sense of Starr (Ann. Math. Statist. (1966), 1173–1185), and the asymptotic order of the “regret” (see Starr and Woodroofe, Proc. Nat. Acad. Sci. 63 (1969), 285–288) is given. Moderate sample behaviour of the procedure using Monte-Carlo techniques is also studied. Finally, the asymptotic normality of the stopping time is proved.     

On bootstrap estimation of the distribution of the studentized mean

It is shown that bootstrap methods for estimating the distribution of the Studentized mean produce consistent estimators in quite general contexts, demanding not a lot more than existence of finite mean. In particular, neither the sample mean (suitably normalized) nor the Studentized mean need converge in distribution. It is unnecessary to assume that the sampling distribution is in the domain of attraction of any limit law.Now at Michigan State University     

A note on sequential estimation of the mean

A sequential procedure for estimating the population mean is proposed. The procedure is shown to have a risk less than that of the existing procedure for a certain class of distributions which depends on the skewness and the kurtosis, and includes several common distributions.     

On approximation of optimal stopping of bayesian sequential test for a normal mean

In this paper, we present a simple and direct approach in which supermartinagles are used to approximate the optimal stopping sets associated with the Bayesian sequential test for normal population means. Several conclusions are given. Project supported by the National Natural Science Foundation of China.     

On confidence sequences for the mean vector of a multivariate normal distribution

Let X₁, X₂,… be idd random vectors with a multivariate normal distribution N(μ, Σ). A sequence of subsets {R_n(a₁, a₂,…, a_n), n ≥ m} of the space of μ is said to be a (1 − α)-level sequence of confidence sets for μ if P(μ R_n(X₁, X₂,…, X_n) for every n ≥ m) ≥ 1 − α. In this note we use the ideas of Robbins Ann. Math. Statist. 41 (1970) to construct confidence sequences for the mean vector μ when Σ is either known or unknown. The constructed sequence R_n(X₁, X₂, …, X_n) depends on Mahalanobis' or Hotelling's according as Σ is known or unknown. Confidence sequences for the vector-valued parameter in the general linear model are also given.     

Purely sequential bounded-risk point estimation of the negative binomial mean under various loss functions: one-sample problem

A negative binomial (NB) distribution is useful to model over-dispersed count data arising from agriculture, health, and pest control. We design purely sequential bounded-risk methodologies to estimate an unknown NB mean \(\mu (>0)\) under different forms of loss functions including customary and modified Linex loss as well as squared error loss. We handle situations when the thatch parameter \(\tau (>0)\) may be assumed known or unknown. Our proposed methodologies are shown to satisfy properties including first-order asymptotic efficiency and first-order asymptotic risk efficiency. Summaries are provided from extensive sets of simulations showing encouraging performances of the proposed methodologies for small and moderate sample sizes. We follow with illustrations obtained by implementing estimation strategies using real data from statistical ecology: (1) weed count data of different species from a field in Netherlands and (2) count data of migrating woodlarks at the Hanko bird sanctuary in Finland.     

On a model of sequential point patterns

A finite point process motivated by the cooperative sequential adsorption model is proposed. Analytical properties of the point process are considered in details. It is shown that the introduced point process is useful for modeling both aggregated and regular point patterns. A possible scheme of maximum likelihood estimation of the process parameters is briefly discussed. V. Shcherbakov is on leave from Laboratory of Large Random Systems, Faculty of Mathematics and Mechanics, Moscow State University, Moscow.     

Methods for improvement in estimation of a normal mean matrix

This paper is concerned with the problem of estimating a matrix of means in multivariate normal distributions with an unknown covariance matrix under invariant quadratic loss. It is first shown that the modified Efron-Morris estimator is characterized as a certain empirical Bayes estimator. This estimator modifies the crude Efron-Morris estimator by adding a scalar shrinkage term. It is next shown that the idea of this modification provides a general method for improvement of estimators, which results in the further improvement on several minimax estimators. As a new method for improvement, an adaptive combination of the modified Stein and the James-Stein estimators is also proposed and is shown to be minimax. Through Monte Carlo studies of the risk behaviors, it is numerically shown that the proposed, combined estimator inherits the nice risk properties of both individual estimators and thus it has a very favorable risk behavior in a small sample case. Finally, the application to a two-way layout MANOVA model with interactions is discussed.     

On sequential density estimation

We consider the problem of sequential estimation of a density function f at a point x ₀ which may be known or unknown. Let T _n be a sequence of estimators of x ₀ . For two classes of density estimators f _n , namely the kernel estimates and a recursive modification of these, we show that if N(d) is a sequence of integer-valued random variables and n(d) a sequence of constants with N(d)/n(d) 1 in probability as d 0, then f _N(d) (T _N(d) -f(x ₀) is asymptotically normally distributed (when properly normed). We also propose two new classes of stopping rules based on the ideas of fixed-width interval estimation and show that for these rules, N(d)/n(d) 1 almost surely and EN(d)/n(d) 1 as d 0. One of the stopping rules is itself asymptotically normally distributed when properly normed and yields a confidence interval for f(x ₀) of fixed-width and prescribed coverage probability.     

Minimax estimation of a multivariate normal mean under polynomial loss

Let X be an observation from a p-variate (p ≥ 3) normal random vector with unknown mean vector θ and known covariance matrix

. The problem of improving upon the usual estimator of θ, δ⁰(X) = X, is considered. An approach is developed which can lead to improved estimators, δ, for loss functions which are polynomials in the coordinates of (δ ? θ). As an example of this approach, the loss L(δ, θ) = |δ ? θ|⁴ is considered, and estimators are developed which are significantly better than δ⁰. When

is the identity matrix, these estimators are of the form $\text{δ(X) = (1 ? (}\text{b}\text{(d + |X|}{}^2\text{)}\text{))X}$.     

Orthogonally invariant estimation of the skew-symmetric normal mean matrix

The unbiased estimator of risk of the orthogonally invariant estimator of the skew-symmetric normal mean matrix is obtained, and a class of minimax estimators and their order-preserving modification are proposed. The estimators have applications in paired comparisons model. A Monte Carlo study to compare the risks of the estimators is given.     

On the mean lattice point discrepancy of a convex disc

For a convex planar domain D \cal {D} , with smooth boundary of finite nonzero curvature, we consider the number of lattice points in the linearly dilated domain t D t \cal {D} . In particular the lattice point discrepancy P_D(t) P_{\cal {D}}(t) (number of lattice points minus area), is investigated in mean-square over short intervals. We establish an asymptotic formula for¶¶ ò_{T - L}^{T + L}(P_D(t))²dt \int\limits_{T - \Lambda}^{T + \Lambda}(P_{\cal {D}}(t))^2\textrm{d}t ,¶¶ for any L = L(T) \Lambda = \Lambda(T) growing faster than logT.     

Improved confidence sets for the mean of a multivariate normal distribution

A new class of confidence sets for the mean of a p-variate normal distribution (p3) is introduced. They are neither spheres nor ellipsoids. We show that we can construct our confidence sets so that their coverage probabilities are equal to the specified confidence coefficient. Some of them are shown to dominate the usual confidence set, a sphere centered at the observations. Numerical results are also given which show how small their volumes are.     

Fiducial inference on the largest mean of a multivariate normal distribution

Inference on the largest mean of a multivariate normal distribution is a surprisingly difficult and unexplored topic. Difficulties arise when two or more of the means are simultaneously the largest mean. Our proposed solution is based on an extension of R.A. Fisher’s fiducial inference methods termed generalized fiducial inference. We use a model selection technique along with the generalized fiducial distribution to allow for equal largest means and alleviate the overestimation that commonly occurs. Our proposed confidence intervals for the largest mean have asymptotically correct frequentist coverage and simulation results suggest that they possess promising small sample empirical properties. In addition to the theoretical calculations and simulations we also applied this approach to the air quality index of the four largest cities in the northeastern United States (Baltimore, Boston, New York, and Philadelphia).