Three-term asymptotic expansion: A semi-Markovian random walk with a generalized beta distributed interference of chance

A semi-Markovian random walk process (X(t)) with a generalized beta distribution of chance is considered. The asymptotic expansions for the first four moments of the ergodic distribution of the process are obtained as E(ζ_n) → ∞ when the random variable ζ_n has a generalized beta distribution with parameters (s, S, α, β); ?α, β > 1,?0? ? s < S < ∞. Finally, the accuracy of the asymptotic expansions is examined by using the Monte Carlo simulation method.     

Asymptotic Expansion and Conditional Robustness for the Sample Multiple Correlation Coefficient Under Nonnormality

ABSTRACT

Asymptotic distributions of the standardized estimators of the squared and non squared multiple correlation coefficients under nonnormality were obtained using Edgeworth expansion up to O(1/n). Conditions for the normal-theory asymptotic biases and variances to hold under nonnormality were derived with respect to the parameter values and the weighted sum of the cumulants of associated variables. The condition for the cumulants indicates a compensatory effect to yield the robust normal-theory lower-order cumulants. Simulations were performed to see the usefulness of the formulas of the asymptotic expansions using the model with the asymptotic robustness under nonnormality, which showed that the approximations by Edgeworth expansions were satisfactory.     

Asymptotic approach for a renewal-reward process with a general interference of chance

ABSTRACT

In this study, a renewal-reward process with a discrete interference of chance is constructed and considered. Under weak conditions, the ergodicity of the process X(t) is proved and exact formulas for the ergodic distribution and its moments are found. Within some assumptions for the discrete interference of chance in general form, two-term asymptotic expansions for all moments of the ergodic distribution are obtained. Additionally, kurtosis coefficient, skewness coefficient, and coefficient of variation of the ergodic distribution are computed. As a special case, a semi-Markovian inventory model of type (s, S) is investigated.     

Confidence sets based on the positive part James–Stein estimator with the asymptotically constant coverage probability

The asymptotic expansions for the coverage probability of a confidence set centred at the James–Stein estimator presented in our previous publications show that this probability depends on the non-centrality parameter τ² (the sum of the squares of the means of normal distributions). In this paper we establish how these expansions can be used for a construction of confidence region with constant confidence level, which is asymptotically (the same formula for both case τ→0 and τ→∞) equal to some fixed value 1?α. We establish the shrinkage rate for the confidence region according to the growth of the dimension p and also the value of τ for which we observe quick decreasing of the coverage probability to the nominal level 1?α. When p→∞ this value of τ increases as O(p^1/4). The accuracy of the results obtained is shown by the Monte-Carlo statistical simulations.     

Application domains for the Delta method

The Delta method uses truncated Lagrange expansions of statistics to obtain approximations to their distributions. In this paper, we consider statistics Y=g(μ+X), where X is any random vector. We obtain domains 𝒟 such that, when μ∈𝒟, we may apply the distribution derived from the Delta method. Namely, we will consider an application on the normal case to illustrate our approach.     

A random weighting approach for posterior distributions

ABSTRACT

In Bayesian theory, calculating a posterior probability distribution is highly important but typically difficult. Therefore, some methods have been proposed to deal with such problem, among which, the most popular one is the asymptotic expansions of posterior distributions. In this paper, we propose an alternative approach, named a random weighting method, for scaled posterior distributions, and give an ideal convergence rate, o(n^{( ? 1/2)}), which serves as the theoretical guarantee for methods of numerical simulations.     

Decomposition of Prediction Error in Multilevel Models

ABSTRACT

We present a decomposition of prediction error for the multilevel model in the context of predicting a future observable y _*j in the jth group of a hierarchical dataset. The multilevel prediction rule is used for prediction and the components of prediction error are estimated via a simulation study that spans the various combinations of level-1 (individual) and level-2 (group) sample sizes and different intraclass correlation values. Additionally, analytical results present the increase in predicted mean square error (PMSE) with respect to prediction error bias. The components of prediction error provide information with respect to the cost of parameter estimation versus data imputation for predicting future values in a hierarchical data set. Specifically, the cost of parameter estimation is very small compared to data imputation.     

MODELING AND ESTIMATING VARIANCES IN REGRESSION

ABSTRACT

It is well known that ignoring heteroscedasticity in regression analysis adversely affects the efficiency of estimation and renders the usual procedure for constructing prediction intervals inappropriate. In some applications, such as off-line quality control, knowledge of the variance function is also of considerable interest in its own right. Thus the modeling of variance constitutes an important part of regression analysis. A common practice in modeling variance is to assume that a certain function of the variance can be closely approximated by a function of a known parametric form. The logarithm link function is often used even if it does not fit the observed variation satisfactorily, as other alternatives may yield negative estimated variances. In this paper we propose a rich class of link functions for more flexible variance modeling which alleviates the major difficulty of negative variances. We suggest also an alternative analysis for heteroscedastic regression models that exploits the principle of “separation” discussed in Box (Signal-to-Noise Ratios, Performance Criteria and Transformation. Technometrics 1988, 30, 1–31). The proposed method does not require any distributional assumptions once an appropriate link function for modeling variance has been chosen. Unlike the analysis in Box (Signal-to-Noise Ratios, Performance Criteria and Transformation. Technometrics 1988, 30, 1–31), the estimated variances and their associated asymptotic variances are found in the original metric (although a transformation has been applied to achieve separation in a different scale), making interpretation of results considerably easier.     

The power function and an approximation for testing variance components in the presence of interaction in two-way random effects models

In this paper we consider a two-way layout random-effects model with interaction proportional to the product of two random components and derive the distributions of the sums of squares and the F-ratio under the usual normality assumption for the random effects. Approximations to these distributions by finite series Laguerre polynomial expansions are then developed. Some numerical results are given to illustrate the applications of the theory.     

Asymptotically optimal shrinkage estimates for non-normal data

Motivated by several practical issues, we consider the problem of estimating the mean of a p-variate population (not necessarily normal) with unknown finite covariance. A quadratic loss function is used. We give a number of estimators (for the mean) with their loss functions admitting expansions to the order of p ^?1/2 as p→∞. These estimators contain Stein's [Inadmissibility of the usual estimator for the mean of a multivariate normal population, in Proceedings of the Third Berkeley Symposium in Mathematical Statistics and Probability, Vol. 1, J. Neyman, ed., University of California Press, Berkeley, 1956, pp. 197–206] estimate as a particular case and also contain ‘multiple shrinkage’ estimates improving on Stein's estimate. Finally, we perform a simulation study to compare the different estimates.     

On generalized exponential transformations for proportions

ABSTRACT

Transformation of the response is a popular method to meet the usual assumptions of statistical methods based on linear models such as ANOVA and t-test. In this paper, we introduce new families of transformations for proportions or percentage data. Most of the transformations for proportions require 0 < x < 1 (where x denotes the proportion), which is often not the case in real data. The proposed families of transformations allow x = 0 and x = 1. We study the properties of the proposed transformations, as well as the performance in achieving normality and homoscedasticity. We analyze three real data sets to empirically show how the new transformation performs in meeting the usual assumptions. A simulation study is also performed to study the behavior of new families of transformations.     

Laplace approximations and Bayesian information criteria in possibly misspecified models

We provide general conditions to ensure the valid Laplace approximations to the marginal likelihoods under model misspecification, and derive the Bayesian information criteria including all terms of order O_p(1). Under conditions in theorem 1 of Lv and Liu [J. R. Statist. Soc. B, 76, (2014), 141–167] and a continuity condition for prior densities, asymptotic expansions with error terms of order o_p(1) are derived for the log-marginal likelihoods of possibly misspecified generalized linear models. We present some numerical examples to illustrate the finite sample performance of the proposed information criteria in misspecified models.     

Spatial Regression Models Using Inter-Region Distances in a Non-Random Context

This article considers spatial data z( s ₁), z( s ₂),…, z( s _n ) collected at n locations, with the objective of predicting z( s ₀) at another location. The usual method of analysis for this problem is kriging, but here we introduce a new signal-plus-noise model whose essential feature is the identification of hot spots. The signal decays in relation to distance from hot spots. We show that hot spots can be located with high accuracy and that the decay parameter can be estimated accurately. This new model compares well to kriging in simulations.     

Bayesian hypothesis testing for selected regression coefficients

ABSTRACT

We develop new Bayesian regression tests for prespecified regression coefficients. Simple, closed forms of the Bayes factors are derived that depend only on the regression t-statistic and F-statistic and the usual associated t and F distributions. The priors that allow those forms are simple and also meaningful, requiring minimal but practically important subjective inputs.     

POISSON REGRESSION WITH A PERIODIC FUNCTION

ABSTRACT

Let {y_t } be a Poisson-like process with the mean μ _t which is a periodic function of time t. We discuss how to fit this type of data set using quasi-likelihood method. Our method provides a new avenue to fit a time series data when the usual assumption of stationarity and homogeneous residual variances are invalid. We show that the estimators obtained are strongly consistent and also asymptotically normal.     

Exact inference and prediction for k-sample exponential case under type-ii censoring

The exact inference and prediction intervals for the K-sample exponential scale parameter under doubly Type-II censored samples are derived using an algorithm of Huffer and Lin [Huffer, F.W. and Lin, C.T., 2001, Computing the joint distribution of general linear combinations of spacings or exponen-tial variates. Statistica Sinica, 11, 1141–1157.]. This approach provides a simple way to determine the exact percentage points of the pivotal quantity based on the best linear unbiased estimator in order to develop exact inference for the scale parameter as well as to construct exact prediction intervals for failure times unobserved in the ith sample. Similarly, exact prediction intervals for failure times of units from a future sample can also be easily obtained.     

Asymptotic expansions for sequential confidence levels in inverse linear regression

Non-linear renewal theory is used to derive second order asymptotic expansions for the coverage probability of a fixed-width sequential confidence interval for an unknown parameter xin the inverse linear regression model. These expansions are obtained for a two-stage sequential procedure, proposed by Perng and Tong (1974) for the construction of a confidence interval for x.     

Finite-population prediction under error-in-variables superpopulation models

The prediction problem in finite populations is considered under error-in-variables super population models. The models considered are the usual regression models involving at most two variables, x and y, where both may be measured with error. Properties of some classical predictors are investigated. A Bayesian approach is proposed.     

Expansions for log densities of asymptotically normal estimates

Edgeworth–type expansions are given for the log density and also for the derivatives of the density of an asymptotically normal random variable having the standard expansions for its cumulants. Expansions for the log density are much simpler than for the density. In fact the rth term of the expansion for the log density is a polynomial of degree only r + 2, while that for the density is a polynomial of degree 3r.     

Prediction intervals for ordinary and dual generalized order statistics from two independent sequences

ABSTRACT

Based on the observed dual generalized order statistics drawn from an arbitrary unknown distribution, nonparametric two-sided prediction intervals as well as prediction upper and lower bounds for an ordinary and a dual generalized order statistic from another iid sequence with the same distribution are developed. The prediction intervals for dual generalized order statistics based on the observed ordinary generalized order statistics are also developed. The coverage probabilities of these prediction intervals are exact and free of the parent distribution, F. Finally, numerical computations and real examples of the coverage probabilities are presented for choosing the appropriate limits of the prediction.