Confidence Intervals For Functions Research Articles

Estimation and Prediction of Record Values Using Pivotal Quantities and Copulas

Recently, the area of sea ice is rapidly decreasing due to global warming, and since the Arctic sea ice has a great impact on climate change, interest in this is increasing very much all over the world. In fact, the area of sea ice reached a record low in September 2012 after satellite observations began in late 1979. In addition, in early 2018, the glacier on the northern coast of Greenland began to collapse. If we are interested in record values of sea ice area, modeling relationships of these values and predicting future record values can be a very important issue because the record values that consist of larger or smaller values than the preceding observations are very closely related to each other. The relationship between the record values can be modeled based on the pivotal quantity and canonical and drawable vine copulas, and the relationship is called a dependence structure. In addition, predictions for future record values can be solved in a very concise way based on the pivotal quantity. To accomplish that, this article proposes an approach to model the dependence structure between record values based on the canonical and drawable vine. To do this, unknown parameters of a probability distribution need to be estimated first, and the pivotal-based method is provided. In the pivotal-based estimation, a new algorithm to deal with a nuisance parameter is proposed. This method allows one to reduce computational complexity when constructing exact confidence intervals of functions with unknown parameters. This method not only reduces computational complexity when constructing exact confidence intervals of functions with unknown parameters, but is also very useful for obtaining the replicated data needed to model the dependence structure based on canonical and drawable vine. In addition, prediction methods for future record values are proposed with the pivotal quantity, and we compared them with a time series forecasting method in real data analysis. The validity of the proposed methods was examined through Monte Carlo simulations and analysis for Arctic sea ice data.

Confidence intervals for functions of coefficients of variation with bounded parameter spaces in two gamma distributions

Confidence intervals for functions of coefficients of variation with bounded parameter spaces in two gamma distributions

Functional time series analysis of spatio–temporal epidemiological data

Spatio–temporal statistical models have been proposed for the analysis of the temporal evolution of the geographical pattern of mortality (or incidence) risks in disease mapping. However, as far as we know, functional approaches based on Hilbert-valued processes have not been used so far in this area. In this paper, the autoregressive Hilbertian process framework is adopted to estimate the functional temporal evolution of mortality relative risk maps. Specifically, the penalized functional estimation of log-relative risk maps is considered to smooth the classical standardized mortality ratio. The reproducing kernel Hilbert space (RKHS) norm is selected for definition of the penalty term. This RKHS-based approach is combined with the Kalman filtering algorithm for the spatio–temporal estimation of risk. Functional confidence intervals are also derived for detecting high risk areas. The proposed methodology is illustrated analyzing breast cancer mortality data in the provinces of Spain during the period 1975–2005. A simulation study is performed to compare the ARH(1) based estimation with the classical spatio–temporal conditional autoregressive approach.

Simulation-Based Confidence Intervals for Functions With Complicated Derivatives

In many scientific problems, the quantity of interest is a function of parameters that index the model, and confidence intervals are constructed by applying the delta method. However, when the function of interest has complicated derivatives, this standard approach is unattractive and alternative algorithms are required. This article discusses a simple simulation-based algorithm for estimating the variance of a transformation, and demonstrates its simplicity and accuracy by applying it to several statistical problems.

Estimation of Varying Coefficients for a Growth Curve Model

SYNOPTIC ABSTRACTIn this paper, a new approach of modelling growth curves is developed which uses time-varying coefficients. Since the mean structure of the growth curve model has many unknown parameters depending on both covariates and time trend designs, it can be difficult to understand and interpret estimated parameters. Using varying coefficient functions, the effects of covariates can be evaluated and visualized more easily. The asymptotic functional confidence intervals are derived theoretically and a procedure is proposed to test whether the effects of covariates are significant.

A Simple General Method for Constructing Confidence Intervals for Functions of Variance Components

A simple general method is proposed for constructing confidence intervals for arbitrary functions of variance components in balanced normal-theory models. The concept of “surrogate variables” is introduced as part of the description of the method. “Equal-tail” and “shortest-length” confidence intervals from this method can be computed easily using simulations. As an illustration, the two-way random-effects model is considered. It is shown that the proposed method produces intervals that are comparable in confidence coefficient and average length to those produced by the best existing methods.

On confidence intervals for nonmonotone parametric functions and an application to the squared mean of the normal distribution

Consider the problem of obtaining a confidence interval for some function g(θ) of an unknown parameter θ, for which a (1-α)-confidence interval is given. If g(θ) is one-to-one the solution is immediate. However, if g is not one-to-one the problem is more complex and depends on the structure of g. In this note the situation where g is a nonmonotone convex function is considered. Based on some inequality, a confidence interval for g(θ) with confidence level at least 1-α is obtained from the given (1-α) confidence interval on θ. Such a result is then applied to the n(μ, σ2) distribution with σ known. It is shown that the coverage probability of the resulting confidence interval, while being greater than 1-α, has in addition an upper bound which does not exceed Θ(3z1−α/2)-α/2.

Comparison of confidence intervals for variance components with unbalanced data

We compare the performance of three easily-computed confidence intervals for functions of variance components in the one-way random effect model. The confidence intervals are studied for three underlying generating distributions and a variety of sample sizes and degrees of unbalancedness. Confidence intervals based on the asymptotic covariance matrix for maximum likelihood estimates of the parameters perform poorly for small sample sizes for any function of the between variance component, regardless of the underlying distribution of the random effects.

Transformations for Improving Linearization Confidence Intervals in Nonlinear Regression

Abstract We investigate linear approximation (LA) confidence intervals for functions g(θ) of the parameters θ in a nonlinear regression model. These intervals are almost universally used and generally perform well, but at times have poor coverage probabilities. Using gradient direction plots, we identify transformations of g(θ) that lead to more accurate LA intervals. These include power transformations, whose effectiveness is demonstrated in a variety of nonlinear regression problems via a simulation study. Finally, we show how to use profile t plots and bias indices to suggest transforms to improve LA intervals. The idea is to find a monotone transformation T such that the linearization confidence interval for T(g(θ)) has coverage probability close to its nominal value and then invert this interval to give an accurate interval for g(θ). The transformation T is obtained from a gradient direction plot that may be thought of as an attempt to view the graph of the estimator ĝ of g(θ) in two dimensions. We d...

Diagnostics for Linearization Confidence Intervals in Nonlinear Regression

Abstract We investigate linear approximation (LA) confidence intervals for functions g(θ) of the parameters θ in a nonlinear regression model. These intervals are almost universally used and generally perform well, but at times they have poor coverage probabilities. A diagnostic plot and index are developed to detect these failures. We show how these diagnostics may be used to estimate coverage probabilities and these are used to calibrate the diagnostics. The performance of the coverage probability estimates in a variety of nonlinear regression problems is investigated via simulation; for these problems, they work quite well. Conditions are identified under which the estimates are exact. Finally, we discuss the use of the profile t plot and asymmetry and bias indices as diagnostics for LA intervals and show how to calibrate them in terms of coverage probabilities.

Diagnostics for Linearization Confidence Intervals in Nonlinear Regression

Abstract We investigate linear approximation (LA) confidence intervals for functions g(θ) of the parameters θ in a nonlinear regression model. These intervals are almost universally used and generally perform well, but at times they have poor coverage probabilities. A diagnostic plot and index are developed to detect these failures. We show how these diagnostics may be used to estimate coverage probabilities and these are used to calibrate the diagnostics. The performance of the coverage probability estimates in a variety of nonlinear regression problems is investigated via simulation; for these problems, they work quite well. Conditions are identified under which the estimates are exact. Finally, we discuss the use of the profile t plot and asymmetry and bias indices as diagnostics for LA intervals and show how to calibrate them in terms of coverage probabilities.

Comparison theorems and asymptotic behavior of correlation estimators in spaces of continuous functions. II

This paper is the second part of [12]. Using the comparison theorems which were proved in the first part, the asymptotic normality of the estimator — in a model of a series of several samples — of the correlation function of a stationary Gaussian random process in spaces of continuous functions with weights is established. A method for constructing functional confidence intervals for an unknown correlation function in these spaces is described.

Likelihood-Based Confidence Intervals for Functions of Many Parameters

Likelihood-Based Confidence Intervals for Functions of Many Parameters

Simultaneous Confidence Intervals for the Linear Functions of Expected Mean Squares Used in Generalizability Theory

This paper demonstrates a method, derived by Khuri (1981) , of constructing simultaneous confidence intervals for functions of expected values of mean squares obtained when analyzing a balanced design by a random effects linear model. The method may be applied to obtain confidence intervals for the variance components and other linear functions of the expected mean squares used in generalizability theory, with probability of simultaneous coverage guaranteed to be greater than or equal to the specified confidence coefficient. The Khuri intervals are compared with the approximate intervals obtained by using Satterthwaite’s (1941 , 1946) method in conjunction with Bonferroni’s inequality.