Simultaneous Prediction Intervals Research Articles

One‐sided Simultaneous Prediction Intervals for AR(1) and MA(1) Processes with Exponential Innovations

One‐sided Simultaneous Prediction Intervals for AR(1) and MA(1) Processes with Exponential Innovations

A Comparison of Uncertainty Criteria for Calibration

Calibration consists of using a fitted regression line to estimate the value of an unobserved independent variable x corresponding to an observed dependent variable y. To construct a confidence interval for a single x, Eisenhart introduced a procedure that consists of inverting prediction intervals around the regression line. Numerous other inference procedures have been proposed for multiple-use calibration, in which a single fitted regression line is used repeatedly to estimate many x's, We provide a synthesis of this literature and offer some numerical comparisons. We also attempt to motivate the use of various criteria based on the particular points of view of the various parties involved in determining the calibration or using the results. In addition, we derive probability expressions for computing exact simultaneous prediction intervals that enable the construction of tighter limits than are currently available based on that criterion.

Simultaneous Confidence Intervals in Multiple Regression

Abstract We describe a hybrid method for computing confidence intervals for linear combinations of coefficients in multiple regression, with an emphasis on intervals for fitted values. The hybrid method combines the Bonferroni and Scheffe approaches, and it is applicable when there are both continuous and discrete predictors. It often leads to intervals that are narrower than those produced by the Scheffe method. We also describe how the method can create simultaneous prediction intervals for new observations.

Simultaneous prediction intervals for multinomial logistic regression models

Fomby and Pierce(1986) used a two step procedure to find confidence intervals for the multinomial logistic response probabilities. In the first step the distribution of the estimated parameters is derived. Then, the distribution of response probabilities is computed in the second step. We use special properties of mulitinomial logistic models to derive simultaneous confidence intervals using only the distribution of estimated parameters, which yield more accurate estimates.

Computing factors for uniform simultaneous confidence bands and prediction bands and linear regression

Bowden and Graybill (1966) tabulated factors for simultaneous uniform-width 90% and 95% confidence bands for the true regression line over a specified interval for the independent variables.Trout and Swallow (1979) recognized that these same factors could also be used to construct simultaneous prediction intervals of uniform length.This note derives a computationally convenient integral expression for the confidence level of such bands.New results for simultaneous one-sided limits are also derived.

Simultaneous prediction intervals for all patrwise differences among several future sample means

Hahn (1977) suggested a procedure for constructing prediction intervals for the difference between the means of two future samples from normal populations having equal variance, based on past samples selected from both populations. In this paper, we extend Hahn's work by constructing simultaneous prediction intervals for all pairwise differences among the means of k ≥ 2 future samples from normal populations with equal variances, using past samples taken from each of the k populations. For K = 2, this generalization reduces to Hahn's special case. These prediction intervals may be used when one has sampled the performance of several products and wishes to simultaneously as- sess the differences in future sample mean performance of these products with a predetermined overall coverage probability. The use of the new procedure is demonstrated with a numerical example.

Simultaneous prediction intervals for all distances from the “best”

Hsu (1981, 1984) derived simultaneous confidence intervals for all distances from the “best”. This article proposes simultaneous prediction intervals for all distances from the “best” for future observations under location model. The one-way ANOVA model is investigated as an illustration of the general result. Illustrative tables of critical values are given.

Multiple prediction intervals for time series: Comparison of simultaneous and marginal intervals

AbstractSimultaneous prediction intervals for forecasts from time series models that contain L (L ≤ 1) unknown future observations with a specified probability are derived. Our simultaneous intervals are based on two types of probability inequalities, i.e. the Bonferroni‐ and product‐types. These differ from the marginal intervals in that they take into account the correlation structure between the forecast errors. For the forecasting methods commonly used with seasonal time series data, we show how to construct forecast error correlations and evaluate, using an example, the simultaneous and marginal prediction intervals. For all the methods, the simultaneous intervals are accurate with the accuracy increasing with the use of higher‐order probability inequalities, whereas the marginal intervals are far too short in every case. Also, when L is greater than the seasonal period, the simultaneous intervals based on improved probability inequalities will be most accurate.

Simultaneous prediction intervals for multiple forecasts based on Bonferroni and product-type inequalities

New simultaneous prediction intervals for multiple forecasts from ARIMA models based on the Bonferroni-type and the product-type inequalities are introduced. These prediction intervals are compared with the marginal prediction intervals used in forecasting.

Simultaneous one-sided tolerance intervals for sets of contrasts from normal populations

This paper considers the problem of constructing simultaneous prediction and tolerance intervals for sets of contrasts of normal variables in situations where simultaneous intervals are available. Tables are given with critical values used in simultaneous tolerance bounds for two classes of contrasts: pairwise many-one and profile type.

Simultaneous Tolerance Intervals for Normal Populations With, Common Variance

This article proposes simultaneous tolerance limits for L normally distributed populations with a common variance, assuming that independent random samples are available from the populations. Tables of factors are provided for both one-sided limits and two-sided intervals for the case of equal sample sizes. Approximations and formulas for exact computations are discussed for applications in which the sample sizes differ. The related problem of simultaneous prediction intervals for the L populations is also discussed as a special case of the simultaneous prediction intervals proposed by Hahn (1972). Citations are provided for tables of factors for both one- and two-sided simultaneous prediction intervals.

Simultaneous Tolerance Intervals for Normal Populations with Common Variance

This article proposes simultaneous tolerance limits for L normally distributed populations with a common variance, assuming that independent random samples are available from the populations. Tables of factors are provided for both one-sided limits and two-sided intervals for the case of equal sample sizes. Approximations and formulas for exact computations are discussed for applications in which the sample sizes differ. The related problem of simultaneous prediction intervals for the L populations is also discussed as a special case of the simultaneous prediction intervals proposed by Hahn (1972). Citations are provided for tables of factors for both one- and two-sided simultaneous prediction intervals.

Simultaneous Statistical Inference Concerning the Standardized Mortality Ratios (SMB) of Several Strata in an Epidemiologic Study

AbstractIn the present paper, simultaneous prediction intervals are constructed for some mortality measures involving real data in a recent retrospective epidemiologic study of a cohort of man‐made mineral fiber workers. The object is to compare a test population with a standard population in which the workers are exposed to several levels (say K) of a suspected carcinogen and the number of deaths in the K exposure groups are recorded. The SMR is chosen as a mortality measure for this comparison. If the total number of deaths d. in the entire study is known, the number of deaths in the various strata become dependent random variables. Using univariate statistical procedures to test the significance of the individual SMR's is inappropriate and may give misleading conclusions. The present paper shows how to test the simultaneous statistical significance of the SMR's of the K exposure groups, i.e., to test the simultaneous null hypothesis Ho : E(SMR1) = E(SMR2) =…E(SMRk) = 100. If the null hypothesis Ho is rejected, it is shown how to identify the SMR (or the SMR's) which contributed to the rejection of Ho. The power function of such a test is also constructed for some simple and useful alternative hypotheses.

Simultaneous prediction intervals for regression models with an intercept

Consider the usual linear regression model y = x’β+∊, relating a response y to a vector of predictors x. Suppose that n observations on y together with the corresponding values of x are available , and it is desired to construct simultaneous prediction intervals for k future values of y at values of x which can not be ascertained beforehand. In most applications the regression model contains an intercept. This paper presents two sets of prediction intervals appropriate to this case. The proposed intervals are compared with those of Carlstein (1986), and the improvements are illustrated in the case of simple linear regression.

Simultaneous one-sided prediction intervals for a two-parameter exponential distribution using complete or type II censored data

Simultaneous prediction intervals for observations inl future samples from a two-parameter exponential distribution are considered. The prediction limits depend upon a previously available complete or type II censored sample from the same distribution. An equation is derived from which the prediction factor is determined and exact prediction limits are illustrated with a numerical example.

Simultaneous one-sided Prediction Intervals to Contain all of k Future Means from a Normal Distribution

Given the sample mean and the sample standard deviation S, computed from a random sample of size n from a N(μ, σ2) distribution, factors r = r(k, m, n; γ) are given such that the one-sided prediction interval will contain all of k out of k future means, computed from samples of size m, with (100γ)% confidence. The original n observations, and the additional k means are assumed to be independent samples from the same normal population. Values of r are given for γ = 0.90, 0.95, 0.99; n =8(2)12, 15(5)30, 40(20)100, 250, 500, 1000, ∞; k = 1(1)14(2)20, 25, 30(10)60, 80, 100; and m =1, 5(5)20.

Approximate simultaneous inferential procedures for overall risk assessment of several competing causes in biomedical and epidemiologic studies

The object of the present paper is to develop approximate simultaneous inferential procedures which facilitate overall risk assessment of several competing causes in biomedical and epidemiologic studies when the available data consist only of number of cases or deaths but not the number of individuals at risk. The paper describes how, under some reasonable assumptions, asymptotically precise simultaneous inferences may be based on proportional mortality of morbidity analyses to estimate the indirectly standardized cause-specific risk measures RSMR i involving the corresponding SPMR i and to estimate the externally standardized measures RSRR i involving the corresponding measures S ePMR i . Our paper may be divided into two parts. In the first part, approximate simultaneous Scheffé-type confidence intervals (as d..→∞) are presented for the measures RSMR i about the corresponding SPMR i when the individuals in the study are exposed to several (say M) competing risks of death. The second part of our paper presents approximate simultaneous Scheffé-type prediction intervals (as d..→∞) for the measures RSSR i involving the easily-computable externally standardized measures S ePMR i in the presence of M competing risks of death. The approximate simultaneous inferential techniques developed in the paper are valid regardless of any dependence structure among the competing risk factors. As a special case, if the risks may be assumed as independent, our simultaneous confidence and prediction intervals reduce to those for a single measure, obtained previously by the present authors (1987). The utility of the proportional mortality or morbidity measures SPMR i and S ePMR i is empirically evaluated as predictors of the corresponding cause-specific risk measures RSMR i and RSRR i using the tuberculosis mortality data of Kupper (1978) and the respiratory cancer mortality data of Enterline and Marsh (1983).

Estimation of nektonic insect populations

SUMMARY 1. A water column sampler is described for obtaining absolute population estimates of nektonic insects in shallow waters.2. Relative population estimates were obtained for Corixidae (Hemip‐tera) and Buenoa(Hemiptera: Notonectidae) using a standardized sweep net procedure and compared to absolute estimates from column samples. These two types of estimates were highly correlated among lakes.3. Simultaneous prediction and tolerance intervals for predictions from linear regressions indicated that reliable absolute population estimates generally cannot be expected from relative estimates

Approximate Simultaneous Prediction Intervals for Multiple Forecasts

A general procedure including a simple computational algorithm is presented for the construction of approximate (conservative) simultaneous prediction intervals for L ≥ 2 forecasts generated by an autoregressive integrated moving average (ARIMA) model. For models with a monotone non-increasing autocorrelation function of forecast errors (absolute value) further simplification in computations is obtained. Details are given for two important special cases, an AR(l) model and an exponential smoothing model. These two cases are also illustrated with data on hourly measures of viscosity from a chemical process.

One-Sided Intervals for at Least p of m Observations from a Normal Population on Each of r Future Occasions

In environmental monitoring, one must often use background data obtained before an installation begins operation to produce prediction intervals for future sets of m observations to be taken on each of r future occasions spanning the anticipated lifetime of the installation. A future observation falling outside of its interval will be taken as evidence of a change in the process being monitored; one would like to control the overall risk of false detection of change while maintaining an adequate sensitivity to real changes. Toward this end, the procedure of Fertig and Mann (1977) is extended to produce simultaneous prediction intervals for at least p of m observations to be obtained on each of r future occasions. Tables with superior choices of p for selected m, r, and background sample size are provided; these choices of p provide the best power given the other parameters, where power is defined in terms of the chance of quick detection of a future shift in the process being monitored.