On Expected Lengths of Predictive Intervals (original) (raw)

On prediction interval for independent observations

Applied Mathematical Sciences, 2015

We consider the problem of prediction interval for future observations in the case of normal random variables. Specifically, two prediction intervals are developed with the difference lies on parameter estimators. The resulting estimative prediction intervals have coverage probability as well as expected length bounded to O(n −1). Furthermore, whilst the coverage probability 1 − α is usually given, we propose to set this as parameter. Consequently, the O(n −1) terms in the asymptotic expansion of coverage probability and of expected length will depend not only on model parameter but also α. Our aim is to find an optimal coverage probability. Numerical analysis is carried out to illustrate the unexpected coverage probability and length of estimative prediction interval.

Prediction intervals for future records

Statistics & Probability Letters, 2008

Suppose upper records were observed from a X-sequence of iid continuous random variables, and that another independent Ysequence of iid variables from the same distribution is to be observed. In this paper, we then derive various exact distribution-free prediction intervals for records from the Y-sequence based on the record values from the X-sequence. Specifically, distribution-free prediction intervals for individual records as well as outer and inner prediction intervals are derived based on X-records, and exact expressions for the coverage probabilities of these intervals are also derived. A data representing the records of temperatures is used to illustrate all the results developed here.

Outer and inner prediction intervals for order statistics intervals based on current records

Statistical Papers, 2012

This paper considers the largest and smallest observations at the times when a new record of either kind (upper or lower) occurs. These are called the upper and lower current records and are denoted by R l m and R s m , respectively. The interval (R s m , R l m ) is then referred to as the record coverage. The prediction problem in the two-sample case is then discussed and, specifically, the exact outer and inner prediction intervals are derived for order statistics intervals from an independent future Y -sample based on the m-th record coverage from the X -sequence when the underlying distribution of the two samples are the same. The coverage probabilities of these intervals are exact and do not depend on the underlying distribution. Distribution-free prediction intervals as well as upper and lower prediction limits for spacings from a future Y -sample are obtained in terms of the record range from the X -sequence.

On the Comparison of Interval Forecasts

Social Science Research Network, 2018

We explore interval forecast comparison when the nominal confidence level is specified, but the quantiles on which intervals are based are not specified. It turns out that the problem is difficult, and perhaps unsolvable. We first consider a situation where intervals meet the Christoffersen conditions (in particular, where they are correctly calibrated), in which case the common prescription, which we rationalize and explore, is to prefer the interval of shortest length. We then allow for mis-calibrated intervals, in which case there is a calibration-length tradeoff. We propose two natural conditions that interval forecast loss functions should meet in such environments, and we show that a variety of popular approaches to interval forecast comparison fail them. Our negative results strengthen the case for abandoning interval forecasts in favor of density forecasts: Density forecasts not only provide richer information, but also can be readily compared using known proper scoring rules like the log predictive score, whereas interval forecasts cannot.

On sieve bootstrap prediction intervals

Statistics & Probability Letters, 2003

In this paper we consider a sieve bootstrap method for constructing nonparametric prediction intervals for a general class of linear processes. We show that the sieve bootstrap provides consistent estimators of the conditional distribution of future values given the observed data.

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

Journal of Forecasting, 1991

Simultaneous prediction intervals for forecasts from time series models that contain L ( L 2 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.

Confidence, prediction and tolerance intervals in classical and Bayesian settings

2019

In good statistical practice it is recommended to limit the results not only to point estimates, but to present simultaneously interval estimates. The standard, well known and frequently taught interval in introductory statistical classes, is the condence interval. It is designed to describe a single unknown parameter of a population with some uncertainty. Another type of interval, the prediction interval, is occasionally presented in the context of regression analysis. The aim of the prediction interval is to predict future observation(s) in a population with a dened uncertainty. In contrast to these two intervals, tolerance intervals, capturing a specied proportion P of a population with some uncertainty, are rarely presented. Depending on the context and on the purpose of a study, dierent intervals are recommended. Presenting a condence or a prediction interval, when a tolerance interval would be appropriate, engenders potentially a misuse of the former two intervals. In this thesis the three dierent types of intervals are illustrated with simulated data and two real data sets from veterinary medicine. One data set of normally distributed data originates from a study on body weight loss in obese dogs. The second data set of binomial data comes from a study simulating fractures caused by horse kicks and dierent shoeing materials. The intervals for a normal and a binomial distribution are interpreted from both a classical and a Bayesian perspective. Next to theoretical aspects, practical applications in R are presented. The asymptotic behaviour of the three intervals, subject to increasing sample size, is assessed. With increasing sample size, the width of condence intervals decreases and approaches 0. The widths of the prediction and the tolerance intervals decrease initially, but will reach a stable plateau. The magnitude of the width of these two intervals will be inuenced by the standard deviation and, additionally for the tolerance intervals, the captured proportion P. It became also evident, that the assumption of normality needs to be checked carefully to avoid erroneous intervals. For binomial distributions, Bayesian approaches were found to be superior to Wald normal approximations which should be avoided. As the three types of intervals serve for dierent purposes, the decision which interval to use should be context-driven and clearly justied. In order to avoid confusion and misunderstandings, all three types of intervals should be taught and presented in introductory statistical classes.

Prediction Intervals for k-records in Terms of Current Records

Journal of Statistical Theory and Applications, 2013

In this paper, we discuss the prediction of k-records from the future sequence based on observed current records of either kind upper or lower, record coverage and k-records from the same distribution. It is shown that the coverage probability of all proposed prediction intervals are distribution-free. Exact and explicit expressions for the prediction coefficient of these intervals are obtained. The existence and optimality of these intervals are discussed. At the end, a numerical example is given for illustrating and comparing the proposed procedure.