# Bayesian approach to prediction and the spacings in the exponential distribution

## Summary

A series of independent samples are drawn from a general population with positive variation*f(x,ϕ), x>0*. Based on the Bayesian approach, a general predictive distribution is given, to predict a statistic in the future sample based on the statistics in the earlier samples (or stages). Few general classes of distributions of this type like Koopman-Pitman family, power function family and Burr's class of distributions are considered to show how this procedure works in predicting order statistics in the future sample. Also, the sum of the spacings in the future samples from an exponential population is predicted in terms of similar sum of spacings in the earlier samples. Discussion on the variance of this predictive distribution is dealt with. Finally, an illustrative example with simulated samples from an exponential population gives actual prediction of an order statistic as well as the sum of spacings in the future sample.

## Keywords

Order Statistic BAYESIAN Approach Prediction Interval Predictive Distribution Life Testing## Preview

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## References

- [1]Aitcheson, J. and Dunsmore, I. R. (1975).
*Statistical Prediction Analysis*, Cambridge University Press.Google Scholar - [2]Bancroft, G. A. and Dunsmore, I. R. (1976). Predictive distribution in life tests under competing causes of failure.
*Biometrika*,**63**, 195–217.MATHMathSciNetCrossRefGoogle Scholar - [3]Britney, R. R. and Winkler, R. L. (1974). Bayesian point estimation and prediction,
*Ann. Inst. Statist. Math.*,**26**, 15–34.MATHMathSciNetGoogle Scholar - [4]Burr, I. W. and Cislak, P. J. (1968). On a general system of distributions; I. Its curve-shape and characteristics, II. The sample median,
*J. Amer. Statist. Ass.*,**63**, 627–635.MathSciNetCrossRefGoogle Scholar - [5]Dunsmore, I. R. (1974). The Bayesian predictive distribution in life testing models,
*Technometrics*,**16**, 455–460.MATHMathSciNetCrossRefGoogle Scholar - [6]Dunsmore, I. R. (1976). Asymptotic prediction analysis,
*Biometrika*,**63**, 627–630.MATHMathSciNetCrossRefGoogle Scholar - [7]Faulkenberry, David G. (1973). A method of obtaining prediction intervals,
*J. Amer. Statist. Ass.*,**68**, 433–435.MATHCrossRefGoogle Scholar - [8]Fertig, Kenneth W. and Mann, Nancy R. (1977). One sided prediction intervals for at least
*p*out of*m*future observations from a normal population.*Technometrics*,**19**, 167–178.MATHCrossRefGoogle Scholar - [9]Hahn, Gerald J. (1977). A prediction interval on the difference between two future sample means and its application to a claim of product superiority,
*Technometrics*,**19**, 131–134.MATHCrossRefGoogle Scholar - [10]Hahn, Gerald J. (1972). Simultaneous prediction intervals to contain the standard deviation or ranges of future samples from a normal distribution,
*J. Amer. Statist. Ass.*,**67**, 938–942.MATHCrossRefGoogle Scholar - [11]Kaminsky, Kenneth S. and Nelson, Paul I. (1975). Best linear unbiased prediction of order statistics in location and scale families,
*J. Amer. Statist. Ass.*,**70**, 145–150.MATHMathSciNetCrossRefGoogle Scholar - [12]Kaminsky, Kenneth S., Luks, Eugene M. and Nelson, Paul I. (1975). An urn model and the prediction of order statistics,
*Commun. Statist.*,**4**, 245–250.MATHMathSciNetGoogle Scholar - [13]Kaminsky, Kenneth S. (1977). Comparison of prediction intervals for failure times when life is exponential,
*Technometrics*,**19**, 83–86.MATHMathSciNetCrossRefGoogle Scholar - [14]Kaminsky, Kenneth S. and Nelson, P. I. (1974) Prediction intervals for the exponential distribution using subset of the data,
*Technometrics*,**16**, 57–59.MATHMathSciNetCrossRefGoogle Scholar - [15]Khan, M. Z. (1976). Optimum allocation in Bayesian stratified two-phase sampling when there are
*m*-attributes,*Metrika*,**23**, 211–219.MATHMathSciNetCrossRefGoogle Scholar - [16]Lawless, J. F. (1971). A prediction problem concerning samples from the exponential distribution, with applications to life testing,
*Technometrics*,**13**, 725–730.MATHCrossRefGoogle Scholar - [17]Lawless, J. F. (1972). On prediction intervals for samples from the exponential distribution and prediction limits for system survivals
*Sankhyã*, B,**34**, 1–14.MathSciNetGoogle Scholar - [18]Likes, J. (1974). Prediction of sth ordered observation for two parameter exponential distribution,
*Technometrics*,**16**, 241–244.MATHMathSciNetCrossRefGoogle Scholar - [19]Likes, John (1967). Distribution of some statistics in samples from exponential and power function populations,
*J. Amer. Statist. Ass.*,**62**, 259–272.MathSciNetCrossRefGoogle Scholar - [20]Ling, K. D. (1975). On structural prediction distributions for samples from exponential distribution,
*Nanta Math.*,**7**, 47–52.CrossRefGoogle Scholar - [21]Ling, K. D. (1977). Bayesian predictive distribution for sample from double exponential distribution,
*Nanta Math.*,**10**, 13–19.MATHMathSciNetGoogle Scholar - [22]Lingappaiah, G. S. (1973). Prediction in exponential life testing,
*Canad. J. Statist.*,**1**, 113–117.MATHMathSciNetCrossRefGoogle Scholar - [23]Lingappaiah, G. S. (1974). Prediction in the samples from the Gamma distribution as applied to life testing,
*Aust. J. Statist.*,**16**, 30–32.MATHMathSciNetCrossRefGoogle Scholar - [24]Lingappaiah, G. S. (1978). On the linear combination of spacings and the restricted range in the exponential populations, (submitted).Google Scholar
- [25]Malik, Henrick John (1967). Exact moments of order statistics from a power function distribution,
*Skand. Actuarietidiskr.*,**50**, 64–69.MathSciNetGoogle Scholar