Record ranked set sampling scheme
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Record ranked set sampling scheme Mahdi Salehi · Jafar Ahmadi
Received: 1 March 2013 / Accepted: 17 March 2014 © Sapienza Università di Roma 2014
Abstract A new sampling scheme for generating record-breaking data is introduced and called record ranked set sampling (RRSS). A distribution-free two-sided prediction interval for future order statistics based on RRSS is derived. Numerical computations are obtained for comparing the results with the case based on ordinary records. Also, confidence intervals for quantiles of the parent distribution and prediction intervals for the ordinary records are given. Keywords
Coverage probability · Distribution-free · Prediction interval · Record data
Mathematics Subject Classification (2010)
62G30
1 Introduction Let X 1 , X 2 , . . . be an infinite sequence of continuous random variables and Z n = max{X 1 , . . . , X n } for n ≥ 1; then an observation X j is called an upper record value if Z j > Z j−1 , j > 1. Lower record values are defined analogously. As a convention, let the 1th upper and lower record be taken as L 1 = U1 ≡ X 1 , and denote the nth ordinary upper and lower record by Un and L n , respectively (for n ≥ 1). These type of data arise in a wide variety of practical situations such as industrial stress testing, meteorology, hydrology, sports, and stock market analysis. Interested readers may refer to the book by Arnold et al. [4] and the references contained therein. Independently of X -sequence, suppose that Y1:n , Y2:n , . . . , Yn:n are the order statistics from future Y -sample from the same distribution. For more details concerning on the theory and application of order statistics see for example [5,11]. Oneand two-sample prediction problems applying the data involving record values and order M. Salehi · J. Ahmadi (B) Department of Statistics, Ordered and Spatial Data Center of Excellence, Ferdowsi University of Mashhad, P.O. Box 1159, 91775 Mashhad, Iran e-mail: [email protected] M. Salehi e-mail: [email protected]
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M. Salehi, J. Ahmadi
statistics have discussed by several authors. Our main interest in this context is predicting of record values and order statistics. Among the various authors who worked on this field, one may refer to, [13] who considered the prediction of order statistics in one-sample as well as two-sample cases, and obtained linear point predictors and prediction intervals based on samples from location-scale families. Ahmadi and Doostparast [2] obtained a Bayesian estimation and a prediction for some life distributions based on record values. Ahmadi and Balakrishnan [3] derived a prediction interval for order statistics of a future sample based on upper record statistics. But, in some situations, the only observations that is stored are the last record-breaking data. For example, in athletic data, weather data and Olympic data. So, it is worthwhile to develop the distribution theory for using such a data in order to make an inference or prediction. We intend to introduce a new sampling scheme for generating record-breaking.
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