Record values from NH distribution and associated inference
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Record values from NH distribution and associated inference S. M. T. K. MirMostafaee1 · A. Asgharzadeh1 · A. Fallah2
Received: 4 September 2014 / Accepted: 20 June 2015 © Sapienza Università di Roma 2015
Abstract A new extension of exponential distribution, called NH distribution, was recently introduced by Nadarajah and Haghighi (Statistics 45:543–558, 2011). In this paper, we consider the upper record values from this distribution. We obtain exact explicit expressions as well as several recurrence relations for the single and product moments of record values and then we use these results to compute the means, variances and the covariances of the upper record values. We make use of these calculated moments to find the best linear unbiased estimators (BLUEs) of the location and scale parameters of NH distribution. In addition, based on the observed records, we investigate how to obtain best linear unbiased predictor for the future record values. Confidence intervals for the unknown parameters and prediction intervals for future records are also discussed. Finally, we present an example in order to illustrate the inferential results obtained in this paper and compare the BLUEs with maximum likelihood estimators numerically. Keywords Best linear unbiased estimator · Best linear unbiased predictor · Incomplete gamma function · NH distribution · Record values · Recurrence relations
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S. M. T. K. MirMostafaee [email protected] A. Asgharzadeh [email protected] A. Fallah [email protected]
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Department of Statistics, Faculty of Mathematical Sciences, University of Mazandaran, P. O. Box 47416-1467, Babolsar, Iran
2
Department of Statistics, Faculty of Mathematics and Statistics, Payame Noor University, P. O. Box 19395-3697, Tehran, Iran
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S. M. T. K. MirMostafaee et al.
1 Introduction Suppose that {X i ; i ≥ 1} is a sequence of independent and identically distributed (iid) random variables from an absolutely continuous distribution function (cdf) F(x) and probability density function (pdf) f (x). Suppose further, Tk∗ = max{X 1 , . . . , X k } for k ≥ 1, then X j ∗ is called an upper record value of this sequence if T j∗ > T j−1 for j ≥ 2. By definition, X 1 is also an upper record value which is called the trivial (first) record. If the sequence {U (n); n ≥ 1} is defined by U (1) = 1, U (n) = min{ j : j > U (n − 1), X j > X U (n) } for n ≥ 2, then the sequence {Rn = X U (n) ; n ≥ 1} is called the upper record values. The sequence {U (n); n ≥ 1} is called the upper record times. Lower record values and lower record times can be defined similarly. It is well known that the pdf of the s-th upper record value Rs is given by f Rs (x) =
s−1 ¯ [− log F(x)] f (x), (s − 1)!
(1)
¯ where F(·) = 1 − F(·). Moreover, the joint pdf of the s-th and n-th (s < n) upper record values is given by f Rs ,Rn (x, y) =
s−1 [− log F(y) n−s−1 ¯ ¯ ¯ [− log F(x)] + log F(x)] f (x) f (y), x < y, (2) ¯ (n − s − 1)! (s − 1)! F(x)
see for example [1]. Many scientists specially the statisticians have become interested i
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