Maximum product of spacings prediction of future record values. (English) Zbl 1450.62121
Summary: A spacings-based prediction method for future upper record values is proposed as an alternative to maximum likelihood prediction. For an underlying family of distributions with continuous cumulative distribution functions, the general form of the predictor as a function of the estimator of the distributional parameters is established. A connection between this method and the maximum observed likelihood prediction procedure is shown. The maximum product of spacings predictor turns out to be useful to predict the next record value in contrast to likelihood-based procedures, which provide trivial predictors in this particular case. Moreover, examples are given for the exponential and the Pareto distributions, and a real data set is analyzed.
MSC:
| 62M20 | Inference from stochastic processes and prediction |
| 62G30 | Order statistics; empirical distribution functions |
| 62E15 | Exact distribution theory in statistics |
| 62P12 | Applications of statistics to environmental and related topics |
Keywords:
point prediction; cumulative hazard rate; spacings; maximum observed likelihood predictor; upper record values; exponential distribution; Pareto distributionReferences:
| [1] | Ahmadi, J.; Doostparast, M., Bayesian estimation and prediction for some life distributions based on record values, Stat Pap, 47, 3, 373-392 (2006) · Zbl 1125.62020 · doi:10.1007/s00362-006-0294-y |
| [2] | Ahsanullah, M., Linear prediction of record values for the two parameter exponential distribution, Ann Inst Stat Math, 32, 1, 363-368 (1980) · Zbl 0456.62026 · doi:10.1007/BF02480340 |
| [3] | Anatolyev, S.; Kosenok, G., An alternative to maximum likelihood based on spacings, Econ Theor, 21, 2, 472-476 (2005) · Zbl 1062.62040 · doi:10.1017/S0266466605050255 |
| [4] | Arnold, BC; Balakrishnan, N.; Nagaraja, HN, Records (1998), Hoboken: Wiley, Hoboken |
| [5] | Awad, AM; Raqab, MZ, Prediction intervals for the future record values from exponential distribution: comparative study, J Stat Comput Simul, 65, 1-4, 325-340 (2000) · Zbl 0966.62031 · doi:10.1080/00949650008812005 |
| [6] | Basak, P.; Balakrishnan, N.; Lindqvist, B.; Doksum, K., Maximum likelihood prediction of future record statistic, Chapter 11: Mathematical and statistical methods in reliability, 159-175 (2003), New York: World Scientific Publishing, New York |
| [7] | Burkschat, M., Linear estimators and predictors based on generalized order statistics from generalized Pareto distributions, Commun Stat Theory Methods, 39, 2, 311-326 (2009) · Zbl 1185.62100 · doi:10.1080/03610920902746630 |
| [8] | Chandler, KN, The distribution and frequency of record values, J R Statist Soc B, 14, 220-228 (1952) · Zbl 0047.38302 |
| [9] | Cheng, RCH; Amin, NAK, Estimating parameters in continuous univariate distributions with a shifted origin, J R Stat Soc B, 45, 3, 394-403 (1983) · Zbl 0528.62017 |
| [10] | David, HA; Nagaraja, HN, Order statistics (2003), New Jersey: Wiley, New Jersey · Zbl 1053.62060 |
| [11] | Dunsmore, IR, The future occurrence of records, Ann Inst Stat Math, 35, 1, 267-277 (1983) · Zbl 0522.62027 · doi:10.1007/BF02480982 |
| [12] | Dziubdziela, W.; Kopociński, B., Limiting properties of the k-th record values, Appl Math, 2, 15, 187-190 (1976) · Zbl 0337.60023 · doi:10.4064/am-15-2-187-190 |
| [13] | Ekström, M., On the consistency of the maximum spacing method, J Stat Plann Inf, 70, 2, 209-224 (1998) · Zbl 0941.62027 · doi:10.1016/S0378-3758(97)00185-7 |
| [14] | Ekström, M., Alternatives to maximum likelihood estimation based on spacings and the Kullback-Leibler divergence, J Stat Plann Inf, 138, 6, 1778-1791 (2008) · Zbl 1255.62049 · doi:10.1016/j.jspi.2007.06.031 |
| [15] | Gupta, RC; Kirmani, SNUA, Closure and monotonicity properties of nonhomogeneous Poisson processes and record values, Probab Eng Inf Sci, 2, 4, 475-484 (1988) · Zbl 1134.60313 · doi:10.1017/S0269964800000188 |
| [16] | Gupta, RC; Kirmani, SNUA, On predicting repair times in a minimal repair process, Commun Stat Sim Comput, 18, 4, 1359-1368 (1989) · Zbl 0695.62228 · doi:10.1080/03610918908812825 |
| [17] | Kamps, U., A concept of generalized order statistics, J Stat Plann Inf, 48, 1, 1-23 (1995) · Zbl 0838.62038 · doi:10.1016/0378-3758(94)00147-N |
| [18] | Kamps, U.; Balakrishnan, N.; Brandimarte, P.; Everitt, B.; Molenberghs, G.; Piegorsch, W.; Ruggeri, F., Generalized order statistics, Wiley StatsRef: statistics reference online, 1-12 (2016), Chichester: Wiley, Chichester |
| [19] | Madi, MT; Raqab, MZ, Bayesian prediction of temperature records using the Pareto model, Environmetrics, 15, 7, 701-710 (2004) · doi:10.1002/env.661 |
| [20] | Nadar, M.; Kızılaslan, F., Estimation and prediction of the Burr type XII distribution based on record values and inter-record times, J Stat Comput Sim, 85, 16, 3297-3321 (2015) · Zbl 1510.62128 · doi:10.1080/00949655.2014.970554 |
| [21] | Nagaraja, HN, Comparison of estimators and predictors from two-parameter exponential distribution, Sankhyā B, 48, 1, 10-18 (1986) · Zbl 0613.62039 |
| [22] | Nevzorov, VB, Records: mathematical theory (2001), Providence: American Mathematical Society, Providence |
| [23] | Paul, J.; Thomas, PY, On generalized upper (k) record values from Pareto distribution, Aligarh J Stat, 36, 63-78 (2016) · Zbl 1392.62048 |
| [24] | Ranneby, B., The maximum spacing method: an estimation method related to the maximum likelihood method, Scand J Stat, 11, 2, 93-112 (1984) · Zbl 0545.62006 |
| [25] | Raqab, MZ; Ahsanullah, M.; Raqab, MZ, Exponential distribution records: different methods of prediction, Chapter 16: recent developments in ordered random variables, 239-251 (2007), Hauppauge: Nova Science Publishers, Hauppauge · Zbl 1163.62038 |
| [26] | Raqab, MZ; Alkhalfan, LA; Bdair, OM; Balakrishnan, N., Maximum likelihood prediction of records from 3-parameter Weibull distribution and some approximations, J Comput Appl Math, 356, 118-132 (2019) · Zbl 1418.62212 · doi:10.1016/j.cam.2019.02.006 |
| [27] | Shao, Y.; Hahn, MG, Strong consistency of the maximum product of spacings estimates with applications in nonparametrics and in estimation of unimodal densities, Ann Inst Stat Math, 51, 1, 31-49 (1999) · Zbl 1049.62504 · doi:10.1023/A:1003827017345 |
| [28] | Volovskiy G (2018) Likelihood-based prediction in models of ordered data. Ph.D. thesis, RWTH Aachen University, http://publications.rwth-aachen.de/record/751672/files/751672.pdf |
| [29] | Volovskiy G, Kamps U (2020) Maximum observed likelihood prediction of future record values. TEST, To appear, 10.1007/s11749-020-00701-7 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
