Probability Distribution Modelling of Scanner Prices and Relative Prices Using Theoretical Distributions with Two, Three, Four, and Five Parameters
DOI:
https://doi.org/10.18778/0208-6018.366.02Keywords:
data modeling, scanner data, price distributionsAbstract
This article addresses the problem of proper adjustment of the theoretical probability distribution to the empirical distribution of scanner prices. In the empirical study, we use scanner data from one retail chain in Poland, i.e., monthly data on natural yogurt, yogurt drinks, long grain rice and coffee powder sold in 212 outlets in January and February 2022. Prices and relative prices are modelled using fifty two‑, three‑, four‑, and five‑parameter probability distributions with non‑negative support. Some of them consist of somewhat known distributions which are called their special cases. The study indirectly involves over a hundred of these distributions. Information criteria such as AIC, BIC, HQIC and p‑values of goodness‑of‑fit tests are used for comparative analysis. This article shows that models such as Frechet, Pareto IV and Log‑Logistic could be distinguished as very accurate, which provides a good background for simulation research on price indices or for the construction of the so‑called population price indices. The Appendix presents the cumulative distribution function formulas of the models used and the necessary R codes for conducting the research.
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References
Abdollahi Nanvapisheh A. (2019), A New Five Parameter Distribution: Properties and Applications, “International Journal of Mathematical Modelling and Computations”, vol. 9(3), pp. 201–212.
Akaike H. (1974), A new look at the statistical model identification, “IEEE Transactions on Automatic Control”, vol. 19(6), pp. 716–723. DOI: https://doi.org/10.1109/TAC.1974.1100705
Al Babtain A., Eid A.M., Ahmed A.H.N., Merovci F. (2015), The five parameter Lindley distribution, “Pakistan Journal of Statistics”, vol. 31(4). DOI: https://doi.org/10.1186/2193-1801-4-2
Awodutire P. (2020), Chen Pareto Distribution: Properties and Application, “Pakistan Journal of Statistics and Operation Research”, vol. 16(4), pp. 812–826. DOI: https://doi.org/10.18187/pjsor.v16i4.3418
Bakouch H.S., Saboor A., Khan M.N. (2021), Modified beta linear exponential distribution with hydrologic applications, “Annals of Data Science”, no. 8, pp. 131–157. DOI: https://doi.org/10.1007/s40745-019-00222-7
Barreto Souza W., Morais A.L. de, Cordeiro G.M. (2011), The Weibull geometric distribution, “Journal of Statistical Computation and Simulation”, vol. 81(5), pp. 645–657. DOI: https://doi.org/10.1080/00949650903436554
Barreto Souza W., Santos A.H., Cordeiro G.M. (2010), The beta generalized exponential distribution, “Journal of Statistical Computation and Simulation”, vol. 80(2), pp. 159–172. DOI: https://doi.org/10.1080/00949650802552402
Bebbington M., Lai C.D., Zitikis R. (2007), A flexible Weibull extension, “Reliability Engineering and System Safety”, vol. 92(6), pp. 719–726. DOI: https://doi.org/10.1016/j.ress.2006.03.004
Bemmaor A.C. (1994), Modeling the diffusion of new durable goods: Word of mouth effect versus consumer heterogeneity, [in:] G. Laurent, G.L. Lilien, B. Pras (eds.), Research Traditions in Marketing, Kluwer, Boston, pp. 201–229. DOI: https://doi.org/10.1007/978-94-011-1402-8_6
Białek J. (2015), Construction of confidence intervals for the Laspeyres price index, “Journal of Statistical Computation and Simulation”, vol. 85(14), pp. 2962–2973. DOI: https://doi.org/10.1080/00949655.2014.946416
Białek J. (2022), Elementary price indices under the GBM price model, “Communications in Statistics – Theory and Methods”, vol. 51(5), pp. 1232–1251. DOI: https://doi.org/10.1080/03610926.2021.1938127
Białek J., Beręsewicz M. (2021), Scanner data in inflation measurement: from raw data to price indices, “The Statistical Journal of the IAOS”, no. 37, pp. 1315–1336. DOI: https://doi.org/10.3233/SJI-210816
Białek J., Bobel A. (2019), Comparison of price index methods for CPI measurement using scanner data, 16th Meeting of the Ottawa Group on Price Indices, Rio de Janeiro.
Birnbaum Z.W., Saunders S.C. (1969), A new family of life distributions, “Journal of Applied Probability”, vol. 6(2), pp. 637–652. DOI: https://doi.org/10.2307/3212003
Bourguignon M., Lima M.D.C.S., Leão J., Nascimento A.D., Pinho L.G.B., Cordeiro G.M. (2015), A new generalized gamma distribution with applications, “American Journal of Mathematical and Management Sciences”, vol. 34(4), pp. 309–342. DOI: https://doi.org/10.1080/01966324.2015.1040178
Brandt S. (2014), Data analysis, Springer International Publishing, Switzerland.
Brazauskas V. (2003), Information matrix for Pareto (IV), Burr, and related distributions, “Communications in Statistics Theory and Methods”, vol. 32(2), pp. 315–325. DOI: https://doi.org/10.1081/STA-120018188
Carli G. (1804), Del valore e della proporzione de’metalli monetati, “Scrittori Classici Italiani di Economia Politica”, no. 13, pp. 297–336.
Carrasco J.M., Ortega E.M., Cordeiro G.M. (2008), A generalized modified Weibull distribution for lifetime modelling, “Computational Statistics and Data Analysis”, vol. 53(2), pp. 450–462. DOI: https://doi.org/10.1016/j.csda.2008.08.023
Castillo E., Hadi A.S., Balakrishnan N., Sarabia J.S. (2005), Extreme Value and Related Models with Applications in Engineering and Science, Wiley Interscience, Hoboken.
Chen Z. (2000), A new two parameter lifetime distribution with bathtub shape or increasing failure rate function, “Statistics and Probability Letters”, no. 49, pp. 155–161. DOI: https://doi.org/10.1016/S0167-7152(00)00044-4
Chesneau C., Bakouch H.S., Hussain T. (2018), A new class of probability distributions via cosine and sine functions with applications, “Communications in Statistics Simulation and Computation”, vol. 48(8), pp. 2287–2300. DOI: https://doi.org/10.1080/03610918.2018.1440303
Chhikara R.S., Folks J.L. (1989), The Inverse Gaussian Distribution: Theory, Methodology and Applications, Marcel Dekker, New York.
Cooray K. (2006), Generalization of the Weibull distribution: The odd Weibull family, “Statistical Modelling”, vol. 6(3), pp. 265–277. DOI: https://doi.org/10.1191/1471082X06st116oa
Cordeiro G.M., Ortega E.M., Silva G.O. (2011), The exponentiated generalized gamma distribution with application to lifetime data, “Journal of Statistical Computation and Simulation”, vol. 81(7), pp. 827–842. DOI: https://doi.org/10.1080/00949650903517874
Cordeiro G.M., Castellares F., Montenegro L.C., Castro M. de (2013), The beta generalized gamma distribution, “Statistics”, vol. 47(4), pp. 888–900. DOI: https://doi.org/10.1080/02331888.2012.658397
Drapella A. (1993), The complementary Weibull distribution: unknown or just forgotten?, “Quality and Reliability Engineering International”, vol. 9(4), pp. 383–385. DOI: https://doi.org/10.1002/qre.4680090426
Dutot C.F. (1738), Reflexions Politiques sur les Finances et le Commerce, Les Freres, The Hague.
El Gohary A., Alshamrani A., Al Otaibi A.N. (2013), The generalized Gompertz distribution, “Applied Mathematical Modelling”, vol. 37(1–2), pp. 13–24. DOI: https://doi.org/10.1016/j.apm.2011.05.017
El Gohary A., El Bassiouny A.H., El Morshedy M. (2015), Inverse flexible Weibull extension distribution, “International Journal of Computer Applications”, vol. 115(2), pp. 46–51. DOI: https://doi.org/10.5120/20127-2211
Eltehiwy M., Ashour S. (2013), Transmuted Exponentiated Modified Weibull Distribution, “International Journal of Basic and Applied Sciences”, vol. 2(3), pp. 258–269. DOI: https://doi.org/10.14419/ijbas.v2i3.1074
Felipe R.Sd.G., Edwin M.M.O, Gauss M.C. (2009), The generalized inverse Weibull distribution, “Statistical Papers”, vol. 52(3), pp. 591–619. DOI: https://doi.org/10.1007/s00362-009-0271-3
Gaddum J.H. (1945), Lognormal distributions, “Nature”, vol. 156(3964), pp. 463–466. DOI: https://doi.org/10.1038/156463a0
Ghitany M.E., Al Hussaini E.K., Al Jarallah R.A. (2005), Marshall–Olkin extended Weibull distribution and its application to censored data, “Journal of Applied Statistics”, vol. 32(10), pp. 1025–1034. DOI: https://doi.org/10.1080/02664760500165008
Ghitany M.E., Al Mutairi D.K., Balakrishnan N., Al Enezi L.J. (2013), Power Lindley distribution and associated inference, “Computational Statistics and Data Analysis”, no. 64, pp. 20–33. DOI: https://doi.org/10.1016/j.csda.2013.02.026
Gumbel E.J. (1958), Statistics of Extremes, Columbia University Press, New York. DOI: https://doi.org/10.7312/gumb92958
Hannan E.J., Quinn B.G. (1979), The determination of the order of an autoregression, “Journal of the Royal Statistical Society: Series B (Methodological)”, vol. 41(2), pp. 190–195. DOI: https://doi.org/10.1111/j.2517-6161.1979.tb01072.x
Javed M., Nawaz T., Irfan M. (2019), The Marshall Olkin kappa distribution: properties and applications, “Journal of King Saud University Science”, vol. 31(4), pp. 684–691. DOI: https://doi.org/10.1016/j.jksus.2018.01.001
Jevons W.S. (1865), The variation of prices and the value of the currency since 1782, “Journal of the Statistical Society of London”, no. 28, pp. 294–320. DOI: https://doi.org/10.2307/2338419
Jędrzejczak A., Pekasiewicz D. (2020), Teoretyczne rozkłady dochodów gospodarstw domowych i ich estymacja, Wydawnictwo Uniwersytetu Łódzkiego, Łódź. DOI: https://doi.org/10.18778/8142-899-6
Johnson N.L., Kotz S., Balakrishnan N. (1995), Continuous univariate distributions, vol. 2, John Wiley & Sons, New York.
Kleiber C., Kotz S. (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley Interscience, Hoboken. DOI: https://doi.org/10.1002/0471457175
Kotz S., Nadarajah S. (2000), Extreme Value Distributions: Theory and Applications, Imperial College Press, London. DOI: https://doi.org/10.1142/9781860944024
Lindsey J.K. (2004), Statistical analysis of stochastic processes in time, vol. 14, Cambridge University Press, Cambridge. DOI: https://doi.org/10.1017/CBO9780511617164
Lu W., Shi D. (2012), A new compounding life distribution: the Weibull–Poisson distribution, “Journal of Applied Statistics”, vol. 39(1), pp. 21–38. DOI: https://doi.org/10.1080/02664763.2011.575126
Mahdavi A. (2015), Two Weighted Distributions Generated by Exponential Distribution, “Journal of Mathematical Extension”, vol. 9(1), pp. 1–12.
McDonald J.B. (1984), Some generalized functions for the size distribution of income, “Econometrica”, vol. 52(3), pp. 647–663. DOI: https://doi.org/10.2307/1913469
Nadarajah S., Rocha R. (2016), Newdistns: An R package for new families of distributions, “Journal of Statistical Software”, no. 69, pp. 1–32. DOI: https://doi.org/10.18637/jss.v069.i10
Nakagami M. (1960), The m Distribution – A General Formula of Intensity Distribution of Rapid Fading, [in:] W.C. Hoffman (ed.), Statistical Methods in Radio Wave Propagation, Pergamon, Oxford, pp. 3–36. DOI: https://doi.org/10.1016/B978-0-08-009306-2.50005-4
Okasha H.M., El Baz A.H., Tarabia A.M.K., Basheer A.M. (2017), Extended inverse Weibull distribution with reliability application, “Journal of the Egyptian Mathematical Society”, vol. 25(3), pp. 343–349. DOI: https://doi.org/10.1016/j.joems.2017.02.006
Pal M., Tiensuwan M. (2015), Exponentiated transmuted modified Weibull distribution, “European Journal of Pure and Applied Mathematics”, vol. 8(1), pp. 1–14.
R Core Team (2021), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, https://www.R project.org/ [accessed: 24.11.2023].
Sarhan A.M., Apaloo J. (2013), Exponentiated modified Weibull extension distribution, “Reliability Engineering and System Safety”, no. 112, pp. 137–144. DOI: https://doi.org/10.1016/j.ress.2012.10.013
Sarhan A.M., Zaindin M. (2009), Modified Weibull distribution, “APPS. Applied Sciences”, no. 11, pp. 123–136.
Schwarz G. (1978), Estimating the dimension of a model, “The Annals of Statistics”, vol. 6(2), pp. 461–464. DOI: https://doi.org/10.1214/aos/1176344136
Shahbaz M.Q., Shahbaz S., Butt N.S. (2012), The Kumaraswamy–Inverse Weibull Distribution, “Pakistan Journal of Statistics and Operation Research”, vol. 8(3), pp. 479–489. DOI: https://doi.org/10.18187/pjsor.v8i3.520
Shanker S., Shukla K.K. (2019), A generalization of Generalized Gamma distribution, “International Journal of Computational and Theoretical Statistics”, vol. 6(1), pp. 33–42.
Silver H., Heravi S. (2007), Why elementary price index number formulas differ: Evidence on price dispersion, “Journal of Econometrics”, no. 140, pp. 874–883. DOI: https://doi.org/10.1016/j.jeconom.2006.07.017
Stacy E.W., Mihram G.A. (1965), Parameter estimation for a generalized gamma distribution, “Technometrics”, vol. 7(3), pp. 349–358. DOI: https://doi.org/10.1080/00401706.1965.10490268
Subhradev S., Mustafa C.K., Haitham M.Y. (2018), The Quasi XGamma Poisson distribution: Properties and Application, “Istatistik: Journal of the Turkish Statistical Assocation”, vol. 11(3), pp. 65–76.
Sulewski P., Białek J. (2022), Probability Distribution Modelling of Scanner Prices and Relative Prices, “Statistika: Statistics & Economy Journal”, vol. 102(3). DOI: https://doi.org/10.54694/stat.2022.14
Tieling Z., Min X. (2007), Failure Data Analysis with Extended Weibull Distribution, “Communications in Statistics – Simulation and Computation”, no. 36, pp. 579–592. DOI: https://doi.org/10.1080/03610910701236081
Witkovsky V. (2001), Computing the distribution of a linear combination of inverted gamma variables, “Kybernetika”, vol. 37(1), pp. 79–90.
Yusuf A., Qureshi S. (2019), A five parameter statistical distribution with application to real data, “Journal of Statistics Applications and Probability Letters”, no. 8, pp. 11–26. DOI: https://doi.org/10.18576/jsap/080102
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