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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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.
Google Scholar
Akaike H. (1974), A new look at the statistical model identification, “IEEE Transactions on Automatic Control”, vol. 19(6), pp. 716–723.
Google Scholar
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).
Google Scholar
Awodutire P. (2020), Chen Pareto Distribution: Properties and Application, “Pakistan Journal of Statistics and Operation Research”, vol. 16(4), pp. 812–826.
Google Scholar
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.
Google Scholar
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.
Google Scholar
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.
Google Scholar
Bebbington M., Lai C.D., Zitikis R. (2007), A flexible Weibull extension, “Reliability Engineering and System Safety”, vol. 92(6), pp. 719–726.
Google Scholar
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.
Google Scholar
Białek J. (2015), Construction of confidence intervals for the Laspeyres price index, “Journal of Statistical Computation and Simulation”, vol. 85(14), pp. 2962–2973.
Google Scholar
Białek J. (2022), Elementary price indices under the GBM price model, “Communications in Statistics – Theory and Methods”, vol. 51(5), pp. 1232–1251.
Google Scholar
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.
Google Scholar
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.
Google Scholar
Birnbaum Z.W., Saunders S.C. (1969), A new family of life distributions, “Journal of Applied Probability”, vol. 6(2), pp. 637–652.
Google Scholar
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.
Google Scholar
Brandt S. (2014), Data analysis, Springer International Publishing, Switzerland.
Google Scholar
Brazauskas V. (2003), Information matrix for Pareto (IV), Burr, and related distributions, “Communications in Statistics Theory and Methods”, vol. 32(2), pp. 315–325.
Google Scholar
Carli G. (1804), Del valore e della proporzione de’metalli monetati, “Scrittori Classici Italiani di Economia Politica”, no. 13, pp. 297–336.
Google Scholar
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.
Google Scholar
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.
Google Scholar
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.
Google Scholar
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.
Google Scholar
Chhikara R.S., Folks J.L. (1989), The Inverse Gaussian Distribution: Theory, Methodology and Applications, Marcel Dekker, New York.
Google Scholar
Cooray K. (2006), Generalization of the Weibull distribution: The odd Weibull family, “Statistical Modelling”, vol. 6(3), pp. 265–277.
Google Scholar
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.
Google Scholar
Cordeiro G.M., Castellares F., Montenegro L.C., Castro M. de (2013), The beta generalized gamma distribution, “Statistics”, vol. 47(4), pp. 888–900.
Google Scholar
Drapella A. (1993), The complementary Weibull distribution: unknown or just forgotten?, “Quality and Reliability Engineering International”, vol. 9(4), pp. 383–385.
Google Scholar
Dutot C.F. (1738), Reflexions Politiques sur les Finances et le Commerce, Les Freres, The Hague.
Google Scholar
El Gohary A., Alshamrani A., Al Otaibi A.N. (2013), The generalized Gompertz distribution, “Applied Mathematical Modelling”, vol. 37(1–2), pp. 13–24.
Google Scholar
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.
Google Scholar
Eltehiwy M., Ashour S. (2013), Transmuted Exponentiated Modified Weibull Distribution, “International Journal of Basic and Applied Sciences”, vol. 2(3), pp. 258–269.
Google Scholar
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.
Google Scholar
Gaddum J.H. (1945), Lognormal distributions, “Nature”, vol. 156(3964), pp. 463–466.
Google Scholar
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.
Google Scholar
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.
Google Scholar
Gumbel E.J. (1958), Statistics of Extremes, Columbia University Press, New York.
Google Scholar
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.
Google Scholar
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.
Google Scholar
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.
Google Scholar
Jędrzejczak A., Pekasiewicz D. (2020), Teoretyczne rozkłady dochodów gospodarstw domowych i ich estymacja, Wydawnictwo Uniwersytetu Łódzkiego, Łódź.
Google Scholar
Johnson N.L., Kotz S., Balakrishnan N. (1995), Continuous univariate distributions, vol. 2, John Wiley & Sons, New York.
Google Scholar
Kleiber C., Kotz S. (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley Interscience, Hoboken.
Google Scholar
Kotz S., Nadarajah S. (2000), Extreme Value Distributions: Theory and Applications, Imperial College Press, London.
Google Scholar
Lindsey J.K. (2004), Statistical analysis of stochastic processes in time, vol. 14, Cambridge University Press, Cambridge.
Google Scholar
Lu W., Shi D. (2012), A new compounding life distribution: the Weibull–Poisson distribution, “Journal of Applied Statistics”, vol. 39(1), pp. 21–38.
Google Scholar
Mahdavi A. (2015), Two Weighted Distributions Generated by Exponential Distribution, “Journal of Mathematical Extension”, vol. 9(1), pp. 1–12.
Google Scholar
McDonald J.B. (1984), Some generalized functions for the size distribution of income, “Econometrica”, vol. 52(3), pp. 647–663.
Google Scholar
Nadarajah S., Rocha R. (2016), Newdistns: An R package for new families of distributions, “Journal of Statistical Software”, no. 69, pp. 1–32.
Google Scholar
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.
Google Scholar
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.
Google Scholar
Pal M., Tiensuwan M. (2015), Exponentiated transmuted modified Weibull distribution, “European Journal of Pure and Applied Mathematics”, vol. 8(1), pp. 1–14.
Google Scholar
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].
Google Scholar
Sarhan A.M., Apaloo J. (2013), Exponentiated modified Weibull extension distribution, “Reliability Engineering and System Safety”, no. 112, pp. 137–144.
Google Scholar
Sarhan A.M., Zaindin M. (2009), Modified Weibull distribution, “APPS. Applied Sciences”, no. 11, pp. 123–136.
Google Scholar
Schwarz G. (1978), Estimating the dimension of a model, “The Annals of Statistics”, vol. 6(2), pp. 461–464.
Google Scholar
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.
Google Scholar
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.
Google Scholar
Silver H., Heravi S. (2007), Why elementary price index number formulas differ: Evidence on price dispersion, “Journal of Econometrics”, no. 140, pp. 874–883.
Google Scholar
Stacy E.W., Mihram G.A. (1965), Parameter estimation for a generalized gamma distribution, “Technometrics”, vol. 7(3), pp. 349–358.
Google Scholar
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.
Google Scholar
Sulewski P., Białek J. (2022), Probability Distribution Modelling of Scanner Prices and Relative Prices, “Statistika: Statistics & Economy Journal”, vol. 102(3).
Google Scholar
Tieling Z., Min X. (2007), Failure Data Analysis with Extended Weibull Distribution, “Communications in Statistics – Simulation and Computation”, no. 36, pp. 579–592.
Google Scholar
Witkovsky V. (2001), Computing the distribution of a linear combination of inverted gamma variables, “Kybernetika”, vol. 37(1), pp. 79–90.
Google Scholar
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.
Google Scholar
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