A COMPUTATIONAL STUDY UPON THE BURR 2-DIMENSIONAL DISTRIBUTION

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1 TOME VI (year 8), FASCICULE 1, (ISSN ) A COMPUTATIONAL STUDY UPON THE BURR -DIMENSIONAL DISTRIBUTION MAKSAY Ştefa, BISTRIAN Diaa Alia Uiversity Politehica Timisoara, Faculty of Egieerig Hueoara ABSTRACT: I this paper we geeralyze the oe-imesioal Burr istributio at - imesioal space. We showe that this two-imesioal rule meets all coitios to be a cotiuous probability istributio, a it ca be use i future moelig of certai statistical problems. Statistical ifereces about the shape parameters have bee iscusse. Aalysis of a real ata set has bee performe by givig the expresio of the probability esity fuctio applyig the least squares metho. KEYWORDS: probability istributio, Burr istributio, least squares metho, orer statistics. 1. INTRODUCTION Burr itrouce twelve ifferet forms of cumulative istributio fuctios for moelig ata (see []). I probability theory a statistics, the Burr istributio is a probability istributio use i ecoometrics, which is cocere with the tasks of evelopig a applyig quatitative or statistical methos i the stuy of ecoomic priciples (see [1, 3, 5, 7]). Stuyig several aspects of the oe-parameter Burr-istributio [8, 9] oe ca observe that the Burr-istributio ca be use quite effectively i moelig stregth ata a also moelig geeral lifetime ata. The aim of this paper is to geeralyze the oe-imesioal Burr istributio at - imesioal space. We preset i the Sectio the expresio of -D Burr istributio that we propose a the coitio that this fuctio has to meet i orer to be a cotiuous probability istributio. Havig a set of real iput ata, applyig the least squares metho we obtaie the optimal parameters of the two-imesioal probability esity fuctio for the givig ata set. The metho is expose i Sectio 3. Calculatios have bee oe i MathCa calculatio package. Sectio 4 iclues the results a the coclusios of the paper.. DISTRIBUTIONAL PROPERTIES The Burr istributio has the probability esity fuctio α γ 1 γ B1x, (,, ) x ( x ) ( α+ α β γ = γ α β β + ( where x a α >, β > a γ > must be strictly positive shape parameters [4, 6]. The aspect of the probability esity fuctio of Burr istributio epes o the values of its parameters, as it ca be see i Figure 1. Its cumulative istributio fuctio is epicte i Figure, for several positive shape parameters. Our aim i this paper is to geeralyze the oe-imesioal Burr istributio at - imesioal space. Let B : R + R + R be the probability esity fuctio of two-imesioal Burr istributio, expresse by α1 α γ1 1 γ 1 γ1 γ α1 α β1 β x y B ( x,y, α 1, β1, α, β, γ) = γ1 ( ) ( α ) γ 1 x ( x ) ( α + β + β + () where x, y a α1>, β1>, γ 1>, α >, β >, γ >. 191

2 ENGINEERING. TOME VI (year 8). Fascicule 1 (ISSN ) Figure 1. The Probability Desity Fuctio of the oe Figure. The Cumulative Distributio Fuctio of the imesioal Burr istributio for ifferet shape oe imesioal Burr istributio for ifferet shape parameters parameters I orer to meet our purpose, the fuctio () has to be a probability esity [1, 6], a therefore has to meet the coitios B( x,y, 1, β1, α, β, γ) a ( α (3) B x,y, α1, β1, α, β, γ) xy = 1. (4) Calculatig the ouble itegral for raom strictly positive parameters, the result is Bx (, y, 1.3,.9, 1.1, 1., 1.3, 1.3 ) x y = 1. The mai goal of our stuy is to obtai the expressio of the probability esity for a - imesioal Burr istributio of a givig set of iput values. 3. DATA ANALYSIS AND DISCUSSIONS The set of umeric ata that are goig to be processe is give bellow, where the first two lies represet the values of the iepeet variables x a y, a the last lie represets the iepeet variable u = f / 1. at := augmet( x, y, f) ate := at T ate = Costats α1 >, β1 >, γ1 >, α >, β >, γ > are goig to be etermie usig the least squares metho, that assumes that the sum of the squares of the iffereces betwee the theoretical values of the fuctio a the experimetal values u, shall be miimum. The ecessary coitio is that the fuctio erivatives with respect to α1, β1, γ1, α, β, γ to be ull. We shall fi the parameters with the help of the program we are givig hereiafter: ORIGIN 1 Bx (, y, α1, β1, γ1, α, β, γ ) γ1 γ β1 α1 β α x γ1 1 y γ 1 ( β1 + x γ1 ( α1+ ) ( β + y γ ( α+ := ) i := 1.. legth( x) := legth( x) α1 :=. 5 β 1 :=. 5 γ1 :=. 3 α :=.4 β :=. 5 γ :=. Give γ1 γ β1 α1 β α ( x i ) γ1 1 ( y i ) γ 1 β1 ( x i ) γ1 ( α1+ + β + α1 y i 19 ( α+

3 ENGINEERING. TOME VI (year 8). Fascicule 1 (ISSN ) β1 γ1 α β γ γ1 γ β1 α1 β α x i γ1 γ β1 α1 β α x i γ1 γ β1 α1 β α x i γ1 γ β1 α1 β α x i γ1 γ β1 α1 β α x i β1 + ( x i ) γ1 ( α1+ β + y i 1 1 β1 + ( x i ) γ1 ( α1+ β + y i 1 β1 + ( x i ) γ1 ( α1+ β + y i 1 β1 + ( x i ) γ1 ( α1+ β + y i 1 sol := Fi( α1, β1, γ1, α, β, γ ) sol = α1e := sol 1 β1e := sol γ1e := sol 3 αe := sol 4 βe := sol 5 γe := sol 6 β1 + ( x i ) γ1 ( α1+ β + y i ( α+ ( α+ ( α+ ( α+ ( α+ 4. RESULTS AND INTERPRETATIONS The program coe presete hereibefore evaluates the first erivatives of the probability esity () with respect to parameters α 1, β 1, γ1, α, β a γ. We have starte the proceure of calculatig erivatives with six iitial guess values, far away from the solutio a the program compute the ext values for the Burr probability esity fuctio i two-imesioal space: α1 e = 1.6, β 1 e = , γ 1e =. 5 α e = 53.96, β e = , γe =.69. These values of α 1e, β 1e, γ 1e, αe, βe, γe substitute i expressio () lea to the twoimesioal Burr probability esity fuctio x y Burr( x,y) = x x Figure 3. The Probability Desity Fuctio of the two-imesioal Burr istributio ( ) ( ) 96 (5) The graphic represetatio of the twoimesioal probability esity fuctio which has bee obtaie for the give iput ata set is epicte i Figure 3, with its cotours showe i Figure 4. The correlatio coefficiet a the staar eviatio of the two-imesioal probability fuctio (5) are 193

4 ENGINEERING. TOME VI (year 8). Fascicule 1 (ISSN ) r := 1 ( ) ( u i B x, y α 1e i i, α e, γe ) i ( u i mea ( u ) i r =.887, where mea(u) represets the mea value of the iepeet variable u, a respectively 1 s := u Bx y (,, α1e, β1e, γ1e, αe, βe, γe i ( i i )) s = i Figure 4. The cotours of the probability esity fuctio of the two-imesioal Burr istributio: a 3D view, b D view The graph of the cumulative istributio fuctio that escribes the two-imesioal statistical istributio is give i Figure 5. The -imesioal Burr istributio is a cotiuous probability istributio a it ca be use i moelig certai problems. Figure 5. The cumulative istributio fuctio of the two-imesioal Burr istributio REFERENCES/BIBLIOGRAPHY [1.] Abramowitz, M., Stegu, I.A., Habook of Mathematical Fuctios, Natioal Bureau of Staars, [.] Burr, I.W., Cumulative frequecy istributio, Aals of Mathematical Statistics, 13, 15-3, 194. [3.] Cowa, G., Computig a Statistical Data Aalysis, Uiversity of Loo Lectures [4.] Feller, W., A Itrouctio to Probability Theory a Its Applicatios, 3r e., Joh Wiley & Sos, Ic., 1968 [5.] Maksay, Şt., Bistria, D., Cosieratios upo the vo Mises -imesioal istributio, Joural of Egieerig, Faculty of Egieerig Hueoara, Tome V, Fascicole 3, , 7. [6.] Maksay, Şt., A probabilistic istributio law with practical applicatios, Mathematica Revue D Aalyse umerique et e Theorie e L Approximatio, Tome (45), Nr. 1, pp , 198 [7.] Maria, Kati, V., Peter, E., Directioal Statistics, New York, Wiley,

5 ENGINEERING. TOME VI (year 8). Fascicule 1 (ISSN ) [8.] Raqab, M.Z., Orer statistics from the Burr type X moel, Computers Mathematics a Applicatios, 36, 111-1, [9.] Wolak, F.A., Testig iequality costraits i liear ecoometric moels, Joural of Ecoometrics, 41, 5 35,