Determination of Parameters of the Weibull Distribution by Applying the Method of Least Squares

DETERMINATION OF PARAMETERS OF THE WEIBULL DISTRIBUTION BY APPLYING THE METHOD OF LEAST SQUARES Dobrivoje Ćatić1, Ph.D, Full professor, Jasna Glišović2, Ph.D, Assistant professor, Sandra Veličković3, Ph.D, student, Jasmina Blagojević4, Ph.D, student, Marko Delić5, Ph.D, student UDC:629.3.027.484;519.213.2 INTRODUCTION Swedish scientist, Waloddi Weibull, conducted a series of tests, while researching dynamic durability of materials. When he conducted statistical analysis of gained data, he had found that normal distribution could not be used for modelling of statistical features. By generalisation of exponential distribution and adjustment of mathematical model according to empirical distribution, Weibull had reached a new distribution that today bears his name 1, 2. Undoubtedly, Weibull distribution is used mostly in the area of reliability. This directly comes from its parametric character and wide possibilities to interpret very different laws of random variables by selection of corresponding values of parameters. MATHEMATICAL MODEL OF WEIBULL DISTRIBUTION Depending on number of parameters, there are the two- and three-parameter Weibull distributions. Expression for the survival function for a three-parameter model is 3, 4: where:  t    Rt   e      , t   ,   0,   0,   0, (1) t – is independent variable (time),  - is location parameter (parameter of minimal operation until failure),  - is scale parameter and  - is shape parameter. If location parameter is   0 , the two-parameter Weibull distribution is obtained. Figure 1 presents charts of failure intensity function and density of operation time until failure for the two-parameter Weibull distribution and different values of shape parameter  5, 6. corresponding author: Dobrivoje Ćatić, Faculty of Engineering, University of Kragujevac, Sestre Janjić 6, Kragujevac, caticd@kg.ac.rs 2 Jasna Glišović, Faculty of Engineering, University of Kragujevac, Sestre Janjić 6, Kragujevac, jaca@kg.ac.rs 3 Sandra Veličković, Faculty of Engineering, University of Kragujevac, Sestre Janjić 6, Kragujevac, sandravelickovic@gmail.com 4 Jasmina Blagojević, Faculty of Engineering, University of Kragujevac, Sestre Janjić 6, Kragujevac, jasminamikovic@gmail.com 5 Marko Delić, Faculty of Engineering, University of Kragujevac, Sestre Janjić 6, Kragujevac, delicmarko@gmail.com 1 Volume 41, Number 1, 2015 66 Dobrivoje Ćatić, Jasna Glišović, Sandra Veličković, Jasmina Blagojević, Marko Delić As it may be seen in the chart of failure intensity, h(t), for different values of the shape parameter, , this distribution may be used as approximate model for all three periods of exploitation of machine systems’ elements 5, 6. In addition, if  < 1, the two-parameter Weibull distribution corresponds to hyper-exponential distribution. For  = 1, exponential distribution is gained, while for  = 2, Rayleigh distribution is gained. For greater values of parameter,  , Weibull distribution gets closer to normal distribution. For values of  between 3 and 4, differences between these two distributions are negligible. Nevertheless, it should be noted that theoretical laws are never mathematically identical. Considering the previous and based on the results of approximation of empirical distribution with Weibull distribution, a model of other hypothetical distribution is determined more closely. a) b) Figure 1. Charts of functions of a) failure intensity and b) density of operation time until failure for the two-parameter Weibull distribution In order to approximate the empirical distribution with the two- or three-parameter Weibull distributions, it is necessary to linearize mathematical models of distributions. With the three-parameter model, starting from expression (1), elementary mathematical transformations may change the expression R(t )  1  F (t ) into 4, 5, 7: 1 ln ln    lnt       ln . (2) 1  F t  If the following substitutions are introduced: 1 y  ln ln , (3) 1  F t  a1   , (4) x  lnt    and (5) a0    ln , (6) equation of the shape line is obtained: y  a0  a1  x . (7) A straight line in a coordinate system where x-axis has logarithmic and y-axis has double logarithmic scale may represent expression for reliability function of the threeparameter Weibull distribution. Linearization of reliability function of the two-parameter Weibull distribution is done in similar way. The only difference is that, instead of substitute x  ln(t   ) , a substitute x  ln t is used during transformation of coordinates. Volume 41, Number 1, 2015 Determination of parameters of the Weibull distribution by applying the method of least square 67 DETERMINATION OF PARAMETERS OF THE TWOPARAMETER WEIBULL DISTRIBUTION Determination of parameters of the two-parameter Weibull distribution may be achieved graphically or analytically. Goode and Kao had developed a graphical procedure for determination of Weibull distribution by the aid of probability paper 8. In the two-parameter Weibull model, procedure demands that points with transformed coordinates ( xi , yi ) , that is with coordinates ln t i , ln ln 1 1  F (t i )  are imported on Weibull probability paper (coordinate system with logarithmic scale on x-axis and double logarithmic scale on yaxis) 5. If the arrangement of the points is approximately linear, the two-parameter Weibull distribution may be used for approximation. Procedure continues with approximation of a series of plotted points with a straight line. This approximation may be conducted graphically (by subjective assessment) or analytically (by the least squares method). If approximation is conducted with the least squares method, further course of graphical procedure has no meaning. Namely, based on determined coefficients a 0 and a1 of regression line of the series, distribution parameters are easily achieved from expressions (4) and (6). For graphical determination of scale parameter, , the following condition is used: t    F t   1  Rt   1  exp1  0.632 . (8) Based on this, parameter  is equal to x-coordinate of a point on approximate straight line that has y-coordinate equal to 0.632. According to linearized model of Weibull distribution, shape parameter, , represents inclination of approximate straight line in regard to x-axis. It is determined by drawing a line through point C on probability paper for Weibull distribution that is displayed within example in Figure 4 that is parallel to approximate line until it crosses a vertical line passing through point x  1 . From intersection point, a horizontal line is drawn until it crosses an axis for transformed y-coordinate. Value of parameter  is read directly on this axis with a plus sign. During analytical determination of parameters of the two-parameter Weibull distribution using the least squares method, series of points with coordinates ( xi , yi ) is approximated by a straight line. The best of all approximate straight lines in a form (7) is the one for which the sum of squares of vertical offsets of the points from regression line is the smallest. Determination of coefficients a 0 and a1 with the least squares method is done by previous determination of the expression for a sum of squares of ordinates offsets: S a0 , a1   a0  a1 x1  y1 2  a0  a1 x2  y 2 2  ...  a0  a1 x n  y n 2   a0  a1 xi  yi 2 . n i 1 (9) Necessary and sufficient condition for function S (a0 , a1 ) to achieve the maximum is expressed by equation: Volume 41, Number 1, 2015 Dobrivoje Ćatić, Jasna Glišović, Sandra Veličković, Jasmina Blagojević, Marko Delić 68 n S  2   a 0  a1 xi  y i   0 ,  a0 i 1 n S  2   xi a 0  a1 xi  y i   0 ,  a1 i 1 (10) from where a system of linear algebraic equations for determination of coefficients a 0 and a1 is gained: a 0  n  a1   xi   y i , n n i 1 i 1 n a 0   xi  a1   xi2   xi y i . i 1 i 1 i 1 n n By determination of determinants of equation systems (11): n n  yi  xi  xi  yi D , D0  , D1  2 , 2  xi y i  xi  xi  xi y i  xi  xi (11) (12) values of required coefficients are gained: a0  D0  xi2   yi   xi   xi yi  , D n   xi2   xi 2 n   xi yi   xi  yi D . a1  1  D n   xi2   xi 2 (13) Based on relationship between distribution parameters  ,  and the coefficients of straight line equation, given by expressions (4) and (6), the following is obtained:   a1 and   e a  0 a1 . (14) DETERMINATION OF PARAMETERS OF THE THREEPARAMETER WEIBULL DISTRIBUTION Procedures for determination of parameters of the three-parameter Weibull distribution are essentially different from previously described procedures. If operation time until the first failure occurs of the objects in the observed sample is not small in regard to total operation time, or if when entering the points onto probability paper, their layout is such that may be approximated by quadratic parabola convex upwards (in positive direction of y-axis), in that case, location parameter, , should be used. Value of this parameter is in the interval 0 to t1 , where t1 is operation time until the first failure occurs on objects for a small sample or lower limit of the first interval for a large sample. Value of location parameter, , of the three-parameter Weibull distribution may be determined graphically, graphically-analytically and analytically. During graphical determination of location parameter, , after points with coordinates ln t i , ln ln 1 1  F (t i )  are entered on Weibull probability paper and conclusion that it is the three-parameter distribution is made, value of   0.9  t1 is taken in the first step of the iterative procedure 5, 7. Points are entered onto probability paper again, now with coordinates ln(t i   ), ln ln 1 1  F (t i )  , and distribution of points is Volume 41, Number 1, 2015 Determination of parameters of the Weibull distribution by applying the method of least square 69 analysed. Feedback information for further steps of the procedure is gained based on distribution of points and possibility for approximation with a straight or a curved line, concave or convex upwards. Generally, three possibilities may occur in iterative procedure: ● if approximation curve obtained is still convex upwards,  should be increased, ● if a straight line is obtained,  is determined and procedure continues with determination of the rest of distribution parameters, and finally, ● if concave curve is obtained,  should be decreased. Iterative procedure continues until value of parameter  is obtained, for which the points are located on a straight line. Afterwards, scale parameter, , and shape parameter, , are determined identically as for the two-parameter Weibull distribution. In order to determine position parameter, , faster, graphical-analytical procedure has been developed. After entering the points onto probability paper and subjective approximation by a curved line, position parameter may be determined from the following expression 9: t  t   t 2  t1  , (15)   t2  3 2 t 3  t 2   t 2  t1  where: t1 – is x-coordinate of the first point of the curve ( t min ), t 2 – is x-coordinate of the point on the curve, which y-coordinate is arithmetic mean of y coordinates of points having x coordinates t1 and t 3 , t 3 – is x-coordinate of the last point of the curve ( t max ). Value of t 2 is determined graphically. After determination of position parameter, , points with coordinates ln(t i   ), ln ln 1 1  F (t i )  are entered onto probability paper. Correctness of graphical-analytical procedure should be confirmed with the fact that points follow closely a straight line. After a best-fit straight line for a series of points is drawn, parameters  and  are determined in the same way as in the two-parameter Weibull distribution. Analytical computer determination represents the third possibility for determination of parameter. One of the possibilities for analytical computer determination of parameter  is to approximate a series of points having transformed coordinates with a straight line. The idea is to approximate the points with the second order polynomial by the least squares method: y  a0  a1 x  a 2 x 2 , (16) that is, by square parabola. Approximation of a series of points by the least squares method with square parabola is conducted in a similar way as approximation with a straight line. Firstly, an expression for a sum of squares of offsets is formed: 2   2  a 0  a1 x 2  a 2 x 22  y 2   ... 2  a 0  a1 x n  a 2 x n2  y n   n 2   a 0  a1 xi  a 2 xi2  yi  . S a 0 , a1 , a 2   a 0  a1 x1  a 2 x12  y1 i 1 Volume 41, Number 1, 2015 (17) 70 Dobrivoje Ćatić, Jasna Glišović, Sandra Veličković, Jasmina Blagojević, Marko Delić By partial differentiation of expression (17) by coefficients a 0 , a1 and a 2 , the system of linear algebraic equations for determination of these coefficients is obtained: a0  n  a1   xi  a 2   xi2   yi , n n i 1 n i 1 i 1 a0   xi  a1   xi2  a 2   xi3   xi yi , n n n i 1 n i 1 n i 1 n n i 1 n (18) a0   xi2  a1   xi3  a 2   xi4   xi2 yi . i 1 i 1 i 1 i 1 By determination of determinants of the system of equations (18): n D   xi  xi 2 n D1   xi 2  xi  xi  xi 2  xi 3  yi  xi 2  xi 3  xi  xi y i 4  xi 2 2  xi y i 3  xi , 4  xi  yi D0   xi yi 2  xi y i n D2   xi 2  xi  xi  xi 2 3  xi  xi , 2  xi 3  xi 2  xi 3  xi  xi 4  yi  xi y i , 2  xi y i (19) values of required coefficients are obtained: a 0  D0 D D , a1  1 i a 2  2 . D D D Based on curvature of the obtained squared parabola, that is on sign of coefficient a 2 , it may be concluded whether the approximate curve is convex or concave in a positive direction of vertical curve. In this manner, similarly as in graphical procedure, feedback information is gained for further steps of the procedure. By searching through interval 0 t1 , value  is obtained for which the arrangement of points is closest to the straight line. For analytical determination of parameters of the three-parameter Weibull distribution with help of computer, a procedure shown by algorithm in Figure 2 is used. According to this algorithm, a part of the computer program is formed for modelling the reliability. Searching through intervals of possible values of  may be done in different ways. The algorithm solves this problem by halving the interval 0 t1 and investigating the curvature of the curve for   t1 2 . Based on a sign of second derivative of approximate polynomial (coefficient a 2 ), it may be concluded whether the value  is smaller or larger than the used value. For a 2  0 ,  takes value of a middle of the right interval (   0.75  t1 ), and for a 2  0 ,  takes value of a middle of the left interval (   0.25  t1 ). Procedure continues until the width of the interval, where value of  stands, becomes less than some value given in advance (e.g.  = 0.001). If during iterative procedure, value a 2  0 occurs, this means that a straight line is obtained during approximation of a series of points, that is value of parameter  is determined. If exiting from a cycle is a consequence of reduction of interval’s width, a series of points is approximated by a straight line in order to determine coefficients of regression line. After the coefficients a 0 and a1 are determined, the coefficients  and  are gained from the expression (14). Volume 41, Number 1, 2015 Determination of parameters of the Weibull distribution by applying the method of least square 71 Start n, (t(i), F(i), i=1,n) s5 = 0 i = 1, n y(i) = ln ln (1/(1-F(i))) s5 = s5 + y(i) gd = 0 gg = t(1) eps = 0.001 s1 = 0; s2 = 0; s3 = 0; s4 = 0; s6 = 0; s7 = 0. gama = (gd + gg) / 2 i = 1, n x(i) = ln (t(i) - gama) gd = gama s1 = s1 + x(i) s2 = s2 + x(i)2 s3 = s3 + x(i)3 s4 = s4 + x(i)4 s6 = s6 + x(i)y(i) s7 = s7 + x(i) 2y(i) gg = gama Determination of polynomial coefficients y(x) = a0 + a1x + a2x2 gd, gg, gama, a0, a1, a2 gg - gd > eps ne da <0 >0 a2 =0 a0 = (s2s5 - s1s6) / (ns2 - s12) a1 = (ns6 - s1s5) / (ns2 - s12) beta = a1 eta = exp (- a0 / a1) gama, eta, beta End Figure 2. Algorithm of a program for determination of parameters of the three-parameter Weibull distribution Volume 41, Number 1, 2015 Dobrivoje Ćatić, Jasna Glišović, Sandra Veličković, Jasmina Blagojević, Marko Delić 72 Graphical, graphical-analytical and analytical determination of parameters of the three-parameter Weibull distribution will be illustrated through the example of determining the distribution of operation time until failure of wheel’s drum in the brake system of light commercial vehicles. EXAMPLE Sample of 65 identical drums in braking system of light commercial vehicles is tested for reliability assessment. Number of 7 intervals for grouping the values of random variable is adopted, based on expression z  1  3.3 log n . Calculated width of the interval is t = 40,000 km, based on maximal and minimal values of random variable. Obtained values of operation time until failure of wheel’s drum are grouped in time intervals and shown in Table 1. Parameters of Weibull distribution used as theoretical model of operation time until failure of wheel’s drum should be determined. Table 1. Number of failures of the wheel’s drum in braking system by time intervals Distance travelled 110150 150190 190230 230270 270310 310350 350390 x 103 km 9 13 17 11 8 5 2 Number of failures With application of software for determination of theoretical model of empirical distribution, described in detail in 10, numerical characteristics of statistical series are gained: - mean value tsr = 221,692; - standard deviation σ = 63,429; - median t50 = 214,706; - mode Mo = 206,000; - coefficient of asymmetry Ka = 0.417 and - coefficient of flatness Ke = 2.439. In continuation of a program, based on the procedures for assessment of functional indicators of the distribution of the random variable for a large sample (n> 30), estimated values of the number of correct objects n(t), the reliability R(t), the unreliability F(t), density of operation time until failure f(t) and failure intensity h(t) of wheel’s drum are gained for middles of time intervals and given in Table 2. Table 2: Estimated values of functional indicators of the distribution of the random variable i m(i) ti n(ti) R(ti) F(ti) f(ti) h(ti) 1 2 3 4 5 6 7 9 13 17 11 8 5 2 130.00 170.00 210.00 250.00 290.00 330.00 370.00 60.5 49.5 34.5 20.5 11.0 4.5 1.0 0.93077 0.76154 0.53077 0.31538 0.16923 0.06923 0.01538 0.06923 0.23846 0.46923 0.68462 0.83077 0.93077 0.98462 Volume 41, Number 1, 2015 0.34615E-02 0.50000E-02 0.65385E-02 0.42308E-02 0.30769E-02 0.19231E-02 0.76923E-03 0.37190E-02 0.65657E-02 0.12319E-01 0.13415E-01 0.18182E-01 0.27778E-01 0.50000E-01 Determination of parameters of the Weibull distribution by applying the method of least square 73 Illustrations of graph charts of estimated values of density of operation time until failure, f(t), and failure intensity, h(t), wheel’s drum, in the form of polygons and histograms, are given in Figure 3. In rough assessments, theses graph charts may serve for determination of hypothetical distribution models. a) b) Figure 3. Graphical display of estimated values of: a) density and b) intensity of wheel’s drum failure  Graphical determination of parameters of Weibull distribution using probability paper. Usually, at the first step of graphical solving of task, approximation of a series of points is done by the two-parameter Weibull distribution. By transformation of coordinates and using the expressions xi  ln t i and yi  ln ln1 /1  F t i  and by entering the points onto probability paper for Weibull distribution (Figure 4), arrangement of points is gained that may be approximated by curve 1. Since the approximate function is convex, it means that the location parameter, , is positive. In the second step, series of points are approximated by the three-parameter Weibull distribution, with   0.9  t1  99,000 km. In this case, time, t1 is a lower limit of the first interval. By repeated calculation of xi coordinates according to expression xi  ln(t i   ) and by entering the points onto probability paper, arrangement of points is obtained that may be approximated by curve 2. Since obtained curve is concave,  should be smaller. Iterative procedure continues with decreasing values for  with step 0.1  t1 . Thus, for   0.7  t1  77,000 km approximately linear arrangement of point on probability paper is obtained. Parameter  value is equal to xcoordinate of the point on approximate straight line having y-coordinate 0.632, that is  = 163,000 km. Shape parameter, , is determined as cathetus of right-angled triangle, whose other cathetus is equal to 1. Parameter value is read on auxiliary vertical axis for transformed coordinate y. In this particular case, value  = 2.38 is obtained.  Graphical-analytical determination of location parameter . In order to determine value of location parameter , according to expression (15), it is necessary to graphically determine the value for t 2 . According to definition, t 2 is a value of x-coordinate of a point on approximate curve 1, whose y-coordinate is equal to arithmetic mean of y-coordinates of points with x-coordinates equal to t1 and t 3 . Thus, according to Volume 41, Number 1, 2015 74 Dobrivoje Ćatić, Jasna Glišović, Sandra Veličković, Jasmina Blagojević, Marko Delić Figure 4, orientation value of t 2 = 200,000 km is obtained. By using the expression (15), value of location parameter  = 81,000 km, is obtained, which is nearly equal to the value obtained during graphical problem solving. By transformation of coordinates for calculated value for , arrangement of points that may be approximated by a straight line is obtained. Figure 4. Determination of parameters of the three-parameter Weibull distribution using probability paper  Analytical computer determination of parameters of Weibull distribution using the least squares method. By approximation of empirical distribution of operation time until failure of wheel’s drum by the three-parameter Weibull distribution and by using a computer program whose algorithm is presented in Figure 3, after 18 iterations by halving the intervals and determination of a sign of second derivative a 2 , the parameters of the distribution are Volume 41, Number 1, 2015 Determination of parameters of the Weibull distribution by applying the method of least square 75 obtained: location parameter, γ = 76,115 km, scale parameter, η = 164,161 km and shape parameter, β = 2.355. Based on this, the expression for probability of faultless operation of wheel’s drum is:  t    Rt   e       t 76,115     e  164,161  2.355 y(i) . (20) In order to determine validity of approximation, graphical testing over probability paper for Weibul distribution and nonparametric testing were conducted using tests of Kolmogorov, Pearson and Romanovsky. Table 3 presents values of transformed x and y coordinated for particular value γ = 76,115 km and Figure 5 presents arrangement of points on probability paper for Weibull distribution. Table 3. coordinates No. 1 2 3 4 5 6 7 ti 130.0 170.0 210.0 250.0 290.0 330.0 370.0 Values F(ti) 0.0692 0.2385 0.4692 0.6846 0.8308 0.9308 0.9846 of transformed x(i) 3.987 4.542 4.897 5.158 5.365 5.537 5.683 y(i) -2.635 -1.300 -0.457 0.143 0.575 0.982 1.429 1,6 1,2 0,8 0,4 0 -0,4 -0,8 -1,2 -1,6 -2 -2,4 -2,8 3,8 4,0 4,2 4,4 4,6 4,8 5,0 5,2 5,4 5,6 5,8 x(i) Figure 5. Arrangement of points on probability paper for Weibull distribution Linear arrangement of the points in Figure 5 suggests that the approximate model satisfies conditions of graphical testing. For testing of hypothetical distribution model, according to Kolmogorov test, it is necessary to determine the greatest absolute value of difference between theoretical model and estimated values of distribution functions of operation time until failure. Table 4 contains a segment of output list of a program that relates to this part. Figure 6 presents graphical representation of deviations of theoretical approximate model, Ft (t ) , from empirical distribution, Fe (t ) . As it may be seen from Table 4, the largest deviation of theoretical model from empirical distribution is for the result No. 5 and amounts to 0.0143. For number of samples, n = 65 and given level of significance for Kolmogorov’s test, α = 0.20,   1.07 , permitted value of difference is:  1.07 Dn     0.1327. 65 n Since the maximal deviation is less than permitted value of difference, Weibull approximate distribution satisfies the Kolmogorov’s test for adopted level of significance. By application of Pearson’s test procedure, value of  2  0.9711 as a measure of deviation between empirical and approximate distribution is obtained. For number of Volume 41, Number 1, 2015 Dobrivoje Ćatić, Jasna Glišović, Sandra Veličković, Jasmina Blagojević, Marko Delić 76 Table 4. Deviations of Weibull approximate curve from estimated values of distribution function of operation time until failure No. 1 2 3 4 5 6 7 ti 130.0 170.0 210.0 250.0 290.0 330.0 370.0 Fe(ti) 0.0692 0.2385 0.4692 0.6846 0.8308 0.9308 0.9846 Ft(ti) 0.0700 0.2352 0.4613 0.6818 0.8451 0.9387 0.9806 delta 0.0007 0.0032 0.0079 0.0028 0.0143 0.0080 0.0040 Distribution function F(t) intervals z  7 and number of distribution parameters l  3 , number of degrees of freedom equals: k  z  l 1  7  3 1  3 . 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 110 150 190 230 270 310 350 390 Distance travelled [1000 km] Figure 6. Graphical representation of deviations of Weibull approximate distribution from empirical distribution By application of Pearson’s test procedure, value of  2  0.9711 as a measure of deviation between empirical and approximate distribution is obtained. For number of intervals z  7 and number of distribution parameters l  3 , number of degrees of freedom equals: k  z  l 1  7  3 1  3 . Based on calculated values for  2 and number of degrees of freedom, k, and by looking at the table for  2 distribution, it may be concluded that Weibull distribution may be accepted as approximate model at level of significance   0.80 . Comparable value for Romanovsky’s test is: 2 k 0.9711  3  0.828 , 2k 23 which is smaller than 3 and it means that Weibull distribution meets criterion of Romanovsky’s test. Ro   CONCLUSIONS Procedure for graphical determination of location parameter, , of the threeparameter Weibull distribution is long lasting and liable to errors due to imprecise entering of points onto probability paper, subjectivity during assessment of drawing of approximate straight lines and curves and impossibility to precisely read the parameter value. Since graphic-analytic procedure is largely based on graphical representation of points on probability paper and corresponding approximations, everything that has been said on graphical method applies also to graphical analytical method. Program determination of parameters of the three-parameter Weibull distribution enables gaining desired accuracy of the results with great speed. Thanks to that and known features of this distribution regarding possibility of approximation of empirical distribution Volume 41, Number 1, 2015 Determination of parameters of the Weibull distribution by applying the method of least square 77 of random variables, in great number of cases, Weibull distribution is optimal solution, with respect to other theoretical models. Based on the graphic of the estimated values of failure intensity of wheel’s drum in braking system of light commercial vehicles, it can be concluded that these are failure modes that occur during object aging. Theoretical approximate model of Weibull distribution satisfies graphics tests and analytical nonparametric tests with a high level of significance. REFERENCES 1 Weibull W., A statistical theory of the strength of material, Ing. Vetenskapa Acad. Handlingar 151, 1–45, 1939. 2 Weibull W., A statistical distribution function of wide applicability, Journal of Applied Mechanics, 18, 293–297, 1951. 3 British standard BS 5760, Part 2: Guide to the Assessment of Reliability. Reliability of Systems, Eequipments and Components, British Standards Institution, London, 1981. 4 Smith JD., Reliability, Maintainability and Risk, Practical methods for engineers, Butterworth Heinemann, Oxford, Sixth edition, 2001. 5 Kapur KC. & Lamberson LR., Reliability in Engineering Design, John Wiley & Sons, New York, 1977. 6 Dodson B., The Weibull Analysis Handbook. Second Edition, ASQ Quality Press, Milwaukee, Wisconsin, 2006. 7 Chin-Diew Lai DN, Pra Murthi, Min Xie., Weibull Distributions and Their Applications, in Springer Handbook of Engineering Statistics, Hoang Pham (Ed.), Springer-Verlag, London, pp. 63-78, 2006. 8 Kao HJ., A summary of some new tehniques on failure analysis, Proceedings of the sixth National Symposium on Reliability and Quality Control in elektronics, p. 190201, 1960. 9 Ivanovic G., Stanivukovic D., Reliability of technical systems - Collection of solved problems (in Serbian), Faculty of Mechanical Engineering, Belgrade, 1983. 10 Catic D., Development and application of reliability theory methods (in Serbian), Monograph, The Faculty of Mechanical Engineering from Kragujevac, Kragujevac, 2005. Volume 41, Number 1, 2015 View publication stats