Kurtosis of the Topp-Leone distributions

Transcription

1 Kurtosis of the Topp-Leone distributions Samuel Kotz Department of Engineering Management and Sstems Engineering, George Washington Universit, 776 G Street, N.W. Washington D.C. 5, USA kotz@gwu.edu Edith Seier Department of Mathematics, P.O.Box East Tennessee State Universit, Johnson Cit, TN 376, USA seier@etsu.edu Abstract The kurtosis of the Topp - Leone (T-L) famil of distributions is explored b means of the spread-spread function to compare it with the left triangular distribution that originates the T-L famil. Based on the second derivative of the spread-spread function, intervals of values of the parameter b are identified regarding the behavior of kurtosis. Ke Words: Triangular distribution, Spread function, Spread-spread plots, Convexit. Introduction Kurtosis as a measure of flat-toppedness of a probabilit densit function of a continuous random variable was introduced b Pearson (95) but it is currentl understood as related to its center and tails. For a comprehensive discussion of kurtosis see Zenga (6). A famil of distributions, introduced b Topp and Leone (955) over 5 ears ago was recentl resurrected b Nadarajah and Kotz (3) and further studied b Kotz and Van Dorp (). The Topp-Leone (T-L) famil of distributions is generated from the left triangular distribution b elevating the CDF to a power b>, not necessaril an integer, that becomes the parameter of the new distribution.

2 When new probabilit distributions are defined it is interesting to stud their characteristics. The insight we can get about them can be of help to professionals looking to use those distributions as models. Kurtosis is also related to the performance of tests for variabilit and normalit. In this note we shall explore the kurtosis of the T-L famil of distributions, comparing the distribution with parameter b with the left triangular distribution. Since the T-L distributions are skewed, the spread-spread plot introduced b Balanda and MacGillivra (99) seems to be an appropriate tool for comparison. The convexit of the spread-spread function is studied b means of its second derivative and three intervals of values of the parameter b, (identified in a first exploration of the convexit of the function) : <b<, <b<, and b are examined. In the final section an appraisal of the kurtosis is carried out b several alternative kurtosis measures.. The spread function and spread-spread plots in Topp - Leone distributions The pdf of the Topp and Leone distributions is f(x) =bx b ( x)( x) b for x ; with the cdf F (x) = x b ( x) b, the inverse cdf being b. Evidentl for b =, the T-L is reduced to the left triangular distribution. Balanda and MacGillivra (99) defined the spread function to be S H = H (.5+u) H (.5 u) for u.5, where H is the inverse cdf of the distribution. The spread-spread plot introduced b Balanda and MacGillivra (99) assigns S H and S F for two different distributions H and F to the and x axes respectivel, the distribution H is considered to have larger kurtosis than F iff the curve in the graph (S H SF ) is convex. For the Topp and Leone distribution with parameter b, the spread function becomes: =H (.5+u) H (.5 u) = b.5 u b.5+u () Evidentl for b =, the spread function becomes that of the left triangular distribution given b S =.5+u.5 u () It is of interest for applications to compare, in terms of kurtosis, the T-L distributions with parameter b = with the left triangular distribution b

3 means of the spread-spread plot. Let = S, from () and () we have u = S () =.5 and the corresponding spread-spread function appearing in the spread-spread plot to compare a distribution with parameter b with the distribution with parameter b = becomes (for <<): (S ()) = (.5( )) /b (.5( + )) /b (3) As it was alread alluded, for a given value of b, the convexit of the line (S ()) indicates that the T-L distribution with parameter b possesses a larger kurtosis than the left triangular distribution; if the line is concave the opposite would be true. If (S ()) is neither concave nor convex, the two distributions are not comparable in terms of kurtosis. To determine whether the (S ()) is convex we calculate the second derivative of (3) with respect to. This results to be =( (S ())) = M(ABC +AB C +A BC (DEG +DE G+D EG)) () where k = b.5,m = k/(b), r = t, t =,v = k b r, u = k b +r, A =/ v, B =( r) /b, C =( / t) t, D =/ u, E = ( + r) /b and G = C. Thus, A = MBCv 3/, G = C, B =(/b )C( r) /b, C = 3 /t 3/ +3/ t, E =(/b )(+r) /b G and D = MEGu 3/. We shall now analze the behavior of the second derivative () for all < <, providing the spread and the spread-spread plots for b<, <b<, and b. 3. Topp-Leone distributions with b< For <b<, the T-L distribution is a J-shape. These distributions were actuall the original Topp and Leone distributions and have been found to have applications in reliabilit. Their authors were searching for distributions to model failure data of equipment and devices. Figure displas the densit function for a number of values of the parameter b =.(.). The smaller is b, the more negativel skewed the distribution becomes. The surface in Figure corresponds to the spread function () for. b and u.5. The sections of the surface obtained b fixing the 3

4 Topp Leone distribution, b =.(.).9 and b= f(x) 5 3 b=.7 b=.9 b=.5 b=.3 b=. b= x Figure : Topp-Leone distributions for b Spread Function for Topp and Leone Distributions,b=.(.).8.6 S( u,b) b.... u.3..5 Figure : Spread function of Topp-Leone distributions for. b value of b show the manner in which the spread function varies as b increases. Figure 3 displas the spread-spread function (3) evaluated for. b and comparing each one of the corresponding T-L distributions with the left triangular distribution. The sections of the surface in Figure 3 obtained b fixing the value of b result in spread-spread plots such as those in Figure A. For low values of b, the spread-spread plot depicts a convex line, but as b approaches to, the spread-spread plot becomes almost linear (it is linear for b= because here we are comparing the left triangular distribution with itself.) It is of course of interest to compare the Topp-Leone distributions among themselves and not onl with the left triangular distribution. Figure b (where spread-spread plots compare and +. ), seem to suggest that the

5 Spread Spread comparing TP (.<=b<=) with the left triangular distribution.8 Sb(S ()) b Figure 3: Spread-Spread function (S ()) to compare T-L distributions with. b with the left triangular distribution. Topp -Leone distributions are indeed kurtosis ordered for the most part of the interval <b<, the smaller the value of the parameter b the larger the kurtosis. The behavior in the vicinit of the extremes of the interval [,] requires special attention. Observing the change in shape of the curve S along the values of b, we can conclude that kurtosis increases ver fast as b decreases in the interval [.,.3], graduall increases as b decreases in the interval [.3,.8), and seems to change ver little after that. The issue of the kurtosis when b is ver close to ( <b<.5) seems to be quite delicate. Figure displas the extreme form of the spread function for ver low values of the parameter b. To stud the convexit of the spread-spread function its second derivative is displaed in Figure 5. Observe that when b is ver close to, the second derivative takes value for a substantial range of values of. Figure 5 confirms the convexit (S ()) when. b.88. However rather unexpectedl the second derivative takes on values slightl below zero for a range of values of when.89 b<. The interval of values of for which the second derivative of (S ()) is negative is located in the lower part of the interval [, ] and it prolongs as b approaches from the left. When b =.95 (last plot in Figure 5), the second derivative takes on a negative value for <.38, however those negative values are ver close to. For example, the minimum value attained b the second derivative for b =.95 is.8. Since the second derivative () has a complicated expression and its value is expected to be close to zero in the vicinit of 5

6 a.spread Spread plot for TL distributions with b=.(.).9 vs b=.8.6. b=.3 b=.5 b=.7 b=.9. b= S b.spread Spread plot, for b vs b+. with b=.(.) b=.6 b=. b= Figure : Spread-spread plots for pairs of T-L distributions b =, the negative value might be the consequence of the accumulation of rounding off errors. Due to this behavior of the spread-spread function it ma be safer not to claim that the Topp-Leone distribution with.89 b< has higher kurtosis than the left triangular distribution.for b., the second derivative takes on value for a substantial range of values of. From all the figures in this section it is evident that in the interval. b.8, the T-L distributions have a smooth spread function and strictl larger kurtosis than the left triangular distribution. Moreover the T-L distributions seem to be ordered with respect to kurtosis (the lower the value of the parameter b, the higher the kurtosis is), and small changes in b can provide a ver substantial change in kurtosis. The curvature of the spread-spread functions in Figure seem to indicate that the change in kurtosis for equal increments in the parameter b decreases as b increases in [.,.8].. Topp-Leone distributions for <b< Figure 6 displas the densit function of the Topp-Leone distributions for b = (.). Unlike the distributions in Figure (for <b<), the mode here is not and f(mode) < ; f(mode) first decreases and then increases. As b increases in this interval the distribution becomes less skewed. 6

7 8 b=. 5 b=.5 8 b= b=. b= b=. b= b=.5 b= Figure 5: Second derivative of Spread-Spread function (S ()) for <b< b= Topp Leone distribution, b =(.).8.6 b=. b=. b=.3.. f(x) x Figure 6: Topp-Leone distributions for b Figure 7 seems to indicate that the spread functions for the different values of b in the interval [, ] are quite similar. This is confirmed b Figure 8 (to be compared with Figure a), which displas the spread-spread plot comparing the T-L distributions with b =.(.) with the left triangular distribution. The second derivative in expression () is displaed in Figure 9 for <b< ; it takes negative values for some values of thus indicating that (S ()) is not convex and hence one can not claim that the T-L distributions with the parameter in the interval [, ] posses higher or lower kurtosis than the left triangular distribution. The values of for which the second derivative takes negative values are at the right side of the interval <<, in the vicinit of. 7

8 Spread Function for Topp and Leone Distributions, b in [,].8.6 S( u,b) b u Figure 7: Spread function for Topp-Leone distributions b b=. b=. b=.3 b=. b=.5 b=.6 b=.7 b=.8 b=.9 S S S Figure 8: Spread-Spread plots to compare T-L distributions <b< with the left triangular distribution. 5. Topp-Leone distributions for b The T-L densities for a number of values of b are displaed in Figure. Here both the mode and f(mode) increase when the parameter b increases. A comparison with figures and 6 clearl indicates the change in shape from J shaped to distributions similar to the exponential and then unimodal Gamma tpe distributions. Skewness takes value in the vicinit of b =.56, is positive for b<.56 and negative for b>.57. The sharp turn in the spread function in Figure becomes more noticeable as b increases. Figure suggest that for b, the Topp-Leone distribution has larger kurtosis than the left triangular distribution. This is confirmed b the second derivative of the spread-spread function in () 8

9 . b=..5 b=.3 b= b=.7.5 b=.8.5 b= Figure 9: Second derivative of Spread-Spread function (S ()) for <b< 5 Topp Leone distribution, b =().5 f(x) b= b=8 b=6 b= b= b= b=8 b=6 b=.5 b= x Figure : Topp and Leone distributions with b displaed in Figure 3 for b. Note that for larger values of b, the second derivative takes rather high values for in the vicinit of = and ver low ones in the vicinit of =. Figures and also seem to indicate that as b increases the difference in kurtosis between T-L distributions and the left triangular distribution is located mainl in the tails rather than at the peak since the curvature is larger at the right side of the spread-spread plot. The trend noticed in the shape of the spread and the spread-spread functions as b increases in figures and, continues to be valid when b takes much larger values. Figure suggests that the distributions seem to be ordered according to the value of b, the larger the value the higher the kurtosis. Figure is to be compared with Figure B; the change in the kurtosis as b increases beond does not seem to be as prominent as when b decreases below.8. 9

10 Spread Function for Topp and Leone Distributions, b>.8.6 S( u,b) b u Figure : Spread function for b 6. Kurtosis measures Based on the analsis of the spread-spread function we know the ranges of values of the parameter b for which the T-L distribution has higher kurtosis than the left triangular distribution (b.88 and b ). Also as b decreases below.88 the kurtosis increases, and the behavior of the kurtosis for values of b in the vicinit of and ma be somewhat delicate. Indeed, small changes in b in the interval (.,.5) ma result in large changes in kurtosis. For b we would expect small increments in the kurtosis as b increases, while in the interval <b< the comparison of the T-L distributions with the left triangular distribution in terms of kurtosis is just meaningless since the corresonding spread-spread function is not convex. It mabe of interest to examine the behaviour of kurtosis measures for the T-L distributions. B now it is well known that it is often misleading to represent kurtosis b a single number. Pearson s kurtosis appears much more frequentl in the literature dealing with statistical distributions; however other measures of kurtosis do exist. The distributions that we were studing are skewed so that the kurtosis measures defined for smmetric distributions are not applicable here. We shall use 3 kurtosis measures that are expected values of functions of the standardized variable z, z =(x µ)/σ, where µ and σ are the mean and standard deviation of the distribution respectivel. One of those measures is the well known Pearson s coefficient β = E[z ] (Pearson, 95). The other two measures to be considered are those proposed b Seier and Bonett(3)

11 b= b= b=6 b=8 b= b= b= b=6 b=8 S S S Figure : Spread-Spread plots to compare T-L (b ) with the left triangular distribution. : K (e) =E[5.73 exp( z )] and K () = E[.8( z )], the latter being a scaled version of the robust kurtosis defined b Stavig(98). These three measures are of the form E[g(z)] where g(z) is a continuous even function, strictl monotonic at each side of z=. However, g(z) =z (corresponding to Pearson s β ) increases as z departs from giving more importance to the tails of the distribution while the g(z) appearing in the other two give more importance to the middle. All three measures attain value 3 for the normal (Gaussian) distribution, but the ranges of values are different. It should perhaps be noted that the L-kurtosis, a measure of kurtosis quite popular in hdrolog, provides for a number of distributions values that tend to be highl correlated with the value of K (e);r =.99 for a set of smmetric distributions (Seier and Bonett, 3) and r =.93 for a set of 3 skewed distributions. Whether this is also valid for the T-L distributions is to the best of our knowledge an open problem. From Kotz and van Dorp () the values of the mean and the variance are: µ = b Γ(b+)Γ(b+) and σ = Γ(b+) ( b+ )B(b+,b+)[ b B(b+,b+)] b+3 B(b+,b+)B(.5 b+,b+), where B =Γ(a)Γ(b)/Γ(a + b) and B is the incomplete beta function. Strictl speaking we shouldn t quantif the T-L distributions kurtosis <b<because based on the distribution functions we can not reall assert that the have higher or lower kurtosis than the left triangular, however we plot the values of β, K (e) and K () for. b in Figure 5. Each row of the figure corresponds to one of the 3 measures, each column

12 b= 5 b=.5 6 b= b=5 b= 5 b= Figure 3: Second derivative of Spread-Spread function (S ()) for b Spread Spread plot, for b+ vs b with b=() b=8 b= Figure : Spread-spread to compare pairs of T-L distributions with b units apart (b ) corresponds to different intervals of values of the parameter b. Use b = as a reference point since we are comparing all the distributions with the originating left triangular distribution. For values of b in the vicinit of we have noticed some problems with K (e), and in a lesser extent with K (). A program to calculate these measures b means of numerical integration reveals some problems. The value for K (e) peaks for b=.6 and then decreases as b goes (backwards) towards. It is onl for b=. that K (e) attains a value lower than the one for the left triangular (.899). K () peaks at b=.5 and goes down as b goes backwards towards, however its values are alwas higher than the one corresponding to the left triangular distribution. The coefficient β does not experience problems in that region:

13 . b. b b β K(e) K().5 5 b.5 b. 6 b Figure 5: Values of the three kurtosis measures for different intervals for the parameter b its value continues to increase as b approaches. On the other hand, both K (e) and K () confirm our previous comments about kurtosis based on the spread-spread function for b> : here the kurtosis of the T-L distributions is greater than the kurtosis of the left triangular distribution, and seems to increase slightl as b increases. However, β takes values lower than. (the value corresponding to the left triangular distribution) for the whole range of values <b<.8. It is understandable that the measures would disagree in their behavior in the interval <b< since the distributions here are not kurtosis ordered with respect to the left triangular distribution. Figures,6 and vividl demonstrate the movement of the T-L densit encompassing a wide range of forms. Table summarizes the values of Pearson s skewness and kurtosis coefficients and with parameter b in the range < b Additional explorations indicate that the kurtosis coeficient β converges to approximatedl 3. as the parameter b increases indefinitel. 7. Conclusions The Topp-Leone famil of distributions originated from the left triangular distribution b elevating the cdf to the power b>. Kurtosis of this famil of 3

14 Table : Pearson s skewness and kurtosis b β β distributions is studied as a function of parameter b. Since the distributions are non-smmetric, the spread-spread function is an appropriate tool for this investigation. Small values of b, in the vicinit of, cause the spread function to be almost constant in some intervals, at the same time taking sharp turns. For b<.5 the second derivative of the spread-spread function takes value for part of the domain of the variable. The value of b = presents the extreme situation in which all the mass is concentrated at one point. Values of b close to do not produce convex spread-spread functions when the distribution is compared with the left triangular. Neglecting the extreme and the non-comparable situations we feel comfortable to assert that for. b.8 a Topp-Leone distribution has larger kurtosis than the left triangular distribution and that the lower the value of b in that interval is, the higher the kurtosis is. Small changes in the parameter b in the interval (.,.5) produce noticeable changes in kurtosis. For the values of b in the interval (, ), the comparison between kurtosis of the T-L distribution and the left triangular distribution is non-feasible. For b>, the distributions have larger kurtosis than the left triangular, however in this range even large increments in the value of the parameter b result in minor changes in kurtosis.

15 References Balanda, K.P. and MacGillivra, H.L (99) Kurtosis and Spread. Can. J. Stat 8,7-3. Kotz, S. and Van Dorp, J.R. (5) Beond Beta- Other Continuous Families of Distributions with Bounded Support and Applications. World Scientific, Singapore. Nadarajah, S. and Kotz, S (3) Moments of some J-shaped distributions, Journal of Applied Statistics 3, Pearson, K. (95) Skew Variation, a Rejoinder. Biometrika,69-. Seier, E. and Bonett, D.G. (3) Two families of kurtosis measures. Metrika 58, Stavig, G.R. (98) A Robust Measure of Kurtosis. Perceptual and Motor Skills 55, 666 Topp, C.W. and Leone, F.C. (955). A Famil of J-shaped frequenc functions, Journal of the American Statistical Association 5, 9-9. Van Zwet, W.R.(96) Convex transformations of random variables. Mathematics Centre Tract 7, Mathematisch Centrum Amstrerdam, Amsterdam. Zenga, M. (6). Kurtosis. In: Encl. of Stat. Sciences,6, pp (second edition) S. Kotz et.al (editors), J. Wile, Haboken, NJ. 5