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For the generalized normal distribution, two simulation studies have been conducted in order to investigate the small sample behavior of the estimators. Rahman and Gokhale 1996 found that the method of moments MM and ML estimators for m and sr perform similarly for r lt 2, whereas for r gt 2 the ML estimator of r seems to perform better than its MM counterpart for small samples. For r lt 2 the situation is reversed. Agro 1995 noted that for samples of size n lt 100 there is sometimes no...
References Piw
D'Addario, R. 1934 . Sulla misura della concentrazione dei redditi. Rome Poligrafico dello Stato. D'Addario, R. 1936 . Le trasformate Euleriane. In Annali dell'Istituto di Statistica dell'Universita di Bari, Vol. 8. Bari Macri Editore. D'Addario, R. 1936 . Sulla curva dei redditi di Amoroso. In Annali dell'Istituto di Statistica dell'Universita di Bari, Vol. 10. Bari Macri Editore. D'Addario, R. 1939 . La curva dei redditi sulla determinazione numerica dei parametri della seconda equazione...
[1 b
Hence, it is a Singh-Maddala distribution with a 1. The Lomax distribution is perhaps more widely known as the Pareto II distribution this term is used, for example, by Arnold 1983 and is related to the classical Pareto distribution via X Lomax b, q X b Par b, q . It is also a Pearson type VI distribution. Lomax 1954 considered it a suitable model for business failure data. There is not as much variety in the possible basic shapes of the Pareto II distribution after all, it is just a shifted...
[1 xbq1 0 x b6138
the three-parameter beta distribution. The GB1 is related to the GB2 distribution via the relation X - GB2 a, b,p, q - GB1 a, b,p, q . This generalizes a well-known relationship between the Bl and B2 distributions. The c.d.f.'s of the GBl and Bl distributions cannot be expressed in terms of elementary functions. However, in view of 6.8 , they are available in terms of Gauss's hypergeometric function 2F1, in the form McDonald, 1984 In analogy with the GB2 case discussed in Section 6.1, they can...
Champernowne Distribution
Champernowne 1937, 1952 considered the distribution of log-income Y logX, also termed income power, as the starting point and assumed that it has a density function of the form f y -TT7- TT' 1 lt y lt 1, 7-12 where a, T, y0, n are positive parameters, n being the normalizing constant and hence a function of the others. The reader will hopefully consult the biography of Champernowne presented in the appendix to learn more about this colorful person. The function given by 7.12 defines a...
Benktander Distributions
Starting from the observation that empirical mean excess functions point to distributions that are intermediate between the Pareto and exponential distributions, the Swedish actuary Gunnar Benktander 1970 discussed two new loss models. Whereas for the exponential distribution the mean excess function is given by A being the exponential scale parameter, in the Pareto case we have However, empirically one observes mean excess functions that are increasing but at a decreasing rate. Two...
A3 Max Otto Lorenz
Born September 19, 1876, Burlington, Iowa. Died July 1, 1959, Sunnyvale, California. Max Otto Lorenz was the son of Carl Wilhelm Otto and Amalie Marie Brautigam Lorenz. His father was born in Essen, Germany, in 1841. In 1851 his father's parents came to America and after a short sojourn in New Jersey took up permanent residence in Burlington, Iowa. For a number of years Otto Lorenz maintained a wholesale and retail grocery store and afterwards became a successful businessman engaging in...
x0[1 2cos 9xx0a xx02a
asin 9 x x0 a 1 9x0 cos 9 x x0 a 2 sin2 9 , Parameters a and x0 play the same role as in the case where l 1. Unfortunately, the new parameter 9 evades simple interpretation Champernowne 1952 noted that it may be regarded as a parameter for adjusting the kurtosis of the distribution of log income. Harrison 1974 observed that, for 9 p, the distribution approaches a point mass concentrated at x0. For 9 0, the distribution becomes the one with l 1, that is, the Fisk distribution. The role of the...
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Here b b1 a is a scale and a, p are shape parameters. This was introduced by Amoroso 1924-1925 as the family of generalized gamma distributions. Amoroso originally considered a four-parameter variant defined by X m, m IR, but we shall confine ourselves to the three-parameter version 5.2 . We use the notation X GG a, b, p . It is sometimes convenient to allow for a lt 0 in 5.2 one then simply replaces a by a in the numerator. For clarity, we shall always refer to a generalized gamma distribution...
Info Gjx
Sarabia, Castillo, and Slottje 1999 of the Ortega et al. 1991 and Chotikapanich 1993 as well as the Rao-Tam 1987 and Chotikapanich 1993 models, they concluded that the additive models perform distinctly better than either constituent model and moreover yield a satisfactory fit over the entire range of income. Among the many further proposals, we should mention the work of Maddala and Singh 1977b who suggested expressing the Lorenz curve as a sum of powers of u and 1 u. Holm 1993 proposed a...
Multivariate Lognormal Distribution
The most natural definition of a multivariate lognormal distribution is perhaps in terms of a multivariate normal distribution as the joint distribution of log Xi, i 1, , k. This approach leads to the p.d.f. xl, , xk - L--exp 1 log x m TS 1 log x m L where x x1, , xk T, logx logx1, , logxk T, m m1, , mk T, and S sj . If X X , , Xk T is a random vector following this distribution, this is denoted as X LNk m, S . From the form of the moment-generating function of the multivariate normal...
Further Paretotype Distributions
Krishnan, Ng, and Shihadeh 1990 proposed a generalized Pareto distribution by introducing a more flexible polynomial form for the elasticity called polynomial Pareto curves by Krishnan, Ng, and Shihadeh and comprising the Pareto type I for b1 bk 0 distribution as a special case. When we apply their linear specification b2 bk 0 , which is very close to Pareto's third proposal 3.7 , to two data sets, it turns out that the estimates of the new parameter b1 are rather small, confirming Pareto's...
Lorenz Curve And Inequality Measures 1
Unfortunately, the lognormal Lorenz curve cannot be expressed in a simple closed form it is given implicitly by L u F F-1 u - a2 , 0 lt u lt 1. 4.16 It follows directly from the monotonicity of F that the Lorenz order is linear within the family of two-parameter lognormal distributions, specifically This basic result can be derived in various other ways First, as was noted above, the parameter exp is a scale parameter and hence plays no role in connection to the Lorenz ordering. Thus, for Xi LN...
Lorenz Curve And Inequality Measures
The Lorenz curve, which exists whenever a gt 1, is given by L u 1 - 1 - u 1-1 a, 0 lt u lt 1. 3.51 As was already mentioned in the preceding chapter, it follows that Pareto Lorenz curves never intersect and that, for X Par x0, a , provided a,- gt 1, i 1,2. There is an interesting alternative but less direct argument leading to this result. Arnold et al. 1987 observed that every distribution F corresponding to an unbounded random variable and possessing a strongly unimodal density generates an...
Hazard Rates Mean Excess Functions And Tailweight
Researchers in the actuarial sciences have addressed the problem of distinguishing among various skewed probability distributions given sparse observations at the right tail. Specifically, it has been suggested to employ what Benktander 1963 called the mortality of claims r x 1 FX , gt 0, 2.56 and what Benktander and Segerdahl 1960 called the average excess claim e x E X - x X gt x -- , x gt 0, 2.57 for distinguishing among potential models. See Benktander 1962, 1963 and Benktander and...
Sampling Theory of Lorenz Curves and Inequality Measures
In view of the long history of inequality measurement, it is rather surprising that only comparatively recently the asymptotics of time-honored tools such as the Lorenz curve and associated inequality measures have been investigated. A possible explanation is that in earlier literature income data were often believed to come from censuses. Nowadays it is however generally acknowledged that most data are, in fact, obtained from surveys although not necessarily from simple random samples . In...
Early History
As was already mentioned in Chapter 1, Pareto 1895, 1896 observed a decreasing linear relationship between the logarithm of income and the logarithm of Nx, the number of income receivers with income greater than x, x gt x0, when analyzing income reported for income tax purposes. Hence, he specified where A, a gt 0. Normalizing by the number of income receivers N Nx0, one obtains Almost immediately, public interest was aroused and other economists began to criticize the idea of a universal form...
History And Genesis
The history of the Pareto distribution is lucidly and comprehensively covered in the above-mentioned monograph by Arnold 1983 . We shall therefore provide selected highlights of his exposition supplemented by several additional details that have emerged in the last 20 years or so and describe the very few earlier historical sources not covered in Arnold 1983 . The history of Pareto distributions is still a vibrant subject of modern research. The Web site sponsored by the University of Lausanne...
Stochastic Process Models For Size Distributions
Interestingly enough, income and wealth distributions of various types can be obtained as steady-state solutions of stochastic processes. The first example is Gibrat's 1931 model leading to the lognormal distribution. He views income dynamics as a multiplicative random process in which the product of a large number of individual random variables tends to the lognormal distribution. This multiplicative central limit theorem leads to a simple Markov model of the law of proportionate effect. Let...
Info Xgz
Equation 2.24 is a direct generalization of 2.22 . It is of some at least theoretical interest that income distributions can be characterized in terms of these generalized Gini coefficients, which is equivalent in view of 2.22 to a characterization in terms of the first moments of the order statistics. As we shall see in the following chapters, most parametric models for the size distribution of incomes possess heavy polynomial tails, so only a few of the moments exist. However, these...
Definition
The classical Pareto distribution is defined in terms of its c.d.f. where a gt 0 is a shape parameter also measuring the heaviness of the right tail and x0 is a scale. The density is F-1 u xo 1 - u -1 a, 0 lt u lt 1. 3.3 We shall use the standard notation X Par x0, a . In his pioneering contributions at the end of the nineteenth century, Pareto 1895, 1896,1897a suggested three variants of his distribution. The first variant is the classical Pareto distribution as defined in 3.1 . Pareto's...
Markov Processes Leading to the Pareto Distribution
Champernowne 1953 demonstrated that under certain assumptions the stationary income distribution of an appropriately defined Markov process will approximate the Pareto distribution irrespectively of the initial distribution. Champernowne viewed income determination as a discrete-time Markov chain Income for the current period the state of the Markov chain depends only on one's income for the last period and a random influence. He assumed that there is some minimum income x0 and that the income...