We provide a new stochastic representation for a Kumaraswamy random variable with arbitrary non-negative parameters. The representation is in terms of maxima and minima of independent distributed standard uniform components and extends a similar representation for integer-valued parameters.
The result is further extended for generalized classes of distributions obtained from a “base” distribution function Fviz.G(x) = H(F(x)), where H is the CDF of Kumaraswamy distribution.
Open Peer Review Details | |||
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Manuscript submitted on 7-3-2018 |
Original Manuscript | Kumaraswamy Distribution and Random Extrema |
There is a growing literature on generalized distributions based on Kumaraswamy distribution [1G. Aryal, and Q. Zhang, "Characterizations of Kumaraswamy Laplace distribution with applications", Econ. Qual. Control, vol. 31, pp. 59-70.
[http://dx.doi.org/10.1515/eqc-2016-0009] ]. They are obtained from a “base” distribution with the cumulative distribution function (CDF) F as the CDF G (a generalized version of F) viz.
(1) |
where
(2) |
is the CDF of the Kumaraswamy distribution with parameters α, β > 0. The latter from now is denoted by K_{α,β}. In terms of random variables, the relation (1) can be stated as
(3) |
where Y~G and T~K_{α,β}. This generalization of F was proposed in Cordeiro and de Castro [2G.M. Cordeiro, and M. de Castro, "A new family of generalized distributions", J. Stat. Comput. Simul., vol. 81, pp. 883-898.
[http://dx.doi.org/10.1080/00949650903530745] ], where the authors developed its basic properties and presented generalizations of normal, Weibull, gamma, Gumbel, and inverse Gaussian distributions. Many other generalized distributions following this scheme have been developed since then, including recent works of de Pascoa et al. [3M.A.R. de Pascoa, E.M.M. Ortega, and G.M. Cordeiro, "The Kumaraswamy generalized gamma distribution with application in survival analysis", Stat. Methodol., vol. 8, pp. 411-433.
[http://dx.doi.org/10.1016/j.stamet.2011.04.001] ], Nadarajah et al. [4A. Di Crescenzo, "Some results on the proportional reversed hazards model", Stat. Probab. Lett., vol. 50, pp. 313-321.
[http://dx.doi.org/10.1016/S0167-7152(00)00127-9] ], Mameli [5N. Eugene, C. Lee, and F. Famoye, "The beta-normal distribution and its applications", Commun. Stat. Theory Methods, vol. 31, pp. 497-512.
[http://dx.doi.org/10.1081/STA-120003130] ] and Aryal and Zhang [6M.C. Jones, "Kumaraswamy distribution: A beta-type distribution with some tractability advantages", Statist. Method., vol. 6, pp. 70-81.
[http://dx.doi.org/10.1016/j.stamet.2008.04.001] ]. However, papers on the topic focus mainly on rather elementary properties and do not provide any significant theoretical interpretation of the construction. One exception is an interpretation for integer-valued α and β through maxima and minima of independent and identically distributed (IID) random components (e.g., Jones [7T.J. Kozubowski, and K. Podgórski, "Certain bivariate distributions and random processes connected with maxima and minima", Working Papers in Statistics 2016:9, Department of Statistics, School of Economics and Management, Lund University, 2016a. [Published online in Extremes, 2018, DOI 10.1007/s10687-018-0311-2].], Nadarajah et al. [4A. Di Crescenzo, "Some results on the proportional reversed hazards model", Stat. Probab. Lett., vol. 50, pp. 313-321.
[http://dx.doi.org/10.1016/S0167-7152(00)00127-9] ], Nadarajah and Eljabri [8T.J. Kozubowski, and K. Podgórski, "Transmuted distributions and random extrema", Stat. Probab. Lett., vol. 116, pp. 6-8.
[http://dx.doi.org/10.1016/j.spl.2016.04.001] ]). Indeed, assuming that α = m, β = n are positive integers, we find that x^{m} is the CDF of the maximum of m IID standard uniform variables, with the corresponding survivor function (SF) being 1 - x^{m}. Thus, the quantity (1 - x^{m})^{n} in (2) is the SF of the minimum of n such random variables, with H being the corresponding CDF. In other words, we have the following stochastic representation of T~K_{m,n,}
(4) |
where {U_{i,j}} are IID standard uniform variables. This property, discussed in Jones [7T.J. Kozubowski, and K. Podgórski, "Certain bivariate distributions and random processes connected with maxima and minima", Working Papers in Statistics 2016:9, Department of Statistics, School of Economics and Management, Lund University, 2016a. [Published online in Extremes, 2018, DOI 10.1007/s10687-018-0311-2].], motivated the name minimax for this distribution. Below we extend this interpretation to the general Kumaraswamy distribution as well as to its generalization (1).
To obtain a representation of a Kumaraswamy random variable in terms of min/max, we shall use the following basic result (e.g., Kozubowski and Podgórski, [9P. Kumaraswamy, "A generalized probability density function for double bounded random processes", J. Hydrol., vol. 46, pp. 79-88.
[http://dx.doi.org/10.1016/0022-1694(80)90036-0] ]), which relates the distributions of min/max of IID components with random number of terms to the relevant Probability Generating Function (PGF).
Lemma 2.1. If N is an random variable, independent of the sequence {X_{i}} of IID random variables with the CDF F, then the CDFs of the random variables
(5) |
are given by F_{X}(x) = G_{N}(F(x)) and F_{Y}(y) = 1 - G_{N}(1 - F(y)) , respectively, where G_{N} is the PGF of N.
A notable application of this result, discussed in Kozubowski and Podgórski [9P. Kumaraswamy, "A generalized probability density function for double bounded random processes", J. Hydrol., vol. 46, pp. 79-88.
[http://dx.doi.org/10.1016/0022-1694(80)90036-0] ], concerns a special case where the variable N = N_{α} in (5) has the Sibuya distribution (Sibuya, [10A.J. Lemonte, W. Barreto-Souza, and G.M. Cordeiro, "The exponentiated Kumaraswamy distribution and its log-transform", Braz. J. Probab. Stat., vol. 27, pp. 31-53.
[http://dx.doi.org/10.1214/11-BJPS149] ]), given by the PGF
(6) |
and the Probability Mass Function (PMF)
(7) |
where necessarily , with the boundary case α = 1 corresponding to a unit mass at n = 1. This variable represents the number of trials till the first success in an infinite sequence of independent Bernoulli trials, where the probability of success varies with the trial, and for the nth trial equals α/n. Here, because of the special form of the PGF of the Sibuya random variable N_{α}, if the latter represents the random number of terms in Lemma 2.1, the CDFs of the random variables X and Y in (5) are given by F_{X}(x) = 1 - [1 - F(x)]^{α} and F_{Y}(Y) = [F(x)]^{α}, respectively. This leads to our first result for the Kumaraswamy distribution with the parameters restricted to the unit interval (0, 1).
Proposition 2.1. Assume that {U_{i,j}} are IID standard uniform variables, have Sibuya distributions with respective parameters , and all the variables on the right-hand-side of (8) are mutually independent. Then the variable
(8) |
has the Kumaraswamy distribution K_{α, β}.
The above result can be re-formulated for any Kumaraswamy generalized random variable obtained viz. (3), providing a meaningful interpretation of this construction in terms of maxima and minima of IID components with the “parent” CDF F.
Proposition 2.2. Let N_{α}^{(i)}, be IID Sibuya variables with parameter , and independent of another Sibuya variable N_{β}, with parameter . Further, for a given CDF F, let Y be a random variable with the CDF G defined by (1), where H = K_{α, β} ; with . Then we have
(9) |
where the {X_{i,j}} are IID random variables with CDF F, independent of N_{α}^{(i)} and N_{β}.
Remark 2.1. Let B_{α, β} denote beta distribution given by the PDF
(10) |
Then, if either α = 1 or β = 1, the Kumaraswamy distribution K_{α, β} coincides with B_{α, β}. Thus, Proposition 2.1 specialized to this case, leads to stochastic representations of such beta-distributed random variables in terms of maxima and minima, i.e. if for the {U_{i}} are IID standard uniform random variables, independent of Sibuya-distributed N_{α} and N_{β} with we define
(11) |
then B_{1}~B_{α, 1} and B_{2}~B_{1, β}. Further, if either α = 1 or β = 1, the generalized distributions obtained viz. (1) are special cases of beta-generalized distributions popularized by Eugene et al. [11V. Mameli, "Kumaraswamy skew-normal distribution", Stat. Probab. Lett., vol. 104, pp. 75-81.
[http://dx.doi.org/10.1016/j.spl.2015.04.031] ], arising when H in (1) is the CDF corresponding to (10). Thus, Proposition 2.2 specialized to this case shows that if a random variables Y is defined viz. (3), where either T~B_{α, 1} or T~B_{1, β} with , then we have
(12) |
respectively, where the {X_{i}} are IID random variables with CDF F, independent of Sibuya-distributed N_{α} and N_{β}.
An extension of Proposition 2.1 to the case where the parameters are no longer restricted to the unit interval is straightforward. To formulate the result, we shall split a positive number r into r = {r} + < r >, where
(13) |
We shall also need the following result taken from Kozubowski and Podgórski [9P. Kumaraswamy, "A generalized probability density function for double bounded random processes", J. Hydrol., vol. 46, pp. 79-88.
[http://dx.doi.org/10.1016/0022-1694(80)90036-0] ], where we use the standard convention that the min and max over an empty set are understood as ∞ and -∞, respectively.
Proposition 2.3. If X and Y have CDFs given by 1 - (1 - F(x))^{r} and [F(x)]^{r}, respectively, where F is a CDF and , then they admit the stochastic representations
(14) |
where N_{< r >} has the Sibuya distribution (7) with parameter α = < r > and is independent of the IID {X_{j}} with the CDF F.
When we set r = β and apply the first representation in (14) to a Kumaraswamy random variable T~K_{α, β}, , we obtain
(15) |
where N_{< β >} has Sibuya distribution with parameter < β > and {X_{j}} is IID with the CDF x^{α}. In turn, when we set r = α and apply the second representation in (14) to each X_{j} in (15), we obtain
(16) |
where the {U_{i,j}} are IID standard uniform variables and the N_{< α >}^{(j)} are IID and Sibuya distributed with parameter < α >, independent of the {U_{i,j}}. By combining (15) with (16), we obtain the following result.
Proposition 2.4. Let T~K_{α, β} with general α, β > 0. Then we have
(17) |
where the {U_{i,j}} are IID standard uniform variables, N_{< α >}^{(j)} and N_{< β >} have Sibuya distributions with respective parameters < α > and < β >, and all the variables on the right-hand-side of (17) are mutually independent.
Remark 2.2. If the parameter α of T~K_{α, β} is a positive integer, , then we have {α} = m - 1, < α > = 1, so that the Sibuya variables N_{< α >}^{(j)} are equal to 1 almost surely. Thus, in the same notation, the representation (17) turns into
(18) |
On the other hand, if the parameter β of T~K_{α, β} is a positive integer, , then we have {β} = n - 1, < β > = 1, and the Sibuya variable N_{< β >} is equal to 1 almost surely, in which case the representation (17) reduces to
(19) |
Moreover, if both parameters of T~K_{α, β} are positive integers, so that in (18) we have and in (19) we have , then both, (18) and (19), turn into the representation (4) derived by Jones [7T.J. Kozubowski, and K. Podgórski, "Certain bivariate distributions and random processes connected with maxima and minima", Working Papers in Statistics 2016:9, Department of Statistics, School of Economics and Management, Lund University, 2016a. [Published online in Extremes, 2018, DOI 10.1007/s10687-018-0311-2].].
Remark 2.3. The stochastic representation (17) can be extended to the case of exponentiated Kumaraswamy (EK) distribution studied by Lemonte et al. [12A.W. Marshall, and I. Olkin, A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families., vol. 84, Biometrika, .], which is given by the CDF
(20) |
This can be achieved by another application of Proposition 2.3 on top of Proposition 2.4, leading to
(21) |
where W has EK distribution (20), γ = {γ} + < γ > with , and N_{< γ >}has Sibuya distribution (7) with parameter α = < γ >, and is independent of the IID Kumaraswamy random variables {T_{j}} with the CDF (2). In turn, each of the Kumaraswamy variables T_{j} admits the stochastic representation (17) involving minima and maxima of IID standard uniform variables.
Finally, we extend Proposition 2.4 to generalized random variables obtained from Kumaraswamy distribution viz. (3), which links this construction with maxima and minima of IID components with the “parent” CDF F.
Proposition 2.5. Let Y be a random variable with the CDF G defined by (1), where H = K_{α, β} with α, β > 0. Then we have
(22) |
where the {X_{i,j}} are IID random variables with CDF F, N_{< α >}^{(j)} and N_{< β >} have Sibuya distributions with respective parameters < α > and < β >, and all variables on the right-hand-side of (22) are mutually independent.
Let us consider again the stochastic representation of T~K_{m, n} given by (4). As discussed in Nadarajah et al. [4A. Di Crescenzo, "Some results on the proportional reversed hazards model", Stat. Probab. Lett., vol. 50, pp. 313-321.
[http://dx.doi.org/10.1016/S0167-7152(00)00127-9] ] (Nadarajah and Eljabri, [8T.J. Kozubowski, and K. Podgórski, "Transmuted distributions and random extrema", Stat. Probab. Lett., vol. 116, pp. 6-8.
[http://dx.doi.org/10.1016/j.spl.2016.04.001] ]), T can be thought of as a lifetime of a system consisting of n sub-systems connected in a series, where each sub-system is composed of m IID components with standard uniform lifetimes. If the lifetimes are non-negative and IID with the CDF F, then similar interpretation holds for the random variable Y obtained viz. (3) with T~K_{m, n}. The results presented above provide a similar interpretation for a general T~K_{α, β} as well as for a non-negative variable Y obtained viz. (3) with T~K_{α, β}, where the number of IID components is a random variable.
In connection with this interpretation of (4), one can instead assume that the n sub-systems are connected in parallel rather than in a series, while the IID standard uniform components {U_{i,j}} within each sub-system are connected in a series. In this case, the total lifetime of the system will admit the representation
(23) |
It is easy to see that in (23) has the same distribution as 1 - T, where T~K_{m, n}. More generally, let us consider the random variable , where T~K_{α, β}, which we shall call a dual of the latter, and denote its distribution by . Clearly,
(24) |
Remark 2.4. Interestingly, the class of distributions on the unit interval defined viz. (24) coincides with the class of distributions whose CDFs are given by the quantile functions corresponding to the Kumaraswamy CDF (2).
All the results for the Kumaraswamy and Kumaraswamy-generated distributions presented above have analogs in terms of the dual distributions (24) and the generalizations of F obtained viz.
(25) |
It turns out that they are obtained by swapping the minima with the maxima in the results for the Kumaraswamy and Kumaraswamy-generated distributions. We shall state them, without proofs, for distributions (25) since other results are obtained by setting F (x) = x. As before, we start with the case .
Proposition 2.6. Let F be a random variable with the CDF G defined by (25) with have Sibuya distributions with respective parameters and . Then
(26) |
where the {X_{i,j}} are IID random variables with CDF F, independent of the Sibuya variables.
An extension to the general case presented below can be derived from Proposition 2.3.
Proposition 2.7. Let Y be a random variable with the CDF G defined by (25) with α , β > 0. Then
(27) |
where the {X_{i,j}} are IID random variables with CDF F, independent of two independent Sibuya variables N_{< α >}^{(j)} and N_{< β >} with parameters < α > and < β >, respectively.
Remark 2.5. If Y in Propositions 2.6 and 2.7 has a dual Kumaraswamy distribution with the CDF (24), then the stochastic representations still apply with the {X_{i,j}} being IID standard uniform variables.
Remark 2.6. If the underlying variables are non-negative, our results have an interpretation in terms of series-parallel or parallel-series systems of IID components where the number of components can be random (and Sibuya distributed). For example, the quantity Y in Proposition 2.2 represents the lifetime of ra andom number N_{β} of sub-systems connected in parallel, where each sub-system i consists of a random number N_{α}^{(i)} of components connected in a series. Similarly, the quantity Y in Proposition 2.6 represents the lifetime of N_{β} sub-systems connected in a series, where each sub-system consists of a random number of components connected in parallel. The other results have analogous, albeit not as straightforward, interpretations.
Remark 2.7. As discussed above, the stochastic representations for Kumaraswamy and related distributions are built on min/max interpretations of distributions given by the survival function or the CDF F^{r} with non-negative r, where F is a “base” CDF. These two transformations are known as the proportional hazard and the proportional reversed hazard transforms (or models, see, e.g., Wang [13S. Nadarajah, G.M. Cordeiro, and E.M.M. Ortega, "General results for the Kumaraswamy-G distribution", J. Stat. Comput. Simul., vol. 82, pp. 951-979.
[http://dx.doi.org/10.1080/00949655.2011.562504] ] and Di Crescenzo [14S. Nadarajah, and S. Eljabri, "The Kumaraswamy GP distribution", J. Data Sci., vol. 11, pp. 739-766.]), since the hazard rates of distributions given by F and are proportional (as are the the reversed hazard rates of distributions given by F and F^{r}). It also is worth noting that these two generalizations of F are examples of the so-called distorted distributions, which in general are obtained through a transformation q(F) where q is a continuous and increasing bijective function on the unit interval onto itself (see, e.g., Navarro et al. [15M. Sibuya, "Generalized hypergeometric, digamma, and trigamma ditributions", Ann. Inst. Stat. Math., vol. 31, pp. 373-390.
[http://dx.doi.org/10.1007/BF02480295] ], Navarro [16J. Navarro, "Stochastic comparisons of generalized mixtures and coherent systems", Test, vol. 25, pp. 150-169.
[http://dx.doi.org/10.1007/s11749-015-0443-5] ], Navarro and Gomis [18J. Navarro, and M.C. Gomis, "Comparisons in the mean residual life order of coherent systems with identically distributed components", Appl. Stochastic Models Bus. Industry, vol. 32, no. 1, pp. 33-47.
[http://dx.doi.org/10.1002/asmb.2121] ] and references therein). Consequently, using this connection and the results for distorted distributions one can study preservation of reliability aging classes and related properties for Kumaraswamy-transformed distributions obtained viz. (1) or (25).
By means of Lemma 2.1, several other classes of generalized distributions based on a “parent” CDF F, which were introduced in recent years, can also be shown to arise from random maxima and minima of IID components with distribution F. For example, as shown by Kozubowski and Podgórski [18J. Navarro, and M.C. Gomis, "Comparisons in the mean residual life order of coherent systems with identically distributed components", Appl. Stochastic Models Bus. Industry, vol. 32, no. 1, pp. 33-47.
[http://dx.doi.org/10.1002/asmb.2121] ], if N has a Bernoulli distribution with parameter shifted up by one, so that its PGF is
(28) |
we obtain the so-called transmuted version of F, defined by
(29) |
Another example is provided by the so-called Marschall-Olkin generalized-F distributions, which were made popular by Marshall and Olkin [19F.J. Samaniego, System Signatures and their Applications in Engineering Reliability., Springer: New York, .
[http://dx.doi.org/10.1007/978-0-387-71797-5] ] who introduced them through minima and maxima of IID components (following distribution F) with a geometric number of terms N.
Remark 2.8. The CDF (29) is a generalized mixture of F and F^{2} (where the weights are allowed to be negative, see, e.g., Navarro, [16J. Navarro, "Stochastic comparisons of generalized mixtures and coherent systems", Test, vol. 25, pp. 150-169.
[http://dx.doi.org/10.1007/s11749-015-0443-5] ]) as well as usual mixture of F^{2} and 1 - (1 - F)^{2} (with non-negative weights (1 - α)/2 and (1 + α)/2, respectively). The latter representation as a mixed system (e.g., Samaniego, [20S. Wang, "Insurance pricing and increased limits ratemaking by proportional hazards transforms", Insur. Math. Econ., vol. 17, pp. 43-54.
[http://dx.doi.org/10.1016/0167-6687(95)00010-P] ]) allows an alternative interpretation of (29) in terms of lifetimes of series/parallel systems with IID components.
Not applicable.
The authors declare no conflict of interest, financial or otherwise.
We thank two referees for their remarks and additional references. Kozubowski’s research was partially supported by the European Union’s Seventh Framework Programme for Research, Technological Development and Demonstration under grant agreement no 318984 - RARE. Podgórski’s research was supported by Swedish Research Council Grant Dnr: 2013-5180.
[1] | G. Aryal, and Q. Zhang, "Characterizations of Kumaraswamy Laplace distribution with applications", Econ. Qual. Control, vol. 31, pp. 59-70. [http://dx.doi.org/10.1515/eqc-2016-0009] |
[2] | G.M. Cordeiro, and M. de Castro, "A new family of generalized distributions", J. Stat. Comput. Simul., vol. 81, pp. 883-898. [http://dx.doi.org/10.1080/00949650903530745] |
[3] | M.A.R. de Pascoa, E.M.M. Ortega, and G.M. Cordeiro, "The Kumaraswamy generalized gamma distribution with application in survival analysis", Stat. Methodol., vol. 8, pp. 411-433. [http://dx.doi.org/10.1016/j.stamet.2011.04.001] |
[4] | A. Di Crescenzo, "Some results on the proportional reversed hazards model", Stat. Probab. Lett., vol. 50, pp. 313-321. [http://dx.doi.org/10.1016/S0167-7152(00)00127-9] |
[5] | N. Eugene, C. Lee, and F. Famoye, "The beta-normal distribution and its applications", Commun. Stat. Theory Methods, vol. 31, pp. 497-512. [http://dx.doi.org/10.1081/STA-120003130] |
[6] | M.C. Jones, "Kumaraswamy distribution: A beta-type distribution with some tractability advantages", Statist. Method., vol. 6, pp. 70-81. [http://dx.doi.org/10.1016/j.stamet.2008.04.001] |
[7] | T.J. Kozubowski, and K. Podgórski, "Certain bivariate distributions and random processes connected with maxima and minima", Working Papers in Statistics 2016:9, Department of Statistics, School of Economics and Management, Lund University, 2016a. [Published online in Extremes, 2018, DOI 10.1007/s10687-018-0311-2]. |
[8] | T.J. Kozubowski, and K. Podgórski, "Transmuted distributions and random extrema", Stat. Probab. Lett., vol. 116, pp. 6-8. [http://dx.doi.org/10.1016/j.spl.2016.04.001] |
[9] | P. Kumaraswamy, "A generalized probability density function for double bounded random processes", J. Hydrol., vol. 46, pp. 79-88. [http://dx.doi.org/10.1016/0022-1694(80)90036-0] |
[10] | A.J. Lemonte, W. Barreto-Souza, and G.M. Cordeiro, "The exponentiated Kumaraswamy distribution and its log-transform", Braz. J. Probab. Stat., vol. 27, pp. 31-53. [http://dx.doi.org/10.1214/11-BJPS149] |
[11] | V. Mameli, "Kumaraswamy skew-normal distribution", Stat. Probab. Lett., vol. 104, pp. 75-81. [http://dx.doi.org/10.1016/j.spl.2015.04.031] |
[12] | A.W. Marshall, and I. Olkin, A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families., vol. 84, Biometrika, . |
[13] | S. Nadarajah, G.M. Cordeiro, and E.M.M. Ortega, "General results for the Kumaraswamy-G distribution", J. Stat. Comput. Simul., vol. 82, pp. 951-979. [http://dx.doi.org/10.1080/00949655.2011.562504] |
[14] | S. Nadarajah, and S. Eljabri, "The Kumaraswamy GP distribution", J. Data Sci., vol. 11, pp. 739-766. |
[15] | M. Sibuya, "Generalized hypergeometric, digamma, and trigamma ditributions", Ann. Inst. Stat. Math., vol. 31, pp. 373-390. [http://dx.doi.org/10.1007/BF02480295] |
[16] | J. Navarro, "Stochastic comparisons of generalized mixtures and coherent systems", Test, vol. 25, pp. 150-169. [http://dx.doi.org/10.1007/s11749-015-0443-5] |
[17] | J. Navarro, Y. del Águila, M.A. Sordo, and A. Suárez-Llorens, "Preservation of reliability classes under the formation of coherent systems", Appl. Stochastic Models Bus. Industry, vol. 30, pp. 444-454. [http://dx.doi.org/10.1002/asmb.1985] |
[18] | J. Navarro, and M.C. Gomis, "Comparisons in the mean residual life order of coherent systems with identically distributed components", Appl. Stochastic Models Bus. Industry, vol. 32, no. 1, pp. 33-47. [http://dx.doi.org/10.1002/asmb.2121] |
[19] | F.J. Samaniego, System Signatures and their Applications in Engineering Reliability., Springer: New York, . [http://dx.doi.org/10.1007/978-0-387-71797-5] |
[20] | S. Wang, "Insurance pricing and increased limits ratemaking by proportional hazards transforms", Insur. Math. Econ., vol. 17, pp. 43-54. [http://dx.doi.org/10.1016/0167-6687(95)00010-P] |