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Twisted Wang Transform Distribution

Frank Xuyan Wang

Author

Frank Xuyan Wang

Title

Twisted Wang Transform Distribution

Description

A contribution to the 2019 China International Conference on Insurance and Risk Management.

Twisted Wang Transform Distribution

Frank Xuyan Wang

Validus Research Inc., 187 King Street South Unit 201, Waterloo, Ontario, Canada N2J 1R1

The twisted wang transform distribution family, defined as the composition of parameter shifted inverse CDF function with CDF function, are found to be most suitable for matching low shape factor distributions. Among them the best form for matching a hard empirical distribution from reinsurance portfolio loss, is the Exponential Fractional Extra Power 0 Distribution in (0,1) with CDF :

-q

(1-

)

The simplest form of this family is the Transformed Hyperboloic Tangent Distribution with CDF:

1+q

which has analytical formulas for the moments. The twisted wang transform distribution family are compared and confirmed to be superior than all other known distribution families.

9:33 AM 8/2/2018

Introduction

Background

In our previous study [A], it is found that a distribution with high CV is hard to simulate with fixed simulation iterations. And such a distribution in extreme case is represented by a bi-valued distribution. To interpolate a bi-valued distribution by smooth function, the hyperbolic tangent function

-x

is considered. With transformation to re-lay the range to

(0,1)

, and shift the central point, we get:

1+q

-2x

. When compositing a transformation of the domain from

(-∞,∞)

also to (0,1):

Log

1-x

, we get

1+q



1-x



. Considering that the power function is

(0,1)(0,1)

and

(0,∞)(0,∞),

we can add different power to

and etc. to add asymmetry to the distribution(for experiment on power function, see [B]). The final transformation is

1+q

:(0,1)(0,1).

(

)

When

k=l&&m=1

1+q

,we name it Transformed Hyperbolic Tangent (THT) distribution, whose moments can be expressed explicitly by hypergeometric function. When

k=1

and with some reparametrization,

1+q

(1-x)

,we name it Alternative Transformed Hyperbolic Tangent (ATHT) distribution.When

k≠l&&m=1

1+q

, we name it Generalized Transformed Hyperbolic Tangent (GTHT) distribution, and when

k≠l&&m≠1

, with reparametrization by

1+q

(1-

)

,we name it Generalized Transformed Hyperbolic Tangent Extended (GTHTE) distribution: even though their moments cannot be explicitly expressed by known functions, we can use numerical integration to get them.

In [B], a shape factor (SF) is defined using kurtosis divided by squared skewness, which can measure the intensity of the asymmetry and steepness of the distribution PDF when the SF is approaching 1; where it is also observed that the finite interval distribution generally work better than infinite range distribution for fitting small SF distributions, thus point to study for distributions just in the unit interval. Although the so created THT in our first round of study cannot fit such cases, but GTHT and GTHTE tested work better.

The construction of GTHTE of compositing CDF with Quantile function also largely fall in the general Wang Transformation schema [C], while the most general Wang Transformation can use different type of distribution for CDF and Quantiles, we just use the CDF and Quantile function of the same distribution family but with different parameters. Also note that the positive real numbers domain is scale invariant, and the whole real number domain is scale and shift invariant which is a major twisting in the Wang Transformation. We use the parameter deviation of CDF and Quantile in place of a direct central shift. The power function transformation is also positive real number domain invariant and the unit interval domain invariant, we use these twist too to add asymmetry to the distribution.

Apply similar idea to the most natural form of transformation from

(-∞,∞)(0,1):

-x



-x



,1-



, the first gives power function, the second gives GTHTE, and the third gives function below:

1-

(1-

)



(

)

This is a generalized Kumaraswamy distribution (GK), called EKw in [D].

Apply similar idea to the most natural form of transformation from

(0,∞)(0,1):1-

-x



1+x

, the first still gives generalized Kumaraswamy distribution, the second gives function below:

1+(q-1)

(

)

We name it the Twisted Wang Transform (TWT) distribution, even though all the GTHTE type and generalized Kumaraswamy distribution belongs to this category, but only (3) do not have alternative names.

Research Objective

Our research objective is to find probability distribution that can fit distributions with small shape factors. The Quantile function of loss for a given probability in the reinsurance industry, inverse function of the CDF, is called VaR; and the average quantile above a given loss is called TVaR. The TVaR curve for all the probabilities is called the EP curve. The fitness of the distribution is judged by the matching of the EP curves. For risk management purpose, only the higher end of the EP curve is of interest. We will use three points of the EP curve, TVaR of probability 0.96, 0.99, and 0.996, as our main matching objective. A loss reference portfolio for the North American Tornado Hail peril (NATH) is used as a test case that can be used to compare the well known probability distribution families, as well as our tailor made probability distributions, to see what kind of distribution functions is better to describe the small SF distributions. For the top distribution thus found, conduct further research of its EP curve change tendency with respect to its parameters changes.

Previous Results

The NATH has SF 1.83 and

{Mean,StandardDeviation,Skewness,Kurtosis}={7418611.10904006,9517336.93024634,5.99378199789956,65.8901734355745}

. Its TVaR for the three probabilities is

{41113929.8424838,68867612.8345741,98579409.238445}

. Even though only the mean, standard deviation and the three TVaRs are directly used in reinsurance pricing and equity allocating, the study in [B] shows that the skewness is the most important factor that affect the EP curve shape, the kurtosis and the shape factor follows. Optimization with the additional constraints of matching these characteristics yield better results than just fit the three TVaRs. Distribution with finite range of values fit better, as well as distribution with infinite range values but are truncated to finite interval. For example, the truncated generalized gamma (TGG) distribution has error of 3.37%, and the largest deviation of the whole spectrum of TVaR in practise is only 12.5%.

Table

NATH 0.96 0.99 0.996 TVaR Fit Error

Distribution	Fit Error
GTHTE	1.52
TGG	3.37
GTHT	3.79
TLG	4.4
NIG	4.6
GB2	5.0
GB1	5.0
GH	5.0
GnH	5.4
DoubleLog	5.59
FD4	5.8
Kumaraswamy	6.0
EIDL	6.48
GG	6.5
Meixner	7.3
LogGamma	7.6
VG	8.0
Beta	8.7
EIG	9.4

Fit error is the maximum of the absolute of the ratio of fitted TVaR to input TVaR minus 1.

In table 1, the TLG is truncated Log Gamma distribution, GH is the generalize hyperbolic distribution, DoubleLog distribution has CDF

∫

(-Log[1-t])

(-Log[t])

t

∫

(-Log[1-x])

(-Log[x])

x

. EIDL is probability distribution in

(1,∞)

with PDF proportional to

(Log[x])

-Log1.-



-c

.All other distributions are well known and are studied in [B].

Method

Numerical integration, optimization [E], and root finding are used to calculate the VaR and TVaR for given distribution; the calculated numbers can be regarded as the theoretical upper limit that simulation can reach, where the simulated numbers are what are seen in practise. If quantile function

Q(t)

can be solved in explicit form, we use the following formula for TVaR:

∫

Q(t)t

1-q

(

)

When only the PDF is known or the CDF is hard to find inverse, we use

FindRoot

to calculate VaR, and use the following formula for TVaR:

∫

VaR

xPDF(x)x

1-q

(

)

Numerical experiments confirm that the skewness and kurtosis are intrinsically correlated with EP curve shape. Optimize by matching skewness and kurtosis yield better TVaR match than doing the TVaR match directly. Combining the two objectives together yield even better results.The seems redundant or repetitive objectives actually help with finding the optimum. This is also observed in [F-G]. The combination will be our objective function for optimization.

Dimension reduction method is used to overcome the numerical optimization difficult, starting from distribution with fewer parameters, gradually add more parameters in. We use Mathematica

Table

function, or one dimensional plot to find the initial value approximate position or region, and use Mathematica

FoldList

function to do one dimensional iterative search when fix a parameter, and change that parameter gradually using previously find solution as consequent starting point.

Local minimization

FindMinimum

or one dimensional line search by

FoldList

is faster, so it is good for find parameters approximate regions, i.e., should it be large or very small. But the local minimization seems never find the true solution, and at times give the wrong conclusion that fit is impossible, if the true solution is outside of the conceivable and tested ranges. So the slower global minimization function

NMinimize

and iterative through

NestWhile

will be always used.

Results

Through extensive numerical experiment, the best fit of the broad twisted Wang transform distribution family is in table 2:

Table

.Fit power of TWT families of distributions

Distribution	Fit error	Maximum fit error
GTHTE	1.52	16.15
PTHT	1.77	5.11
THT	2.09	8.55
GTHT	2.52	11.89
APTHT	2.55	11.29
TGG	3.37	12.54
THT	3.81	5.58
PK	4.95	46.99
GK	5.10	48.04
Kumaraswamy	6.09	41.09
GG	6.52	42.76
Beta	8.61	14.23
TWT	15.05	25.25

Fit error is the maximum of the absolute of the ratio of fitted TVaR to input TVaR minus 1 for probability 0.96 to 0.996.

The maximum fit error in table 2 is the maximum of the absolute of the ratio of fitted TVaR to input TVaR minus 1 for probability 0.5 to 0.99999; it is usually attained near probability 0.9999 or 0.99999.

In table 2, the Beta, GG, TGG, Kumaraswamy, and TGG are for comparison, where Kumaraswamy used only 4 parameters, and TGG used only 5 parameters. The PTHT is the productive transformed hyperbolic distribution, with CDF

1+q

(1-

)(1-

)

; the APTHT is the alternative productive transformed hyperbolic distribution, with CDF

1+q

(1-

)

(1-x)

. They are alteration of the GTHTE distribution by a similar productive factor. An additional extension of the Kumaraswamy distribution (KK), called KwKw in [D], with CDF

1-

(1-

)



, is not included since we cannot find a better fit than GK from it, perhaps due to numerical optimization difficulties with KK. The PK is the productive Kumaraswamy distribution, with CDF

1-

(1-

)

1-

(1-

)



, similar in spirit to PTHT.

The GTHTE and TWT are special case of the twisted Wang transform extended distribution TWTE, with CDF

(1-

)

, since

1+(q-1)

l-k

. The alternative generalization for GTHT, ATHT with CDF

1+q

(1-x)

would be another choice in place of GTHT, only that GTHT have a special case of THT that have analytical representation for all moments, while the ATHT do not:

M[h]=

q-Hypergeometric2F11,

h+k

-1+q



-1+q

=1-

hHypergeometric2F11,

h+k

,2+

-1+q



(h+k)q

HurwitzLerchPhi1-

,1,



q-1

(

)

There are three basic types of function forms: power, logarithm, and exponential. The exponential is widely used in distributions, such as the TGG, EIG,NIG, and GH. We also tried the logarithms in DoubleLog and EIDL. They seems do not stand out than purely power function embedded in fractional construction. The exponential form will be studed further later, we will focus on power function at first, and comparing them thereafter. For the three types of power construction, or the TWTE types of distribution, the GTHTE type performs best, followed by Kumaraswamy type. For GTHTE type, the alternative form of factor

(1-x)

, such as in ATHT, APTHT, seems do not work as well as the form

(1-

)

. To test or confirm this observation, other distributions ATHT(alternative THT), PTHT2(productive THT with 2 additional terms), THTE(THT extended), THTE1 (in table 3) are formed.

Table

Probability distributions to study

Distribution	CDF
GTHTE	1 1+q m (1- k x ) l x
PTHT	1 1+q (1- k x )(1- m x ) l x
THT	1 1+q 1- k x k x
GTHT	1 1+q 1- k x l x
APTHT	1 1+q (1- k x ) m (1-x) l x
KK	1- m 1- l 1- q (1- k x )  
PK	1- q (1- k x ) 1- l (1- m x ) 
GK	l 1- q (1- k x ) 
Kumaraswamy	1- q (1- k x )
TWT	m q k x 1+(q-1) l x
TWTE	1 o n x +q m (1- k x ) l x
TWTE0	1 n 1+q m (1- k x ) l x
TWTE1	1 n x +q m (1- k x ) l x
ATHT	1 1+q k (1-x) l x
PTHT2	1 1+q (1- k x )(1- m x )(1- n x ) l x
THTE	1 m 1+ l q 1- k x k x
THTE1	1 1+ l q 1- k x k x
APTHTE0	1 n 1+q (1- k x ) m (1-x) l x
APK	1- q (1- k x )  l x
EF	-q 1- k x l x E
EFP	m x -q 1- k x l x E
ELP	-q k (-Log[x]) E
ERLP	q Log[1- k x ] E
ERLPE	- q l -Log[1- k x ] E
THTP	l x 1+q 1- k x k x
* TWTE1	n x 1+q m (1- k x ) l x
GTHTP	m x 1+q 1- k x l x
PTHT2P	o x 1+q (1- k x )(1- m x )(1- n x ) l x

In table 3, the APK is the alternative productive Kumaraswamy distribution, to check the effectivity of productive CDF formed from two CDF with one the power function

. The EF is the exponential fractional distribution, EFP is the exponential fractional power distribution, ELP is the exponential logarithmic power distribution, ERLP is the exponential reciprocal logarithmic power distribution, ERLPE is the exponential reciprocal logarithmic power extended distribution: all these distributions are testing exponential form functions, and the GTHT can be regarded as the first order expansion of the EF distribution. The ALIG is the alternative log InverseGaussian distribution: 1-Exp[-InverseGaussian]; whose PDF has a main factor like the ERLP CDF. The TWTE1* is a reformulation of TWTE1 distribution with opposite signs of allowable region of the parameter

We find that even if starting from the best solution GTHTE get, the TWTE1 cannot find a better solution. If search afresh by its own, TWTE1 will get much worse solution. So either the addition of the

post numerical difficult for optimization, or that term is not compatible with small SF distributions. The lesson from this is that a broader distribution family may not be as good as a narrower distribution family for distribution fit in practise. The new round of study starting from best solution from THT, GTHTE and etc. yield results in table 4:

Table

.Fit power of TWT families of distributions more runs

Distribution	Fit error	Maximum fit error
EFP	1.412	8.33
TWTE0	1.515	9.51
APTHT	1.519	15.07
GTHTE	1.523	8.59
GTHTP	1.523	13.24
GTHTE	1.523	16.15
TWTE1	1.525	16.10
PTHT2P	1.550	6.49
* TWTE1	1.558	6.12
THTP	1.620	5.16
GTHT	1.644	7.72
PTHT2	1.651	4.61
PTHT	1.767	5.11
EF	1.785	16.65
THT	2.086	8.55
THTP	2.193	6.38
THTE	2.237	9.37
THTE1	2.240	10.31
THTE	2.326	7.95
PTHT2	2.555	10.70
ERLPE	2.653	12.34
GTHTE	3.175	5.35
ATHT	3.271	30.59
TGG	3.375	12.54
APTHTE0	3.525	35.36
APK	3.548	9.47
THTP	3.727	5.92
ELP	3.788	37.42
THT	3.814	5.58

Fit error is the maximum of the absolute of the ratio of fitted TVaR to input TVaR minus 1 for probability 0.96 to 0.996.

It is a commonly observed phenomena that a special case of a broad distribution which is the optimum is usually a repellant detractor: most optimization starting from initial value deviated from the optimum will not converge to it but will diverge away from it. The TWTE1 vs GTHTE, GTHT vs THT, and NIG vs GH are such cases. The THTE, THTE1 have bad performance than THT. The GTHTE result start from best PTHT2 may be a rare exception, it actually improved than optimization from GTHTE alone. The steps used in the numerical optimizations determined the results to a large extent. But in general, the simpler the CDF form, the easier the optimization process, and the better the results.

Another observation is that a distribution fit starting from another related distribution best fit or starting from a simplified objective best fit will usually give sub-optimum solution, not as good as one starting without using any of such hints. ATHT vs THT is such an example.

The vast numerical experiment with various TWT types of distribution seems suggest that the fit capability limit of them is about 1.5% residual error. It is an indication that types of distribution with function form other than fraction and power need to be considered. Further attempt of the TWT types distribution as well as distributions with exponential or logarithmic function factor are inserted in table 3 and table 4.

THT Distribution study

The command:

Tooltip[ContourPlot[{sktht[q,k]4.99378199789956`,sktht[q,k]5.99378199789956`,sktht[q,k]6.99378199789956`,kttht[q,k]60.8901734355745`,kttht[q,k]65.8901734355745`,kttht[q,k]70.8901734355745`},{k,2.0,3.0},{q,0.0001,0.0009},ContoursAutomatic,ContourLabelsNone,ContourStyle{Red,Orange,Pink,Blue,Purple,Magenta},FrameLabelAutomatic,ImageSize1200,FrameTrue,FrameTicksAll,GridLinesAutomatic,PlotLabel{sktht,kttht},PlotLegends{4.99,5.99,6.99,60.89,65.89,70.89},PlotRange->Automatic],Dynamic[MousePosition["Graphics"]]]

plot the THT distribution skewness and kurtosis, Figure 1. This plot shows the parameters change tendency, to keep the same shape characteristics of skewness and kurtosis,an increasing

need a decreasing

. This observation for THT and TWTE family of distribution in general, is critical for an iterative approach to get better solution, using each solution as the boundary constraints for the next run.

/23/18 09:24:14 Out[255]=

	4.99
	5.99
	6.99
	60.89
	65.89
	70.89

Figure

.Contour Plot of THT Skewness and Kurtosis.

Discussion

When numerical experiment find some emerging pattern, it is desirable to make further study with symbolic calculation, in case analytical formula is available, which can confirm the numerical results, and is also a better form for representing and sharing the knowledge gain from numerical experiments.

Even though a formula approach may not find better solution than numerical approach, the speed improvement will be significant. The THT distribution is such an example, using numerical integration took 1913.64 seconds, while using the following formulas for VAR, TVAR and etc. took 3.36 seconds, more than 500 times faster:

quantiletht=

(1+(1/x-1)/q)

;tvartht[q_?NumericQ,k_?NumericQ,x_?NumericQ]:=

kqHypergeometric2F11,2,2+

,1-q-q

-1+k



1+(-1+q)x



Hypergeometric2F11,2,2+

,x-qx

(1+k)(1-x)

(

)

meantht=

q-Hypergeometric2F11,

,1+

-1+q



-1+q

;sdtht=

q-Hypergeometric2F11,

,1+

-1+q



+(-1+q)q-Hypergeometric2F11,

2+k

-1+q



(-1+q)

(

)

skewtht=

Abs[-1+q]

q-Hypergeometric2F11,

,1+

-1+q



-3(-1+q)q-Hypergeometric2F11,

,1+

-1+q

q-Hypergeometric2F11,

2+k

-1+q

+

(-1+q)

q-Hypergeometric2F11,

3+k

-1+q



(-1+q)

3/2

q-Hypergeometric2F11,

,1+

-1+q



+(-1+q)q-Hypergeometric2F11,

2+k

-1+q



(

)

kurttht=-3

q-Hypergeometric2F11,

,1+

-1+q



+6(-1+q)

q-Hypergeometric2F11,

,1+

-1+q



q-Hypergeometric2F11,

2+k

-1+q

-4

(-1+q)

q-Hypergeometric2F11,

,1+

-1+q

q-Hypergeometric2F11,

3+k

-1+q

+

(-1+q)

q-Hypergeometric2F11,

4+k

-1+q



q-Hypergeometric2F11,

,1+

-1+q



-(-1+q)q-Hypergeometric2F11,

2+k

-1+q



(

)

Should the numerical optimization find the true optimum, the order we see in table 2 would be different. We know GTHT must be better than THT, but the optimization difficult gives the opposite results.

Further Study of the TWTE Type distributions

After eliminated using Logarithm type function as the ingredients, we focus on using power function and fractional functions, as well as exponential function as a factor, utilizing the comparison to GTHT vis-à-vis EF where the first can be considered as a first order expansion of the second, and the second as the exponential counterpart of the first form, we made an extensive search of all such forms of distributions. To overcome the limitation of nested interval or region method of the optimization, we use random seed NelderMead simplex method for large number of seeds. The optimization results is still very much dependant on the variable ranges used, too wide or too narrow range will all gives incorrect results. But all the TWTE type distributions seems have similar ranges of the comparable parameters when attend the optimum, so the correct range find of one distribution can be used as a reference for the other distribution. If a wider distribution cannot find a better solution, one possible reason is incorrect variable range. The results is present in table 5.

Table 5.Final order of TWT families of distributions fit power for NATH

Distribution Symbol	Distribution Name	CDF Formula	Minimum Maximum fit error
EFEP	Exponential Fractional Extra Power Distribution	n x -q m (1- k x ) l x E	2.6618
EFEP0	Exponential Fractional Extra Power 0 Distribution	-q m (1- k x ) l x E	2.77629
PTHT	Productive Transformed Hyperbolic Tangent Distribution	1 1+q (1- k x )(1- m x ) l x	3.14909
PTHTP	Productive Transformed Hyperbolic Tangent Power Distribution	n x 1+q (1- k x )(1- m x ) l x	3.18394
PTHT2P	Productive Transformed Hyperbolic Tangent 2 Power Distribution	o x 1+q (1- k x )(1- m x )(1- n x ) l x	3.1874
EFPP0	Exponential Fractional Productive Power 0 Distribution	-q (1- k x )(1- m x ) l x E	3.26582
* TWTE1	Twisted Wang Transform Extended 1 Distribution	n x 1+q m (1- k x ) l x	3.31532
EFPP	Exponential Fractional Productive Power Distribution	n x -q (1- k x )(1- m x ) l x E	3.31686
PTHT2	Productive Transformed Hyperbolic Tangent 2 Distribution	1 1+q (1- k x )(1- m x )(1- n x ) l x	3.40758
GTHTE	Generalized Transformed Hyperboloic Tangent Extended Distribution	1 1+q m (1- k x ) l x	3.4145
THTE	Transformed Hyperbolic Tangent Extended Distribution	1 m 1+q l 1- k x k x	3.88511
THTE1	Transformed Hyperbolic Tangent Extended 1 Distribution	1 1+q l 1- k x k x	3.91962
EFE1	Exponential Fractional Extended 1 Distribution	-q l 1- k x k x E	4.00827
GTHT	Generalized Transformed Hyperboloic Tangent Distribution	1 1+q 1- k x l x	4.05377
EF	Exponential Fractional Distribution	-q 1- k x l x E	4.11104
GTHTP	Generalized Transformed Hyperboloic Tangent Power Distribution	m x 1+q 1- k x l x	4.11306
EFP	Exponential Fractional Power Distribution	m x -q 1- k x l x E	4.13766
EF1	Exponential Fractional 1 Distribution	-q 1- k x k x E	4.22206
EF1P	Exponential Fractional 1 Power Distribution	l x -q 1- k x k x E	4.22221
THTP	Transformed Hyperbolic Tangent Power Distribution	l x 1+q 1- k x k x	4.27351
THT	Transformed Hyperboloic Tangent Distribution	1 1+q 1- k x k x	4.27352

Minimum maximum fit error is the minimum of the maximum of the absolute of the deviation of fitted TVaR to input TVaR in percentage for probability 0.5 to 0.99999 of the best fit.

From table 5 we see the best distribution for NATH is EFEP with an error of 2.66%, while the simplest form distribution THT has an error of 4.27%. We also see that from the fractional form to the exponential form will sometimes improve the fit, such as from THT to EF1, THTP to EF1P, GTHTE to EFEP0, and TWTE1* to EFEP, but in more than half of the other times become worse, such as from GTHTP to EFP, GTHT to EF, THTE1 to EFE1, PTHT to EFPP0, and PTHT2P to EFPP. Similarly, add the extra power term may improve the fit, such as from EFEP0 to EFEP, GTHTE to TWTE1*, and THT to THTP. But more often will make the fit deteriorated, such as from PTHT to PTHTP, EFPP0 to EFPP, GTHT to GTHTP, EF to EFP, and EF1 to EF1P. Here maybe the difficult with the optimization bring about by the additional parameter is a contributing factor.

The best distribution whose quantile function can be expressed through simple function and do not need a numerical solution is THTE with an error of 3.88%. These class include THTE, THTE1, EFE1, EF1, and THT. They may be utilized for numerical simulation which are large and time sensitive.

The k parameter in THTE THTE1 EFE1 EF1 EF1P THTP and THT and l parameter in other TWT distributions are somewhat similar to the alpha parameter for Pareto distribution. In table 2.3 of [H], we see for NATH that parameter should be in range 1.8-2.2. Our best fit parameters for NATH are in table 6.

Table 6.

Best fit to NATH parameter k or l

Distribution	parameter k or l
EFEP	2.648
EFEP0	2.622
PTHT	2.327
PTHTP	2.314
PTHT2P	2.416
EFPP0	2.539
* TWTE1	2.377
EFPP	2.553
PTHT2	2.453
GTHTE	2.414
THTE	2.029
THTE1	2.000
EFE1	2.236
GTHT	2.073
EF	2.343
GTHTP	2.166
EFP	2.300
EF1	2.459
EF1P	2.459
THTP	2.510
THT	2.326

k for THTE THTE1 EFE1 EF1 EF1P THTP and THT and l for other distributions

These k or l parameters are clustered in the range 2.0-2.6. It promote using an universal parameter per each peril in practise to simplify the distribution fitting.

Appendix A

The Mathematica code of the tool for distribution study is included here. Generic program that can be used for any type of distribution with analytic CDF is achieved through variable numbers of arguments function. The difficult with accessing the first argument is solved by a Block indirection. The mixed up of symbolic derivation and numerical calculation and hold attribute is solved by Module structure and a two steps approach:

/15/18 15:58:46 In[58]:=

/15/18 16:00:07 In[61]:=

definedistribution"aptht",

1+q

(1-

)

(1-x)



/15/18 16:00:33 In[62]:=

defineobjectives["aptht"]

Appendix B

An example code for the random seeds search optimization in the THTE1 case:

/27/18 13:39:06 In[5]:=

biSection[func_,init_:1.0,eps_:10.^-6]/;NumericQ[init]&&Positive[eps]:=Module[{c=0.,d=init},While[func[d]<0.,c=d;d=(d+1.)/2.];While[d-c>eps,With[{e=(c+d)/2.},If[func[e]<0.,c=e,d=e]]];(c+d)/2.]

/27/18 13:39:18 In[6]:=

definedistribution"thte1",

1+q



/27/18 13:39:24 In[7]:=

defineobjectives["thte1"]

/27/18 13:39:31 In[8]:=

defineobjectives35[name_]:=Block[{},Symbol["objective3"<>name][pp___?NumericQ]:=Block[{vars,tvar,stats,M,std,S,K,scal,shif,ep},vars=biSection[Function[x,(Symbol["cdf"<>name][pp,x]//Evaluate)-#]]&/@{0.9,0.96,0.98,0.99,0.996,0.998,0.9998,0.9999,0.99999};tvar=NIntegrate[Symbol["pdf"<>name][pp,x]*x//Evaluate,{x,#[[2]],1.}]/(1.-#[[1]])&/@Transpose[{{0.9,0.96,0.98,0.99,0.996,0.998,0.9998,0.9999,0.99999},vars}];stats={Symbol["mean"<>name][pp],Symbol["sd"<>name][pp],Symbol["sk"<>name][pp],Symbol["kt"<>name][pp]}//Evaluate;{M,std,S,K}={7418611.10904006,9517336.93024634,5.99378199789956,65.8901734355745};{scal,shif}={std/stats[[2]],M-stats[[1]]std/stats[[2]]};ep=shif+scaltvar;Max[Max[Abs[ep/{28445066.802226,41113929.8424838,52880882.2899095,68867612.8345741,98579409.238445,123436136.002865,170402622.81885,177904161.5761,203519335.497}-1.0]],Max[Abs[stats[[3;;4]]/{S,K}-1.0]]]];Symbol["objective5"<>name][pp___?NumericQ]:=Block[{init,vars,tvar,stats,M,std,S,K,scal,shif,ep},init=x/.(FindMinimum[Abs[(Symbol["cdf"<>name][pp,x]//Evaluate)-0.9],{x,0.001,0.999},MaxIterations5000][[2]]);vars=FoldList[x/.FindRoot[(Symbol["cdf"<>name][pp,x]//Evaluate)#2,{x,#1},MaxIterations5000]&,init,{0.9,0.96,0.98,0.99,0.996,0.998,0.9998,0.9999,0.99999}]//Rest;tvar=NIntegrate[Symbol["pdf"<>name][pp,x]*x//Evaluate,{x,#[[2]],1.}]/(1.-#[[1]])&/@Transpose[{{0.9,0.96,0.98,0.99,0.996,0.998,0.9998,0.9999,0.99999},vars}];stats={Symbol["mean"<>name][pp],Symbol["sd"<>name][pp],Symbol["sk"<>name][pp],Symbol["kt"<>name][pp]}//Evaluate;{M,std,S,K}={7418611.10904006,9517336.93024634,5.99378199789956,65.8901734355745};{scal,shif}={std/stats[[2]],M-stats[[1]]std/stats[[2]]};ep=shif+scaltvar;Max[Max[Abs[ep/{28445066.802226,41113929.8424838,52880882.2899095,68867612.8345741,98579409.238445,123436136.002865,170402622.81885,177904161.5761,203519335.497}-1.0]],Max[Abs[stats[[3;;4]]/{S,K}-1.0]]]];]

/27/18 13:39:36 In[9]:=

defineobjectives35["thte1"]

/27/18 13:39:56 In[10]:=

{sol=NMinimize[{objective5thte1[q,k,l],0.0005>q>0.,k>2.,l>1.},{q,k,l},MaxIterations5000,Method{"NelderMead","RandomSeed"#,"ShrinkRatio"0.95,"ContractRatio"0.95,"ReflectRatio"2.0}],objective1thte1[q,k,l]/.sol[[2]],thte1epcurve[q,k,l]/.sol[[2]]}&/@Range[100]

/28/18 12:28:17 In[11]:=

SortBy[%,{#[[2]],#[[1,1]],}&]//TableForm[#,TableDepth1]&

/28/18 12:28:19 In[12]:=

SortBy[%,{#[[1,1]],#[[2]]}&]//TableForm[#,TableDepth1]&

Acknowledgments

This research is supported by Validus Research Inc. The author thanks his colleagues for helpful suggestions and feedbacks.

References

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Cite this as: Frank Xuyan Wang, "Twisted Wang Transform Distribution" from the Notebook Archive (2022), https://notebookarchive.org/2022-02-46daciv