Reappraisal on the estimation of parameters of the hybrid lognormal distribution
Shigeru KUMAZAWA
Former JAERI (Predecessor of JAEA)
ABSTRACTThe hybrid log-normal (HLN) distribution is the probability distribution of positive variates X that thetransformation hyb(ρX) = ρX + ln(ρX) are normally distributed with the mean E[hyb(ρX)] and the varianceV[hyb(ρX)]. The HLN distribution model has been applied to various data of biology and social statistics
as well as radiological statistics with the multiple linear regression: normal rank zi vs variables (xi, ln xi) ofdescending ordered data. The paper presents a single regression method of the HLN distribution modelusing the EXCEL functions LINEST and SOLVER, to maximize the R-squared with respect to ρ (> 0),
including the calculation of standard error. A similar approach is feasible to expand the single regression
method of data plotted on the hybrid-hybrid section paper as a comprehensive section paper, whichhybridizes four popular section papers (linear-linear, linear-log, log-log and log-linear) with five additional
new section papers (linear-hybrid, hybrid-log, log-hybrid, hybrid-linear and hybrid-hybrid) that serve toconnect the conventional section papers smoothly.
Keywords: hybrid lognormal distribution, simple regression, hybrid-hybrid section paper.
INTRODUCTION
The Log-normal distribution was introduced by Galton and McAlister in 1879 to provide the replacementof normal distributions that appeared the skewed distribution in vital and social statistics. Since 1965 thelog-normal distribution has been applied to interpret the characteristics of the distribution of annual doses
incurred by workers under the regulatory control (Gale, 1965). In 1977 the United Nations ScientificCommittee on the Effects of Atomic Radiation (UNSCEAR) reported the comprehensive results of thelognormal analysis of occupational annual dose statistics over world-wide nations, which was essential forestablishing the system of dose limitation in the 1977 Recommendations of the International Commission
on Radiological Protection (ICRP). The statistics of occupational annual doses under the regulatory control,
however, has been pointed out to deviate from the often-observed lognormal distribution, because of theeffect of dose limits or levels of radiation control (Gale, 1965; UNSCEAR, 1977).
In 1980 we proposed the hybrid lognormal distribution to replace the skewed lognormal distribution inradiation protection statistics by introducing an exposure control parameter ρ with the inverse unit of dosethat should explain the degree of active feedback dose control depending on the magnitude of exposure(Kumazawa, Shimazaki and Numakunai, 1980; Kumazawa and Numakunai, 1981). Developing thecomputer package of the HLN analysis, the HLN distribution model was applied to complete the report(EPA- 520/1-84-005, 1984) on the U.S. occupational exposure to review comprehensively for the year 1980
including the trends for the years 1960-1985, as one of revision tasks on the Federal radiation protection
guidance for occupational exposure, approved by the President Reagan in January 27, 1987.

In statistics the hybrid lognormal distribution was discussed to clarify the characteristics of the
distribution as well as to verify the genesis of HLN distributions with the martingale central limit theorem,
including examples to show the feasibility of wide application (Kumazawa and Ohashi, 1986). Especiallyin the case of the lognormal suffered by a constraint of reducing the more the occurrence of data against thelarger-value, the HLN distribution is adequate to interpret the data in terms of risk control or some constraintof underlying phenomena incurred by data.
It is natural to use the multiple linear regression for the HLN analysis in practice: the regression model
is defined as 𝑧𝑖 = 𝛼 + 𝛽𝑥𝑖 + 𝛾 ln 𝑥𝑖 + 𝜀𝑖 for the ascending ordered data {𝑥𝑖 |𝑖 = 1, 𝑛} where the normal
cumulative distribution function Φ(𝑧𝑖 ) ≈ prob{𝑋 ≤ 𝑥𝑖 }, which is (𝑖 – 0.375)⁄(n + 0.25) (Blom,1958). In 1945 this regression model was first used to analyze the particle size distribution of products
ground in tube mill (Fagerholt, 1945; A. Hald, 1948). We use the similar model for the HLN analysis withthe awareness of the positive-value for the active control parameter ρ = β / γ per unit dose or unit quantityof data.
Based on the HLN genesis, it is adequate to introduce the hybrid function hyb(𝑥) = 𝑥 + ln(𝑥) and to
define the HLN distribution as hyb(𝜌𝑋) ~𝑁(𝜇, 𝜎 2 ) where μ is 𝐸[hyb(𝜌𝑋)] and 𝜎 2 is 𝑉[hyb(𝜌𝑋)].
The function hyb(𝜌𝑋) is almost the same to the logarithmic function ln(𝜌𝑋) in the region of ρX < 0.1
and almost proportional to the linear function of ρX in the region of ρX > 5 but it is neither logarithmic nor
linear for ρX between 0.1 and 5 or so. The region of ρX from 0.1 to 5 has the special significance toconcentrate our efforts on risk control of occupational exposure to ionizing radiation. To find the effective
region of risk control is to estimate the parameter ρ with the standard error reasonably.
The paper discusses the simple regression model on the HLN distribution using the hybrid function:
putting ℎ𝑖 = hyb(𝜌𝑥𝑖 ) of ascending ordered data {𝑥𝑖 |𝑖 = 1, 𝑛} and their ranks {𝑧𝑖 |𝑖 = 1, 𝑛}of the normal
probability, the regression model is discussed as ℎ𝑖 = 𝜇 + 𝜎𝑧𝑖 + 𝜀ℎ𝑖 or 𝑧𝑖 = 𝛼 ′ + 𝛾ℎ𝑖 + 𝜀𝑧𝑖 . The former
contains the parameter ρ in the dependent variable but 𝜌 is assumed that it might be estimated separately
in advance. This is accomplished by maximizing the R-squared for ρ using the EXCEL functions LINEST
and SOLVER, including the adjustment to replace degrees of freedom from n - 2 to n - 3. The calculationof the standard error of estimate ρ is derived from the F test method of the multiple regression relating to
R2 (Okuno, Kume, Haga and Yoshizawa, 1977).
In radiological statistics a pair of positive-value data {𝑢𝑖 , 𝑣𝑖 | 𝑖 = 1, 𝑛} both often varies ranging fromthe logarithmic to the linear region continuously connected by the hybrid interface region because of theunderlying phenomena due to stochastically multiplicative and additive interactions: e.g., chromosome
aberrations vs radiation dose, etc. Therefore this paper presents, in discussion section, an introductory
method of the simple regression model hyb(𝜈 ∙ 𝑣𝑖 ) = 𝛼 + 𝛽 hyb(𝜏 ∙ 𝑢𝑖 ) + 𝜀2𝐷𝑖 assumed as the known
parameters 𝜏 and 𝜈 but simultaneously solving by the EXCEL functions LINEST and SOLVER to
maximize 𝑅 2 (𝜏, 𝜈) by 𝜏 and 𝜈 with the replacement of degrees of freedom from n - 2 to n - 4. Some
examples are given to clarify the simple regression of the HLN distribution and the hybrid-hybrid section
paper that contains basically nine types of linear relationships.

METHOD
The density function of the hybrid lognormal distribution is given as follows:
𝑓(𝑥) =(hyb(𝜌𝑥) −𝜇)21
(𝜌 + ) exp [−
]
𝑥
2𝜎 2
√2𝜋𝜎
1

(𝑥, 𝜌, 𝜎 > 0).

(1)

Putting 𝑡 = 𝜌𝑥 , we have 𝑓(𝑡)𝑑𝑡 = 𝑓(𝑥)𝑑𝑥 = 𝜙(𝑧)𝑑𝑧 where 𝜙(𝑧) is the density function of thestandard normal cumulative function Φ(𝑧) and hyb(𝜌𝑥) = hyb(𝑡) = 𝑡 + ln(𝑡) = 𝜇 + 𝜎𝑧 or 𝑧 =(hyb(𝑡) −𝜇)⁄𝜎. Then the simple regression model isℎ𝑖 = hyb(𝑡𝑖 ) = 𝜇 + 𝜎𝑧𝑖 + 𝜀ℎ𝑖 ,(2)

𝑖 = 1, 2, ⋯ , 𝑛

where E[𝜀ℎ𝑖 ] = 0, V[𝜀ℎ𝑖 ] = 𝜎ℎ2 , Cov[𝜀ℎ𝑖 , 𝜀ℎ𝑗 ]𝑖≠𝑗 = 0 and 𝜀ℎ𝑖 ~ 𝑁(0, 𝜎ℎ2 ).

For the known parameter ρ

the simple regression model can be solved in the usual way. The unknown parameter ρ needs to discuss on
the simple regression in terms of the estimate of ρ with the standard error because of ℎ𝑖 = hyb(𝜌𝑥𝑖 ).
For the comparison the multiple linear regression model and another simple regression model are given:𝑧𝑖 = 𝛼 + 𝛽𝑥𝑖 + 𝛾 ln 𝑥𝑖 + +𝜀𝑖𝑧𝑖 = 𝛼 ′ + 𝛾ℎ𝑖 + 𝜀𝑧𝑖(𝛽, 𝛾 > 0, 𝜌 = 𝛽 ⁄𝛾 ),( 𝜌, 𝛾 > 0, ℎ𝑖 = hyb(𝜌𝑥𝑖 ) ).

(3)
(4)

To estimate the parameter ρ there are several approaches, e.g. a Bayesian procedure of the estimation of thehybridization parameter ρ for a specific prior distribution for ρ, μ and σ (Groer and Uppuluri, 1991) and the
maximum likelihood estimation of the parameters of the hybrid lognormal distribution (Sont, 1991), etc. Asa practical way this paper is to reasonably solve the simple regression model in Equation (2) with the
unknown parameter ρ. This attains to maximize the coefficient of determination R2 of Equation (2) for ρ.
The R-squared of Equation (2) is derived as follows:2

𝑅 2 (𝜌) =(𝜌𝑆𝑥𝑧 + 𝑆𝑦𝑧 )𝑆𝑧ℎ (𝜌)2= 2,
𝑆ℎℎ (𝜌) 𝑆𝑧𝑧 (𝜌 𝑆𝑥𝑥 + 𝜌𝑆𝑥𝑦 + 𝑆𝑦𝑦 ) 𝑆𝑧𝑧(5)

where 𝑆ℎℎ (𝜌) = ∑𝑛 𝐻𝑖 2 , 𝑆𝑧𝑧 = ∑𝑛 𝑍𝑖 2 , 𝑆𝑧ℎ = ∑𝑛 𝑍𝑖 𝐻𝑖 , 𝑆𝑥𝑥 = ∑𝑛 𝑋𝑖 2 𝑆𝑦𝑦 = ∑𝑛 𝑌𝑖 2 , 𝑆𝑥𝑦 = ∑𝑛 𝑋𝑖 𝑌𝑖 ,𝑆𝑥𝑧 = ∑𝑛 𝑋𝑖 𝑍𝑖 and 𝑆𝑦𝑧 = ∑𝑛 𝑌𝑖 𝑍𝑖 , and deviations 𝐻𝑖 = ℎ𝑖 − 𝐸[ℎ𝑖 ] , 𝑍𝑖 = 𝑧𝑖 − 𝐸[𝑧𝑖 ] , 𝑋𝑖 = 𝑥𝑖 − 𝐸[𝑥𝑖 ]and 𝑌𝑖 = ln 𝑥𝑖 − 𝐸[ln 𝑥𝑖 ].
The estimation of the parameter ρ is the maximum of 𝑅 2 (𝜌) with respect to ρ:𝜕𝑅 2 (𝜌)=0𝜕𝜌∴ 𝜌̂𝑅2 =𝑆𝑥𝑦 𝑆𝑦𝑧 − 𝑆𝑥𝑧 𝑆𝑦𝑦 𝐴𝑧𝑥=,
𝑆𝑥𝑦 𝑆𝑥𝑧 − 𝑆𝑥𝑥 𝑆𝑦𝑧 𝐴𝑧𝑦(6)

where 𝐴𝑧𝑥 𝑎𝑛𝑑 𝐴𝑧𝑦 are the cofactors of 𝑆𝑧𝑥 𝑎𝑛𝑑 𝑆𝑧𝑦 , respectively, of the following matrix 𝑆:

𝑆𝑥𝑥
𝑆＝ (𝑆𝑦𝑥
𝑆𝑧𝑥

𝑆𝑥𝑦 𝑆𝑥𝑧
𝑆𝑦𝑦 𝑆𝑦𝑧 )𝑆𝑧𝑦 𝑆𝑧𝑧(7)

The estimate 𝜌̂𝑅2 in Equation (6) is equal to the estimate of ρ to maximize the R-squared as well as tominimize the sum of squared errors (residuals) in Equations (3) and (4) but it is different to the estimate 𝜌̂ℎto minimize the sum of squared errors 𝑆ℎ𝑒 (𝜌) in Equation (2). However, the sum of squared errors

𝑆𝑧𝑒 (𝜌) in Equation (4) and 𝑆ℎ𝑒 (𝜌) in Equation (2) satisfy the following equation:
1 − 𝑅2 (𝜌) =𝑆ℎ𝑒 (𝜌)𝑆𝑧𝑒 (𝜌)=.
𝑆ℎℎ(𝜌)𝑆𝑧𝑧(8)

For the estimate 𝜌̂𝑅2 as the maximum of 𝑅 2 (𝜌) in Equation (2), Equation (8) results in:̂𝑅2 )𝑆ℎ𝑒 (𝜌

̂𝑅2 ) =
1 − 𝑅 2 (𝜌̂𝑅2 )𝑆ℎℎ (𝜌̂𝑅2 )
𝑆𝑧𝑒 (𝜌|𝑆|=,
𝑆𝑧𝑧
𝐴𝑧𝑧𝑆𝑧𝑧=(9)where |𝑆| is the determinant of S and 𝐴𝑧𝑧 = 𝑆𝑥𝑥 𝑆𝑦𝑦 − 𝑆𝑥𝑦 2, the cofactor of the element 𝑆𝑧𝑧 of matrix S.
The residual variance 𝑉𝑧𝑒 (𝜌̂𝑅2 ) = 𝑆𝑧𝑒 (𝜌̂𝑅2 )⁄(𝑛 − 3) provides the standard error 𝑠𝑒(𝜌̂𝑅2 ) as 𝑆𝑧𝑒 (𝜌) =̂𝑅2 + 𝑠𝑒(𝜌̂𝑅2 )) is obtained as follows:
𝑆𝑧𝑒 (𝜌̂𝑅2 + 𝑠𝑒(𝜌̂𝑅2 )) = 𝑆𝑧𝑒 (𝜌̂𝑅2 ) + 𝑉𝑧𝑒 (𝜌̂𝑅2 ) and 𝑅2 (𝜌) = 𝑅2 (𝜌|𝑆| 𝑛 − 2
𝑆𝑧𝑒 (𝜌) 𝑆𝑧𝑒 (𝜌̂𝑅2 ) + 𝑉𝑧𝑒 (𝜌̂𝑅2 ) 𝑆𝑧𝑒 (𝜌̂𝑅2 ) 𝑛 − 2==== 1 − 𝑅2 ( 𝜌)𝑆𝑧𝑧
𝑆𝑧𝑧
𝑆𝑧𝑧 𝑛 − 3 𝐴𝑧𝑧 𝑆𝑧𝑧 𝑛 − 3
2𝑅2 (𝜌) =(𝜌𝑆𝑥𝑧 + 𝑆𝑦𝑧 )(𝜌2 𝑆𝑥𝑥 + 2𝜌𝑆𝑥𝑦 + 𝑆𝑦𝑦 )𝑆𝑧𝑧

=1−

|𝑆| 𝑛 − 2

𝐴𝑧𝑧𝑆𝑧𝑧 𝑛 − 3

=

𝑐𝑧
.
𝑆𝑧𝑧

(10)

Equation (10) is solved as the quadratic equation with respect to ρ:𝜌2 (𝑐𝑧 𝑆𝑥𝑥 − 𝑆𝑥𝑧 2 ) + 2 𝜌(𝑐𝑧 𝑆𝑥𝑦 − 𝑆𝑥𝑧 𝑆𝑦𝑧 ) + 𝑐𝑧 𝑆𝑦𝑦 − 𝑆𝑦𝑧 2 = 0,𝜌=−(𝑐𝑧 𝑆𝑥𝑦 − 𝑆𝑥𝑧 𝑆𝑦𝑧 ) ± √𝑐𝑧 𝐴𝑧𝑧 {𝑆𝑧𝑧 𝑅 2 (𝜌̂𝑅2 ) − 𝑐𝑧 }∴ 𝑠𝑒(𝜌̂𝑅2 ) =(𝑐𝑧 𝑆𝑥𝑥 − 𝑆𝑥𝑧 2 ),𝜌2 − 𝜌1 √𝑐𝑧 𝐴𝑧𝑧 {𝑆𝑧𝑧 𝑅 2 (𝜌̂𝑅2 ) − 𝑐𝑧 }=.
2
(𝑐𝑧 𝑆𝑥𝑥 − 𝑆𝑥𝑧 2 )(11)

The range between 𝜌1 and 𝜌2 of the simple regression in Equation (2) is given in terms of 𝑅 2 (𝜌) for
maximization or 1 − 𝑅 2 (𝜌) for minimization. Thus, 𝜌 = 𝜌̂𝑅2 + 𝑠𝑒(𝜌̂𝑅2 ) satisfies the following equation:
𝑅 2 (𝜌) = 𝑅 2 (𝜌̂𝑅2 ) −1 − 𝑅 2 (𝜌̂𝑅2 )1 − 𝑅 2 (𝜌̂𝑅2 )𝑜𝑟 1 − 𝑅 2 (𝜌) = 1 − 𝑅 2 (𝜌̂𝑅2 ) +
𝑛−3
𝑛−3

(12)

The standard error of 𝜌̂𝑅2 is to calculate as (𝜌2 − 𝜌1 )⁄2 because 𝜌̂𝑅2 − 𝜌1 is not always equal to𝜌2 − 𝜌̂𝑅2 . In the case of distribution characteristics to be purely lognormal-dominant or normal-dominant,the HLN analysis can be performed to select the estimate 𝜌 = 0.1/𝑥𝑛 for the negative or too much smallvalue of the estimate 𝜌 (lognormal-dominant) or to select the estimate 𝜌 = 5/𝑥1 for the infinitive or too
much large value of the estimate 𝜌 (normal-dominant). Thus, the HLN model provides the analysis
approach of data ranging from the lognormal-dominant to the normal-dominant systematically.

RESULTS OF ANALYSIS
Using the well-known data on the number of words per sentence in 60 sentences taken from a certain of
Toynbee’s “A Study of History” (Wilks, 1948), the left panel in Figure 1 shows the graph of 𝑅 2 (𝜌) so thatit should change from the log-normal (𝜌 → 0) to the normal (𝜌 → +∞) via the hybrid lognormal model by
increasing the value of 𝜌. R2 (𝜌 → 0) is larger than R2 (𝜌 → +∞) but the largest is 𝑅 2 (𝜌 = 𝜌̂𝑅2 ). The rightpanel is the enlarged view of the bold curve on the left panel to show how the standard error se(𝜌̂𝑅2 ) is
decided based on Equation (12).
1.00
0.99

HLN fit to data on words per sentence← LN model

0.97

R 2 (ρ)R 2 (ρ)0.98
R²(ρ)
R²( . )
R² of ρ1_ρ2

0.96
0.95

Esimation of ̂ and ±0.9914R²(ρ)
R² of ρ1_ρ2
R²( . )
R² of se( )( ̂)

0.9912

Normal →0.94
1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01
ρ→

0.9910
ρ1
0.009

ρ2

0.013
ρ→

0.017

She (ρ)She of ρ1_ρ2
She ( . )

0.8
0.6
0.4

Normal →← LN model

0.2
0
1.0E-040.06

1.0E-03 1.0E-02
ρ→

1.0E-01

Normal →She/Shh of ρ
She/Shh of ρ1_ρ2
She/Shh of

0.03← LN model0
1.0E-05

1.0E-03 1.0E-01 1.0E+01
ρ→

0.4

She or Shh (１－R2)

1

She / Shh or １－R2

She / Shh or １－R2

She or Shh (１－R2)

Figure 1. The graph of R2(ρ) in Equation (2) by ρ and the interval of ρ for 𝜌̂𝑅2 ± se (𝜌̂𝑅2 ).
Left panel is R2(ρ) ranging from the LN to the Normal model, and right panel shows how to decide se (𝜌̂𝑅2 ): 𝜌̂ = 𝜌̂𝑅2 .

0.36

She (ρ)She of ρ1_ρ2
She ( . )She of se( . )

0.32

0.28
0.009
0.00930.009

0.0087
0.009

0.013
ρ→

ρ2

0.017

She/Shh of ρ
She/Shh of ρ1_ρ2
She/Shh of
She/Shh of se( . )

0.013
ρ→

ρ2

0.017

Figure 2. Other options to calculate the standard error of the estimate 𝜌̂𝑅2 .
Upper panel: based on She(ρ), not practicable; lower panel: based on 1 - R2(ρ), another good option.

Figure 2 shows other options to calculate the standard error se(𝜌̂𝑅2 ). The upper panel shows a method
based on minimizing 𝑆ℎ𝑒 (𝜌) of Equation (2), which is somewhat complicated to use. The lower panel
shows another good method based on minimizing 1 − 𝑅 2 (𝜌), namely 𝑆ℎ𝑒 (𝜌)⁄𝑆ℎℎ (𝜌).
These examples confirm that the above-mentioned method is valid and reasonable to estimate ρ with thestandard error as the simple regression model. According to the method, the resultant statistics on the dataof words per sentence are 𝜇̂ = 0.4084 ± 0.0096, 𝜎̂ = 0.7972 ± 0.00985, 𝜌̂ = 0.01294 ± 0.00236,
and 𝑅 2 = 0.9912. Figure 3 (a) shows the good fitting to the data all of which are within the 95% prediction
interval. The number of words per sentence might be the result of making sentences to clarify the thought
considering the readability. Interpreting the graph, Toynbee would write the sentences lognormally forwords / sentence less than 8 but hybrid lognormally for words / sentence above 44. Thus, there is apossibility to avoid a long sentence in Toynbee. Figure 3 (b) shows approaching the normality in the longer
sentence. The normality of the residuals is pretty good shown in Figure 2 (d).
The HLN distribution tells us how to cope with increasing risks and the effective range of risk control
in terms of the control parameter ρ. For example, the statistical fluctuation in running speed is important for
football players to maintain flexible mobility as well as to avoid exhaustion. The distribution of runningspeeds during the game shows the HLN distribution characteristics. Eyes speed data during viewing still /moving pictures are also hybrid lognormally distributed. Thus, somewhat similarities exit like the radiationdose in terms of risk management to gain the benefit during a small risk but to avoid the significant risk.
120

Data xi

HLN fit1

x (words / sentence）

hyb (ρx), x (words / sentence)2

95% PI

0
-1

-2
-3

-3

-2

-1

0

1

2

Data xi
HLN fit
95% PI

100
80
60

40
20
0

3

-3

0.4
0.2
0
-0.2
-2
-1
0
1
2
Normal variate z of prob{X ≤ x}(c) Residuals: 𝜀ℎ𝑖 = hyb(𝜌̂𝑅2 𝑥𝑖 ) − 𝜇̂ − 𝜎̂𝑧𝑖 vs 𝑧𝑖

0

1

2

3

3

(b) Normal probability plots of data
Normal z for residuals

Residuals of hyb (ρx)(a) HLN probability plots of data of words/sentence-3-1Normal variate z of prob{X ≤ x}

Normal variate z of prob{X ≤ x}

-0.4

-2

3
2

y = 13.115x + 1E-15
R² = 0.9673

1
0
-1
-2
-3

-0.2

-0.1
0
0.1
Residuals of hyb (ρx)0.2(d) Probability plots of ascending ordered {𝜀ℎ𝑖 }.

Figure 3. The HLN analysis of 60 ascending ordered data on the number of words per sentence.

DISCUSSION
The hybrid lognormal distribution is relating to the multiple linear regression method by Fagerholt (1945)and Hald (1948). We, as stuff of former JAERI, developed it so that it should reasonably interpret how tocope with radiation exposure risk with some uncertainty. Such risk control issues are common inenvironmental exposure statistics, health risk statistics, economic/social statistics, etc. Therefore, the HLN
distribution should be discussed widely by statisticians because of the issues still unsolved.
The variation of the HLN distribution needs for X > a > 0 as hyb(𝜌(𝑋 − 𝑎)) ~𝑁(𝜇, 𝜎 2 ) and for 0 < a< X < b as hyb(𝜌 (𝑋 − 𝑎)⁄(𝑏 − 𝑋)) ~𝑁(𝜇, 𝜎 2 ), the latter of which calls “the hybrid SB distribution” likethe Johnson’s SB distribution. These have been developed to analyze data in radiation protection and nuclear
environmental safety assessment. Radiological data relating to the Fukushima DAIICHI accident shows
examples of the variation to be the HLN distribution. The systematic random number generation for theradiological uncertainty analysis has been developed a system of normal family distributions that contains
the lognormal, hybrid lognormal and positive-value normal distributions in the region defined X > 0, X > a> 0 and 0 < a < X < b. All of these use the simple regression analysis.
The data relating to radiation emission processes, radiation interactions with matters, and radiation dose
to man in various exposure situations are resultant effects due to stochastically multiplicative and additive
interactions. The data other than radiation protection shows the similar effects with simultaneously
multiplicative and additive uncertainty. Depending on the dominancy between multiplicative and additive
components, the variation characteristics changes to be purely logarithmic, to be logarithmic in the lower
side but linearly in the upper side and to be purely linear. The hybrid function found from the development
the HLN distribution covers the wide range of variations from the logarithmic region to the linear region
continuously connected both via the hybrid interface region.
If the hybrid function expresses as a scale like the logarithmic scale, the new scale is defined, called “thehybrid scale.” It consists of three major regions, the logarithmic for ρx < 0.1 of hyb(𝜌𝑥), the truly hybrid
for 0.1 ≤ ρx ≤ 5 of hyb(𝜌𝑥) and the linear region for ρx > 5 of hyb(𝜌𝑥). Suppose the ascending ordereddata of positive value {𝑥𝑖 |𝑖 = 1, 𝑛}, all hyb(𝜌𝑥𝑖 ) are in the logarithmic region for 𝜌 < 0.1⁄𝑥𝑛 and in thelinear region for 𝜌 > 5⁄𝑥1. Thus, the hybrid scale can express the log-, truly-hybrid- and linear-region ofdata with hyb(𝜌𝑥) by selecting a proper value of 𝜌 . Applying the hybrid scale to the vertical andhorizontal axes graduated with a hybrid scale, a new section paper is provided (Figure 4). This is called “thehybrid-hybrid section paper” like the term of the log-log one. In radiation protection, quantities often changeby orders of magnitude that should be modeled by the logarithmic scale but the magnitude of exposure incontrol is managed in the same order of magnitude that should be modeled by the hybrid scale gradually
approaching the linear scale. The hybrid-hybrid section paper contains nine section paper regions of fourconventional ones (linear-linear, linear-log, log-log and log-linear) and five interface ones (linear-hybrid,hybrid-log, log-hybrid, hybrid-linear and hybrid-hybrid). Each of five interface regions is corresponding tofive types of new independent section papers (Figure 4 (b), (c)).

Nine types of section papers

Conventional section papers
linear-linear, linear-log, log-log & log-linear
New section papers: connecting above papers
linear-hybrid, hybrid-log, log-hybrid,hybrid-linear, hybrid-hybrid(a) A hybrid-hybrid section paper(b) Nine types of section papers included in it.

Semi-Log section paper𝑦 = 𝛼 + 𝛽 ln(𝑥) + 𝜀
𝑥 = exp[(𝑦 − 𝛼)⁄𝛽] + ε′

Semi-Hybrid section paper𝑦 = 𝛼 + 𝛽 hyb(𝜏𝑥) + 𝜀
𝑥 = 𝜏 −1 cyb[(𝑦 − 𝛼)⁄𝛽] + ε′

Normal section paper𝑦 = 𝛼 + 𝛽𝑥 + 𝜀
𝑥 = (𝑦 − 𝛼)⁄𝛽 + ε′

Log-Hybrid section paperhyb(𝜈𝑦) = 𝛼 + 𝛽 ln(𝑥) + 𝜀
𝑥 = exp[(hyb(𝜈𝑦) − 𝛼)⁄𝛽] + ε′

Hybrid-Hybrid section paperhyb(𝜈𝑦) = 𝛼 + 𝛽 hyb(𝜏𝑥) + 𝜀
𝑥 = 𝜏 −1 cyb[(hyb(𝜈𝑦) − 𝛼)⁄𝛽] + ε′

Semi-Hybrid section paperhyb(𝜈𝑦) = 𝛼 + 𝛽𝑥 + 𝜀
𝑥 = (hyb(𝜈𝑦) − 𝛼)⁄𝛽 + ε′

Log-Log section paperln(𝑦) = 𝛼 + 𝛽 ln(𝑥) + 𝜀
𝑥 = exp[(ln(𝑦) − 𝛼)⁄𝛽] + ε′

Log-Hybrid section paperln(𝑦) = 𝛼 + 𝛽 hyb(𝜏𝑥) + 𝜀
𝑥 = 𝜏 −1 cyb[(ln(𝑦) − 𝛼)⁄𝛽] + ε′

Semi-Log section paperln(𝑦) = 𝛼 + 𝛽𝑥 + 𝜀
𝑥 = (ln(𝑦) − 𝛼)⁄𝛽 + ε′(c) Nine types of linear equations for nine types section papers organized by the hybrid function
Figure 4. An example of the hybrid-hybrid section paper that contains nine types of section papers.
This section paper allows any linear relationships among linear, logarithmic, power and exponential functions.
The inverse of hyb(x) is denoted as cyb (x), called “the cyb function” that is the hybrid between exp(x) and x.

Because of the tri-regionality of the hybrid scale as log, hybrid and linear regions, the linear relationshipon the hybrid-hybrid section paper sufficiently approximates nine types of linear relationships
corresponding to nine types of section papers shown in Figure 4 (c). For data {𝑥𝑖 , 𝑦𝑖 |𝑖 = 1, 𝑛} sorted by𝑥𝑖 in ascending order, where yi does not mean yi = ln (𝑥𝑖 ) in Equations after (5), a hybrid-hybrid simple
regression model is given by:
𝑣𝑖 = 𝛼 + 𝛽𝑢𝑖 + 𝜀2𝐷𝑖

← hyb(𝜈𝑦𝑖 ) = 𝛼 + 𝛽 hyb(𝜏𝑥𝑖 ) + 𝜀2𝐷𝑖 ( 𝑖 = 1, ⋯ , 𝑛). (13)For the known parameters τ and ν the simple regression model can be solved in the usual way. For the
unknown τ and ν, scaling parameters with the inverse unit of x and y, respectively, the similar approach of
the HLN model is applicable to attain the global maximum of 𝑅 2 (𝜏, 𝜈) using LINEST and SOLVER, with
the nuisance parameters α, β and the residual 𝜀2𝐷𝑖 . This approach is to find the best linearity of plotting
data on the hybrid-hybrid section paper as a comprehensive section paper, despite the linearity of the data
ranging across any several types of nine section papers.

The R-squared of Equation (13) is 𝑅 2 = 𝑆𝑢𝑣 2⁄𝑆𝑢𝑢 𝑆𝑣𝑣 where 𝑆𝑢𝑢 , 𝑆𝑣𝑣 and 𝑆𝑢𝑣 are the sum of squared
deviations and the sum of deviation products for 𝑢𝑖 𝑎𝑛𝑑 𝑣𝑖 , respectively, and using hyb(𝜏𝑥𝑖 ) andhyb(𝜈𝑦𝑖 ),2(𝜏𝜈𝑆𝑥𝑦 + 𝜏𝑆𝑥𝑦𝐿 + 𝜈𝑆𝑦𝑥𝐿 + 𝑆𝑥𝐿 𝑦𝐿 )
𝑆𝑢𝑣 2
𝑅 =
= 2
,
𝑆𝑢𝑢 𝑆𝑣𝑣 (𝜏 𝑆𝑥𝑥 + 2𝜏𝑆𝑥𝑥𝐿 + 𝑆𝑥𝐿 𝑥𝐿 )(𝜈 2 𝑆𝑦𝑦 + 2𝜈𝑆𝑦𝑦𝐿 + 𝑆𝑦𝐿 𝑦𝐿 )2(14)

𝑆𝑢𝑢 = 𝜏 2 𝑆𝑥𝑥 + 2𝜏𝑆𝑥𝑥𝐿 + 𝑆𝑥𝐿 𝑥𝐿 ← ∑(𝜏(𝑥𝑖 − 𝐸[𝑥𝑖 ]) + ln(𝑥𝑖 ) − 𝐸[ln(𝑥𝑖 )])2 = ∑(𝜏𝑋𝑖 + 𝑋𝐿𝑖 )2,
𝑆𝑣𝑣 = 𝜈 2 𝑆𝑦𝑦 + 2𝜏𝑆𝑦𝑦𝐿 + 𝑆𝑦𝐿 𝑦𝐿 ← ∑(𝜈(𝑦𝑖 − 𝐸[𝑦𝑖 ]) + ln(𝑦𝑖 ) − 𝐸[ln(𝑦𝑖 )])2 = ∑(𝜈𝑌𝑖 + 𝑌𝐿𝑖 )2,
𝑆𝑢𝑣 = 𝜏𝜈𝑆𝑥𝑦 + 𝜏𝑆𝑥𝑦𝐿 + 𝜈𝑆𝑥𝐿 𝑦 + 𝑆𝑥𝐿 𝑦𝐿 ← ∑(𝜏𝑋𝑖 + 𝑋𝐿𝑖 )(𝜈𝑌𝑖 + 𝑌𝐿𝑖 ).
Based on equations satisfied 𝜕𝑅 2⁄𝜕𝜏 = 0, 𝜕𝑅 2⁄𝜕𝜈 = 0 and 𝜕 2 𝑅 2⁄𝜕𝜏𝜕𝜈 = 0 , the estimation ofparameters 𝜏 and 𝜈 are given as follows:𝜏̂ =𝑆𝑥𝑥𝐿 𝑆𝑦𝑥𝐿 − 𝑆𝑥𝑦 𝑆𝑥𝐿 𝑥𝐿,
𝑆𝑥𝑦 𝑆𝑥𝑥𝐿 − 𝑆𝑥𝑥 𝑆𝑥𝐿 𝑦

𝑆𝑥𝑦𝐿 𝑆𝑦𝑦𝐿 − 𝑆𝑥𝑦 𝑆𝑦𝐿𝑦𝐿.
𝑆𝑥𝑦 𝑆𝑦𝑦𝐿 − 𝑆𝑦𝑦 𝑆𝑥𝑦𝐿

𝜈̂ =

(15)

The sum of squared errors (residuals) 𝑆2𝐷𝑒 (𝜏̂ , 𝜈̂) and the residual variance 𝑉2𝐷𝑒 (𝜏̂ , 𝜈̂) = 𝑆2𝐷𝑒 (𝜏̂ , 𝜈̂)⁄𝑓𝑒 ,𝑓𝑒 = 𝑛 − 4 , provide the level of reducing 𝑅 2 from the maximum 𝑅 2 (𝜏̂ , 𝜈̂) to 𝑅 2 (𝜏̂ , 𝜈̂) −(1 − 𝑅 2 (𝜏̂ , 𝜈̂))⁄𝑓𝑒 . Then their standard errors se(𝜏̂ ) and se(𝜈̂) are obtained from the value of 𝜏 and 𝜈 tosatisfy the level.
4.0

4.0

3.0

3.0

3.0

2.0

2.0

2.0

Y

Y
1.0

1.0

0.0

0.0

-2

0
2
ln(X)

4

1.0
-3

0
hyb(τX)

3

0.0

0
-2

-6
-8

y = 2.0264x - 3.5759
R² = 0.9799

-2

0

ln(X)

2

-4
-6
-8

4

hyb(νY )0-2

hyb(νY )0-2-4

y = 1.4229x - 1.8828
R² = 0.9993-30hyb(τX)

-6

-8

3

0

0

-2

-2

-6

ln(Y )-4y = 1.7738x - 2.6011
R² = 0.9966

-2

0

ln(X)

2

4

-6

X

6

9

y = 0.8662x - 5.0353
R² = 0.9112

0

3

0

3

X

6

9

6

9

2

2

0

-4

3

-4

2

-2

0

2

2

ln(Y )

hyb(νY )2ln(Y )

Y

4.0

y = 1.2294x - 1.1271
R² = 0.9902-30hyb(τX)

3

-4
-6

X

(a) Data (Edwards et al.) plotted on nine papers(b) Several data plots on hybrid-hybrid section paper

Figure 5. An example of chromosome aberration data plotted on the hybrid-hybrid section paper.
Figure 5 shows examples of the hybrid-hybrid analysis on several experimental chromosomal aberration
data (Indrawati and Kumazawa, 2000). Figure 5 (a) is the nine types of plots on frequencies of dicentrics
observed in human lymphocytes irradiated by x-rays at 1 Gy/min (Edwards et el. 1979). The hybrid-hybrid
plot is the best maximum of 𝑅 2 (𝜏̂ , 𝜈̂). Figure 5 (b) shows several plots of various types of chromosome
aberration data, the linearity of which is attained across different regions of the hybrid-hybrid section paper.

The hybrid-hybrid analysis has been applied to various data in vital and biological data as well as radiation
protection: basic trend of dose reduction due to the Fukushima DAIICHI accident (Kumazawa, 2013),
resuspension factor vs time after deposition (Kumazawa, 2014), sky-shine dose vs photon source distance(Kumazawa, 2015 in Japanese), various types of dose vs response (Kumazawa, 2001, 2015), and others,e.g., hyb[𝜌𝑃⁄(𝑏 − 𝑃)] 𝑣𝑠 𝑎𝑔𝑒 by income category (low to high) cited from the WHO GHO 2015
statistics: P is population (< b).

CONCLUSION
This paper presented a practical method for a single regression model of hybrid log normal distribution.
It also showed that this method is applicable to the calculation of the linear relationship plotted on the
hybrid-hybrid section paper, whose axes are graded in the hybrid scale of log and linear scales.
References
1. F. Galton (1879). “The geometric mean, in vital and social statistics,” Proc Roy Soc 29,365-367.
2. D. McAlister (1879). “The law of the geometric mean,” Proc Roy Soc 29,367-376.
3. H.J. Gale (1965). “The lognormal distribution and some examples of its application in the field ofradiation protection,” United Kingdom Atomic Energy Authority, AERE-R 4736.
4. UNSCEAR (1977). “Annex E Occupational Exposure, Sources and Effects on Ionizing Radiations.”
United Nations Scientific Committee on the Effects of Atomic Radiation, Sales No. E.77.IX.1, NewYork.
5. ICRP (1977). “ICRP Publication 26 Recommendations of the ICRP, Annals of the ICRP 1(3).
6. S. Kumazawa, J. Shimazaki and T. Numakunai (1980). “Numerical algorithm of statistics on the hybrid
lognormal distribution,” the 2nd symposium on the applied statistics, D3-1~10 (Tokyo, Oct 25, 1980).
7. S. Kumazawa and T. Numakunai (1981). “A new theoretical analysis of occupational dose distributions
indicating the effects of dose limits,” Health Phys 41, 465-449.
8. S. Kumazawa, D.R. Nelson and A.C.B. Richardson (1984). “Occupational exposure to ionizing
radiation in the United States  A comprehensive review for the year 1980 and a summary of trends forthe years 1960-1985,” US Environmental Protection Agency, EPA 520/1-84-005.
9. Federal register (1987). “Radiation protection Guidance to Federal Agencies for occupational exposure Recommendations approved by the President,” January 27, 1987.
10. S. Kumazawa and Y. Ohashi (1986). “The hybrid lognormal distribution and Its application (inJapanese),” Jpn J. Appl. Stat. 1581), 1-14.
11. G. Blom (1958). “Statistical estimates and transformed beta variables,” New York: John Wiley & Sons.
12. G. Fagerholt (1945). “Particle size distribution of products ground in tube mill,” dissertation.
13. A. Hald (1948). “Statistical theory with engineering applications.” Cited a slightly expanded version of
the Danish textbook, New York: John Wiley & Sons (1952).
14. T. Okuno, H. Kurume, T. Haga and T. Yoshizawa (1971). “Multiple regression analysis in Japanese,”monograph, Tokyo, JUSE, Ltd.
15. S.S. Wilks (1948). “Elementary statistical analysis,” Princeton, New Jersey, Princeton University Press.
16. I. Indrawati and S. Kumazawa (2000). “Analysis of chromosome aberration data by hybrid-scale model,”Japan Atomic Energy Agency, JAERI-Research 2005-005.
17. A.A. Edwards, D.C. Lloyd and R.J. Purrott (1979). “Radiation induced chromosome aberrations andthe Poisson distribution,” Rad Environ Biophys 16, 89-100.
18. S. Kumazawa (2013). “Distribution of doses to residents evacuated after the Fukushima nuclear power
station accident,” Proc of Int. Sym. on Environmental monitoring and dose estimation of residents after
accident of TEPCO’s Fukushima Daiichi Nuclear Power Station, Topics_5-04.pdf.
19. S. Kumazawa (2014). “Reappraisal of predictive models for resuspension,” Progress in JNST, 4, 875878.