Some results on quantile-based Shannon doubly truncated entropy

Figures & data

Table 1. Relationship between GVQF and GHQF.

Distribution	Quantile Functions	$G(u_1,u_2)$
Exponential	$-{\lambda }^{-1}\log (1-u)$	$\frac{1}{\lambda }\left[\text{̧}{h}_1\right(u_1,u_2)\log Q(u_1)-\text{̧}{h}_2(u_1,u_2)\log Q(u_2)+R(u_1,u_2\left)\right]$
Finite Range	$\frac{1}{a}(1-(1-u{)}^{1/b})$	$\frac{1}{\left(ab\right)}\left[\right(1-aQ\left(u_1\right)\left)\text{̧}{h}_1\right(u_1,u_2)\log Q(u_1)-(1-aQ\left(u_2\right)\left)\text{̧}{h}_2\right(u_1,u_2)\log Q(u_2)+R(u_1,u_2)-a]$
Pareto-II	$\frac{1}{p}\left(\right(1-u{)}^{1/q}-1)$	$\frac{1}{\left(pq\right)}\left[\right(1+pQ\left(u_1\right)\left)\text{̧}{h}_1\right(u_1,u_2)\log Q(u_1)-(1+pQ\left(u_2\right)\left)\text{̧}{h}_2\right(u_1,u_2)\log Q(u_2)+R(u_1,u_2)+p]$
Power	$a{u}^{{}^{1/b}}$	$\log a+\frac{1}{b}\left[Q\left(u_2\right)\text{̧}{h}_2(u_1,u_2)\log \left(\frac{Q\left(u_2\right)}{a}\right)-Q\left(u_1\right)\text{̧}{h}_1(u_1,u_2)\log \left(\frac{Q\left(u_1\right)}{a}\right)-1\right]$
Pareto-1	$a(1-u{)}^{-1/b}$	$\log a+\frac{1}{b}\left[Q\left(u_2\right)\text{̧}{h}_2(u_1,u_2)\log \left(\frac{a}{Q\left(u_2\right)}\right)-Q\left(u_1\right)\text{̧}{h}_1(u_1,u_2)\log \left(\frac{a}{Q\left(u_1\right)}\right)+1\right]$

Table 2. Relation between $\text{̧}M(u_1,u_2)$ and GHQF for various distributions.

Distribution	$\text{Quantile Functions}$	$\text{̧}M(u_1,u_2)$
Exponential	$-{\lambda }^{-1}\log (1-u)$	$\lambda m(u_1,u_2)-\log \lambda$
Finite range	$\frac{1}{a}(1-(1-u{)}^{1/b})$	$-\log ab+(-b+1)E[\log (1-aX\left)\right|u_1<X<u_2]$
Pareto -II	$\frac{1}{p}\left(\right(1-u{)}^{1/q}-1)$	$(q+1)E[\log (1+pQ\left(u\right)\left)\right|u_1<X<u_2]-\log pq$
Power	$a{u}^{{}^{1/b}}$	$\left(\frac{b-1}{b}\right)\left[1+Q\left(u_1\right)\text{̧}{h}_1(u_1,u_2)\log \left(\frac{Q\left(u_1\right)}{a}\right)-Q\left(u_2\right)\text{̧}{h}_2(u_1,u_2)\log \left(\frac{Q\left(u_2\right)}{a}\right)\right]+\log \left(\frac{a}{b}\right)$
Pareto -I	$a(1-u{)}^{-1/b}$	$\left(\frac{b+1}{b}\right)\left[1-Q\left(u_1\right)\text{̧}{h}_1(u_1,u_2)\log \left(\frac{a}{Q\left(u_1\right)}\right)+Q\left(u_2\right)\text{̧}{h}_2(u_1,u_2)\log \left(\frac{a}{Q\left(u_2\right)}\right)\right]+\log \left(\frac{a}{b}\right)$
Weibull	$(-\log (1-u){)}^{1/p}$	$-\log p-(p-1)G(u_1,u_2)+E\left[\right(X{)}^p|u_1<X<u_2]$
Rayleigh	$(-\log (1-u){)}^{1/2}$	$-\log 2-\log G(u_1,u_2)+E\left[\right(X{)}^2|u_1<X<u_2]$

Table 3. Relationships between $\text{̧}{h}_i(u_1,u_2);i=1,2$ and $\text{̧}H(u_1,u_2)$.

Distribution	$\text{Quantile Functions}$	$\text{̧}H(u_1,u_2)$
Exponential	$-{\lambda }^{-1}\log (1-u)$	$\lambda m(u_1,u_2)-\lambda Q\left(u_2\right))-\log \text{̧}{h}_2(u_1,u_2)$
Finite Range	$\frac{1}{a}(1-(1-u{)}^{1/b})$	$(-b+1)E[\log (1-aQ\left(u\right)|u_1<X<u_1]-\log \frac{ab}{\left(\right(1-aQ(u_1{)}^b-((1-aQ(u_2{)}^b}$
Pareto-II	$\frac{1}{p}\left(\right(1-u{)}^{1/q}-1)$	$(q+1)E[\log (1+pQ\left(u\right)\left)\right|u_1<X<u_1]-\log \frac{pq}{\left(\right(1+pQ(u_1{)}^{-q}-((1+pQ(u_2{)}^{-q}}$
Power	$a{u}^{{}^{1/b}}$	$1+G(u_1,u_2)+Q\left(u_1\right)\text{̧}{h}_1(u_1,u_2)\log \left(\frac{Q(u_1}{a}\right)-Q\left(u_2\right)\text{̧}{h}_2(u_1,u_2)\log \left(\frac{Q(u_2}{a}\right)-\log \left[\frac{b}{{\left(\frac{Q\left(u_2\right)}{a}\right)}^b-{\left(\frac{Q\left(u_1\right)}{a}\right)}^b}\right]$
Pareto-1	$a(1-u{)}^{-1/b}$	$1+G(u_1,u_2)-Q\left(u_1\right)\text{̧}{h}_1(u_1,u_2)\log \left(\frac{a}{Q\left(u_1\right)}\right)+Q\left(u_2\right)\text{̧}{h}_2(u_1,u_2)\log \left(\frac{a}{Q\left(u_2\right)}\right)-\log \left[\frac{b}{{\left(\frac{a}{Q\left(u_1\right)}\right)}^b-{\left(\frac{a}{Q\left(u_1\right)}\right)}^b}\right]$

Table 4. Numerical summary of rainfall for seeded clouds and non- seeded clouds.

	Rainfall from non-seeded clouds	Rainfall from seeded clouds
Sample size	26	26
Number of different values	25	25
Sample mean	601.8	1374.85
Sample standard deviation	849.66	1938.533
$S/\sqrt{n}$	166.63	380.18
sample min	1	4.1
Sample max	1202.6	2745.6
Next to min	4.9	7.7
Next to max	830.1	1697.8
$Q_1$	24.4	92.4
$Q_2$	44.2	221.25
$Q_3$	163	430
MQ	93.7	261.2
IQR2	277.2	675.2
$\frac{Q_2-MQ}{IQR2}$	−0.18	−0.06
Upper fence	370.9	936.4
Upper outliers	830.1,1202.6	1656,1697.8,2745.6
$\frac{MIN-MQ}{IQR2}$	−0.33	−0.38
Conclusion	Skew	symmetric

Table 5. Quantile plot table of seeded rainfall.

Rank	$Q\left(u_i\right)$	$u_i$	$Z_i$
1	4.1	.01923	−2.070
2	7.7	.076923	−1.425
3	17.5	.096154	−1.304
.	.	.	.
.	.	.	.
.	.	.	.
24	1656	.90385	1.304
25	1697	.94231	1.425
26	2745.6	.98077	2.069

Table 6. Quantile plot table of non-seeded rainfall.

Rank	$Q\left(u_i\right)$	$u_i$	$Z_i$
1	1.0	.01923	−2.070
2	4.9	.076923	−1.425
3	4.9	.096154	−1.304
4	11.5	.13462	−1.108
.	.	.	.
.	.	.	.
.	.	.	.
24	372.4	.90385	1.304
25	830.1	.94231	1.425
26	1202.6	.98077	2.070