Page 56 - IJPS-8-2

International Journal of
Population Studies Used versus Offered densities of human population

On the [A,B] range, formulas (D-1) and (D-2) enable
It follows that ϕ . ϕ ts ϕ 2 t s . ( s ) one to recover the quantiles of x and x .
t
ϕ
U
O
2 2 []α
, the quantile x
and in turn that For FA FB, U U of x at
U
U
s order α satisfies that
s

G s 2ϕ  ϕ 2 t dt
FA
2 2 lnx ln A , therefore
U

FB FA ln B ln A
s U U
2ϕ ϕ F U
udu
A
B
2 x [] A() F U
B
A
F U
s U A
22ϕ( ) []

α
,
2 The quantile x O of x at order FA FB
O
O
O
satisfies that
By integration, x
FA
s O 1 1 O , therefore

s s v 2 FB FA A x ln B ln A
U
U

G 2 ϕ dv 2 ϕ 1 ln B ln A FA

vdv
wdw
O
U
0 0 2 0 x [] 1 / A x FB FA
O
s O U
2 Φ Φ 0
2 If the linear-log assumption is valid on the full range of
x , then FA 0 and FB 1, yielding that
Lastly, formula (C-5) is obtained on making use of: U U U
s x [] A() () 1 B () (D-3)
B
A


G s G . U A
G v dv
0
0 BA
x
Appendix D: Linear-log model of used density CDF U ln B ln A
Here, it is assumed that, on a range AB, of x, the ln B ln A

used density CDF, F , is an affine linear function of ln x: From FA 0 and FB 1 , x 1 1
O
O
O
u
thus, for xAB, , A B

FB
FA
F x U U (ln x ln) (D-1) 1 1

F A
A .
U
U
ln B ln A .
x [] A B
O
Differentiating with respect to x, the used density PDF
is obtained as a hyperbolic function: Applying (D-3) to order 1 , the median M of x
FA
1 FB 2 U U
U
U
f x ln B ln A for xAB, . satisfies M = AB . Applying it again to orders ¾ and ¼, the
x

U
U
interquartile ratio RIQ satisfies RIQ x U 075. B A/.
From the general relationship (14), the offered density U U x 025.
U
is determined as From the empirical M and RIQ , the A and B parameters
U
U
FA
x FB for France as of 2019 are recovered as
f O for xAB, .
U
U
x

O
x 2 ln B ln A M
A = U = 24 p/km²,
From this stem the offered density CDF on the range: RIQ U
FA
1 1 FB
U
U
F A
F x ( x ) (D-2) BM RIQ= = 14,895 p/km².
O
O
A x O ln B ln A U U
Volume 8 Issue 2 (2022) 50 https://doi.org/10.36922/ijps.v8i2.297