Page 129 - IJPS-11-1

International Journal of
Population Studies Analysis of age-specific fertility in India

measure the stability of R of the model, we can evaluate 2.4.2. Skew-normal type-2 distribution
2
(1-shrinkage), which is defined as: The skew-normal type-2 (or two-piece normal) distribution
is defined as (Johnson et al., 1994):
1shrinkage 1 cvpp R 2 (V)
2
1 z

2

2
where shrinkage 2 cvpp R . c exp I y
2

fy |, , 2

z

2.3.3. F-test exp 1 Iy (
) (VIII)

The F-test is used to verify the significance of R as well as 2
2
the significance of the proposed model. The F-test statistic

is given by (Gujarati, 1998); y
for −∞ < (y,) < ∞ and (σ,ν) > 0; where z ,
ESS R 2 2
F ( m ) 1 ( m ) 1 (VI) and c 1 2 . Note that I(.) is an indicator function,
RSS (1 R )
2
( nm) ( nm) where I(x)=1 if x is true, and I(x) = 0 if x is false.

where m is the total number of parameters of the fitted 2.5. Skew-t distribution
model, n is the number of cases, ESS is the error sum of There are five types of skew-t distributions that are generally
square, RSS is the regression sum of square, and R is the used, namely, skew-t type-1, type-2, type-3, type-4, and
2
coefficient of determination of the model. type-5. Here, we used only three skew-t distributions, that
Further, we considered six selected skewed is, skew-t type-3, type-4, and type-5. Interested readers may
distributions, viz. skew-normal (type-1, type-2), refer to Azzalini (1986) and Azzalini and Capitanio (2003)
skew-t (type-3, type-4, and type-5), and skew-logistic for skew-t type-1 and type-2 distributions, respectively.
distribution, to study the pattern of ASFRs of women Three considered skew-t distributions are briefly discussed
in India. The parameters of the proposed models were below.
estimated by the Rigby & Stasinopoulos (2005) method. 2.5.1. Skew-t type-3 distribution
The models with their probability density functions
(PDFs) are discussed below. The skew-t type-3 is a spliced scale distribution and the
PDF is defined as (Fernández and Steel, 1998):
2.4. Skew-normal distribution 2

In this study, we considered two types of skew-normal fy |, ,, c 1 z 2 Iy( ) 1 2 Iy( ) (IX)

distributions: skew-normal type-1 and skew-normal type-
2, which are explained below briefly. for −∞ < (y,) < ∞ and (σ,ν,γ) > 0; where z = (y−µ)/σ
2.4.1. Skew-normal type-1 distribution 2
and c . Here  is the location
The PDF of skew-normal type-1 distribution is defined as 1 2 B 1

1 2/

,

(Azzanani, 1986): 22
parameter, σ is the scale parameter, ν is the degree of
2

( )
fy |, , fz Fz (VII) freedom and γ is the skewness parameter.

y Here, the mode of the distribution is . The mean and

for −∞ (y,,ν) < ∞ and σ > 0; where z , and variance are given by E(Y) =  + σE(z) and Var(Y) = σ
2
1
2
f (.) and F (.) are the PDF and cumulative distribution Var(z), where Ez 2( 2 1) and
function (cdf) of standard normal distribution respectively. 1 B 1

,

The mean and variance are given by 3 1 22

12/

3

EY 2 2 / 1 2 and Ez

2
2
1

VarY 12 2 1 2 .
/
2

Volume 11 Issue 1 (2025) 123 https://doi.org/10.36922/ijps.1338