An Extension of Karrup–King–Newton Index

Universit ́e de Caen Normandie, LMNO, Campus II, Science 3, Caen 14032, France Department of Statistics, the Islamia University of Bahawalpur, Rahim Yar Khan Campus, Bahawalpur, Pakistan Department of Statistics, the Islamia University of Bahawalpur, Bahawalpur, Pakistan Department of Statistics Government College University Lahore, Lahore, Pakistan Department of Applied Statistics, Financial Mathematics and Actuarial Science, Jaramogi Oginga Odinga University of Science and Technology, Bondo, Kenya


Karup-King-Newton Index: A Measure of
Age Heaping e Karup-King-Newton formula is also called Newton's quadratic formula. It is a well-known fact that age distortion is minimized when the age data are given at a single age and then grouped into the 5-year age interval, but some errors still remain in the5-year age grouping. Age distortion will be further reduced considerably by the10-year age group, as errors that transfer beyond the 5-year age group will be captured.
is method is based on the following assumptions.
(i) e transfer of age misreporting does not stretch the 5-year age group, so we group into the 10-year age group. is method presumes that the error will be removed.
(ii) e patterns of error are the same in the consecutive six 5-year age groups. e limitations of this method are the following ones.
(i) e errors across 10-year age groups, most particularly older ages, are not covered. (ii) ey cannot adjust the values of ages 0-9. (iii) It may smooth out the genuine pattern. (iv) It makes use of the values of ages 0-9 in the computations, but the error pattern in the age group is different from that of ages 10-70.
e used formulas are given below.
where 5 P x is the first of two 5-year age groups compared to the 10-year age group 10 P x .

Modified Karup-King-Newton Index
In the proposed modification, we use the 3-year age group as compared to the 5-year age group or the 10-year age group. e 3-year age interval is better in the sense that it will reduce errors as compared to the 5-year age group. Age misreporting will be further reduced by using 6-year age groupings, as errors that transfer beyond the 3-year age group will be captured. e index is developed on the principle of the original Karup-King-Newton index. e derivation of the index is given below.
Suppose that the population in 3-year age groups is divided into six consecutive groups denoted by W 0 ,W 1, W 2 ,W 3 ,W 4 , and W 5 . e 6-year age group is made up of two consecutive 3year age groups. Let the population be divided into 6-year age groups denoted by V 0 , V 1 , and V 2 such that e difference table using the cumulated populations is given in Table 1.
Note that is equation is equal to the Karup-King multiplier. e values in the above table are found using the method described below.
σ v for age x and x + 6 are � P + V 0 − P/6 � V 0 /6 , where 6 is the six-year interval. σ v for age x+9 and x+6 are σ v for age x+15 and x+12 are � V 2 /12. For the calculation of σ 2 v of first term, we have done the following operation: e same method is applied for next differences.
We will need to find the smoothed values for the 3-year age groups equivalent to W 2 and W 3 , as the center pair of values.
Let us denote with "U" the smoothed value of W 2, so the smoothed value of W 3 will be V 1 -U, since the total of the smoothed value should always be equal to the original population data. Let "σ v " be the divided differences of the age groups.
From Table 1, a smooth pattern in the cumulated age distribution is given as We split 6-year age group into three-year age group.
It is exciting to see that (4) and (5) are also the same as the Karup-King oscillatory multipliers for splitting 6-year age groups into 3-year age groups.

Assumption
(i) e transfer of age (net age misreporting error) does not stretch beyond 3 years, so by grouping into a 6year age group, the method assumes that the error will be cancelled out. (ii) e patterns of error are similar in six successive 3-year age groups.

Limitations
(1) e method cannot be used for calculating adjusted values for people aged 72-74 and above. (2) It may smooth out a genuine pattern.
(3) It makes use of ages 0-4 and ages 5-9 in the computation, whereas error pattern of age groups 10-70 are different from reported age. (4) e decision to use this method or other methods of adjustment is largely arbitrary.

Graphical Representation of Both Indices
To access the modified Karup-King-Newton index, we compare it with the original Karup-King-Newton index. e graphs for both indices are shown in Figure 1 Karup-King-Newton index is moderately easy to calculate, allowing the linearity assumptions.

Final Remarks
In this paper, we used the Karup-King-Newton and modified Karup-King-Newton indices of age heaping to examine the accuracy of age data from Pakistan Census data 1998 and Ghana Census data 2010 and examined the quality of reported data from the population census 1998. In the literature, Tekpli [1] applied five smoothed techniques to Ghana data to check for age misreporting. Also, in the literature, we see that the main problem in age misreporting is lack of education [2][3][4][5][6][7][8][9].
In previous years, researchers used only the Whipple type index, Bachi index, and the Myers blended index to reduce the error in age distribution; no one used the other methods to overcome the error in age misreporting [10,11]. Our proposed Karup-King-Newton index will produce

Age group
Population      Data Availability e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Authors' Contributions
Christophe Chesneau gave the initial idea and proposed the new model. Ronald Onyango investigated properties and carried out calculations. Abdul Majeed Lodhi was responsible for the numerical part of the paper. Jamal Abdul Nasir and Farrukh Jamal mutually wrote and structured the manuscript and also verified and investigated numerical calculations applied to the dataset. Sadaf Khan revised the final version according to the reviewers' comments and also improved the language and presentation of the final version.