EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5449
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A Novel Forced Scrambling Model Under Time-Scaled
Surveys for Estimation of Population Variance

Muhammad Azeem1,∗, Abdul Salam1, Sundus Hussain2

1 Department of Statistics, University of Malakand, Khyber Pakhtunkhwa, Pakistan
2 Department of Statistics, Shaheed Benazir Bhutto Women University, Peshawar, Pakistan

Abstract. In survey sampling, forced randomized response models are special variants of tradi-
tional models employed by researchers in sensitive surveys. The existing forced models are based
on traditional one-time surveys where the respondents are interviewed only at a single time-point.
These existing one-time survey-based models suffer from a serious drawback - the scrambling
process is performed only once by each respondent. The lack of replication usually results in mea-
surement errors which often have a negative influence on the estimators of population parameters.
This study reveals that time-scaled surveys provide more efficient estimates of the population vari-
ance of sensitive variables than the traditional one-time surveys, under forced randomized response
models. An Exponentially Weighted Moving Average (EWMA) estimator is used to estimate the
population variance, based on the responses obtained at different time points. Further, a new forced
randomized scrambling model is also proposed and the improvement over the available models is
observed.

2020 Mathematics Subject Classifications: 62D05, 62F10

Key Words and Phrases: Forced model, EWMA estimator, scrambled response, time-scaled
surveys, traditional surveys

1. Introduction

In sample surveys, Warner [27] presented the idea of getting scrambled responses in
sensitive sample surveys. Gjestvang and Singh [7] introduced a forced scrambling model,
using fixed responses. Later on, Diana and Perri [4] presented a linear model, achieving
further improvement over the previous scrambling models. Murtaza et al. [17] analyzed
the use of correlated variables under randomized response techniques. Azeem and Salam
[3] recently presented a new model based on the mixture of direct response and scrambled
response. Gupta et al. [10] suggested an optional scrambling technique for application in
sensitive surveys. Pushadapu et al. [21] and Narjis and Shabbir [18] also explored different

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5449

Email addresses: azeemstats2017@gmail.com (M. Azeem),
asalamuom@gmail.com (A. Salam), sundashusain@sbbwu.edu.pk (S. Hussain)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



M. Azeem, A. Salam, S. Hussain / Eur. J. Pure Appl. Math, 18 (1) (2025), 5449 2 of 19

aspects of randomized scrambling models.
The available scrambling models are all based on one-time surveys, where the data is
obtained at one time-point. In practice, some survey situations may require data collec-
tion repeatedly at different points in time. Time-scaled surveys can provide estimates of
the parameters at different time points. Further, time-scaled surveys are also useful in
measurement of the frequency and/or duration of different events of interest. For details
about the different applications of time-scaled surveys, readers may refer to the studies
of Duncan and Kalton [5] and Fuller [6]. More recently, Alomair and Iftikhar [1] also
discussed different real-world uses of time-based sample surveys. Qureshi et al. [22] also
used time-scaled surveys to develop efficient estimators.
The exponential weighted moving average (EWMA) technique was first suggested by
Roberts [23] for monitoring the process mean using control charts. The study of Haq [11]
improved the EWMA method by developing Hybrid EWMA (abbreviated as HEWMA)
control charts. These methods assign weights to the data obtained at different time-points
in such a manner that the recent-most data set gets the highest weightage. The weightage
decreases as the time points get older. The data gathered at the oldest time-point gets
the smallest weightage. As far as estimation of parameters is concerned under time-scaled
surveys, the studies of Noor-ul-Amin [19, 20] analyzed novel memory-type estimators un-
der time-scaled sample surveys.
In addition to central tendency metrics, survey researchers often need to analyze the vari-
ability among population units, e.g. the variation in the monthly income of individuals,
the variation in the taxes paid, and the variation in the number of cigarettes consumed,
etc. Since these variables are sensitive, so a randomized response data collection method
may be more appropriate than direct questioning. Once the data collection process is
completed, a suitable variance estimator can be used to estimate the population variance.
Over the past fifty years, many variance estimators have been proposed by researchers un-
der different sampling methods. Isaki [12] introduced a ratio-type variance estimator using
non-sensitive variable. Gupta et al. [9] analyzed variance estimation of sensitive variables
under the existence of auxiliary information. Recently, Saleem et al. [24], Kumar et al.
[14], and Azeem et al. [2] also suggested variance estimators under one-time sample sur-
veys. Shahzad et al. [25] studied variance estimation using calibration-based estimation
methods. Jewsbury [13] introduced new variance estimation techniques and showed the
validity of the proposed methods using simulation. Mahdizadeh and Zamanzade [15, 16],
also studied variance estimation under ranked set sampling design. Sluijterman et al. [26]
studied optimal implementation approaches of a mean variance estimation network.
In the traditional randomized response methods, each survey participants performs the
scrambling process at one particular time-point to report his/her scrambled response. The
numerical value of the reported response may be smaller or larger than the true response,
depending on the random number chosen by the respondent. The use of random numbers
in the scrambling process may result in large differences between the reported response and
the true response. The lack of replication in one-time surveys may make the measurement
errors even worse. To overcome this issue, time-scaled surveys may be employed where
the scrambling process is replicated at many time points. Such replication of scrambling



M. Azeem, A. Salam, S. Hussain / Eur. J. Pure Appl. Math, 18 (1) (2025), 5449 3 of 19

process may balance out the under-reporting and over-reporting over time. This process
ensures that reliable data is obtained from the survey respondents yet protecting their
privacy.
This manuscript is outlined as follows:
An introduction to the notations has been given in Section 2 along with the assumptions
used in the current study. Section 3 introduces EWMA estimators, with Section 4 provid-
ing some available scrambling models. The proposed new forced response model has been
presented in Section 5. The results of a real-word survey have been presented in Section 6.
The findings of the performance-evaluation have been presented in Section 7. A detailed
discussion about the results of the current study has been presented in Section 8. Finally,
some recommendations for future researchers have been provided in Section 9.

2. Notations and Assumptions

Before proceeding further, let us introduce some notations. Let the population and
sample size be denoted byN and n, respectively. Let Y be the main variable of interest and
let the additive and multiplicative random variables be denoted by S and T , respectively.
Moreover, we use the notations µY , µS , and µT to denote the means of Y , S, and T ,
respectively, for the population data. Likewise, σ2

Y , σ2
S , and σ2

T are the notations for
the population variance of Y , S, and T , respectively, in the subsequent sections. Let us
also employ some assumptions in line with those used by the previous researchers. These
assumptions include: µS = 0, and µT = 1. Further, to facilitate privacy protection, we
use the assumption of independence among all possible combinations of the variables.
In addition to the main variable, this paper also analyzes ratio estimators of variance
under forced response models. For this, we use an auxiliary variable denoted by X which
is assumed to have correlation with the main variable. Let the sample and population
mean of the auxiliary variable X be denoted by x̄ and µX , respectively, with s2X and
σ2
X denoting the variances for the sample and population data, respectively. Further, for

positive numbers, ‘r’ and ‘s’, we can define the moment ratio as:

λTS =
ηTS

η
r
2
20η

s
2
02

, (1)

ηTS =

∑N
i=1(xi − µX)s(yi − µY )

r

N − 1
. (2)

3. EWMA-type Estimators

In time-scaled surveys, we can compute Exponentially Weighted Moving Average
(EWMA) type estimators for estimating the population parameters. Consider a smoothing
constant ‘c’ which is predefined by the researcher, we can define an EWMA-type estimator
of the mean of variable Y as follows:

Ût = cz̄ + (1− c)Ût−1, (3)



M. Azeem, A. Salam, S. Hussain / Eur. J. Pure Appl. Math, 18 (1) (2025), 5449 4 of 19

where 0 < c < 1, and z̄ denotes the sample mean calculated from the observed scrambled
response. An EWMA-type variance estimator is expressed as follows:

V̂t = cs2Z + (1− c)V̂t−1. (4)

In (4), s2Z denotes the sample variance computed from the scrambled response. An EWMA
estimator of the finite population variance of variable X is presented as:

Ŵt = cs2X + (1− c)Ŵt−1. (5)

An EWMA ratio-type variance estimator of the variance of Y can be written as:

Rt = V̂t
σ2
X

Ŵt

. (6)

In the subsequent sections, different estimators of the population variance of the sensitive
main variable Y have been discussed.

4. Available Scrambling Models

Let us revisit two of the existing randomized response scrambling models and the
underlying estimators.

4.1. Gjestvang and Singh [7] Forced Model (R1)

Z =


Y, with probability P1,

TY, with probability P2,

F, with probability P3 = 1− P1 − P2,

(7)

where Z denotes the scrambled response.
Under traditional one-time sample surveys, a simple unbiased variance estimator of Y is:

s2Y =
1

n− 1

n∑
i=1

(yi − ȳ)2. (8)

An unbiased variance estimator of the variance of Z is as follows:

s2Z =
1

n− 1

n∑
i=1

(zi − z̄)2. (9)

An unbiased variance estimator of the variance of X is as follows:

s2X =
1

n− 1

n∑
i=1

(xi − x̄)2. (10)



M. Azeem, A. Salam, S. Hussain / Eur. J. Pure Appl. Math, 18 (1) (2025), 5449 5 of 19

In traditional sample surveys, a simple variance estimator of a sensitive variable using the
model (R1) is given by:

s2Y (GS) =
s2Z − (p1 + p2)(1− p1 − p2)(z̄ − F )2 − p2σ

2
T z̄

2

p1 + p2(σ2
T + 1)

. (11)

In traditional sample surveys, a ratio-type estimator of Y is:

RGS =
s2Z − (p1 + p2)(1− p1 − p2)(z̄ − F )2 − p2σ

2
T z̄

2

p1 + p2(σ2
T + 1)

σ2
X

s2X
. (12)

In time-scaled surveys, a ratio-type EWMA estimator of Y model is expressed as follows:

Rt(GS) =
V̂t − (p1 + p2)(p1 − p2)(Ût − F )2 − p2σ

2
T Û

2
t

p1 + p2(σ2
T + 1)

σ2
X

Ŵt

. (13)

4.2. Saleem et al. [24] Model (R2)

The Saleem et al. [24] scrambling model is given by the following equation:

Z = γ(Y + αS) + (1− γ)T (Y + αS), (14)

where 0 < γ < 1 , with α denoting some pre-defined constant.
In order to simplify the comparison, we use α = 1 in equation (14) to get:

Z = γ(Y + S) + (1− γ)T (Y + S). (15)

In traditional sample surveys, a simple variance estimator based on the Saleem et al. [24]
model is expressed as:

s2Y (S) =
s2Z − (1− γ)2σ2

T (σ
2
S + z̄2) + z̄2

(1− γ2)σ2
T

. (16)

Under traditional one-time sample surveys, a ratio-type estimator of Y is as follows:

RS =
s2Z − (1− γ)2σ2

T (σ
2
S + z̄2) + z̄2

(1− γ2)σ2
T

σ2
X

s2X
. (17)

Under time-scaled sample surveys, the ratio-type EWMA variance estimator of Y using
the Saleem et al. [24] model is expressed as follows:

Rt(S) =
V̂t − (1− γ)2σ2

T (σ
2
S + Û2

t ) + Û2
t

(1− γ2)σ2
T

σ2
X

Ŵt

. (18)



M. Azeem, A. Salam, S. Hussain / Eur. J. Pure Appl. Math, 18 (1) (2025), 5449 6 of 19

5. Proposed Forced Scrambling Model (R3)

Motivated by the study of Saleem et al. [24], the following forced scrambling model is
proposed.

Z =

{
γ(Y + TS) + (1− γ)(TY + TS), with probability P

F, with probability 1− P ,
(19)

where 0 < γ < 1, and F denotes a pre-determined forced response, such that E(F ) = F ,
and V ar(F ) = 0. A simple mean estimator under the proposed model is given as follows:

µ̂P =
z̄ − (1− P )F

P
. (20)

In order to derive variance of Z, we take expectation on equation (19) to get:

E(Z) = P [γE(Y + TS) + (1− γ)E(TY + TS)] + (1− P )E(F ).

We can use the assumptions of Section 2 to get:

E(Z) = PµY + (1− P )F. (21)

Using the result of equation (21), we can easily show that:

E(µ̂P ) = µY . (22)

Also,

E(Z2) = P
(
E [γ(Y + TS) + (1− γ)(TY + Y S)]2

)
+ (1− P )E(F 2).

Using the assumptions discussed in Section 2, the variance of Z can be obtained as:

V ar(Z) = Pσ2
Y

[
γ2 + (1− γ)2(σ2

T + 1) + (1− γ)2σ2
S + 2γ(1− γ)

]
+ γ2P

[
(σ2

T + 1)σ2
S + µ2

Y

]
+ (1− γ)2P [(σ2

T + 1)µ2
Y + µ2

Y σ
2
S ] + 2γ(1− γ)P [µY σ

2
S + µ2

Y ] + (1− P )F 2

− [PµY + (1− P )F ]2,

or,

σ2
Z = Pσ2

Y

[
γ2 + (1− γ)2(σ2

T + σ2
S + 1) + 2γ(1− γ)

]
+ γ2P

[
(σ2

T + 1)σ2
S + µ2

Y

]
+ (1− γ)2P [(σ2

T + 1)µ2
Y + µ2

Y σ
2
S ] + 2γ(1− γ)P [µY σ

2
S + µ2

Y ]

− P 2µ2
Y + P (1− P )(F − 2µY )F.

(23)

Applying variance on equation (20) leads to:

V ar(µ̂P ) = V ar

[
1
n

∑n
i=1 Zi − (1− P )F

P

]
=

1

n2P 2

n∑
i=1

V ar(Zi). (24)



M. Azeem, A. Salam, S. Hussain / Eur. J. Pure Appl. Math, 18 (1) (2025), 5449 7 of 19

Using equation (23) in equation (24) gives:

V ar(µ̂P ) =
1

nP
[σ2

Y {γ2 + (1− γ)2(σ2
T + σ2

S + 1) + 2γ(1− γ)}

+ γ2{(σ2
T + 1)σ2

S + µ2
Y }+ (1− γ)2{(σ2

T + 1)µ2
Y + µ2

Y σ
2
S}

+ 2γ(1− γ){µY σ
2
S + µ2

Y } − Pµ2
Y + (1− P )(F − 2µY )F ].

(25)

Solving equation (23) for σ2
Y yields:

σ2
Y =

1

PA
[σ2

Z − γ2P{(σ2
T + 1)σ2

S + µ2
Y } − (1− γ)2P{(σ2

T + 1)µ2
Y + µ2

Y σ
2
S}

− 2γ(1− γ)P (µY σ
2
S + µ2

Y ) + P 2µ2
Y − P (1− P )(F − 2µY F )],

(26)

where,
A = γ2 + (1− γ)2(σ2

T + σ2
S + 1) + 2γ(1− γ). (27)

A basic variance estimator under traditional surveys may be obtained by using estimators
in place of population parameters to yield:

s2Y (R) =
1

PA
[s2Z − γ2P{(σ2

T + 1)σ2
S + z̄2} − (1− γ)2P{(σ2

T + 1)z̄2 + z̄2σ2
S}

− 2γ(1− γ)P (z̄σ2
S + z̄2) + P 2z̄2 − P (1− P )(F − 2z̄)F )].

(28)

In the case of traditional one-time surveys, a ratio-type estimator of the variance of Y is:

RP =
1

PA
[s2Z − γ2P{(σ2

T + 1)σ2
S + z̄2} − (1− γ)2P{(σ2

T + 1)z̄2 + z̄2σ2
S}

− 2γ(1− γ)P (z̄σ2
S + z̄2) + P 2z̄2 − P (1− P )(F − 2z̄)F ]

σ2
X

s2X
.

(29)

In the case of time-scaled surveys, a ratio-type EWMA estimator of Y using the proposed
model is:

Rt(P ) =
1

PA
[V̂t − γ2P{(σ2

T + 1)σ2
S + Û2

t } − (1− γ)2P{(σ2
T + 1)Û2

t + Û2
t σ

2
S}

− 2γ(1− γ)P (Ûtσ
2
S + Û2

t ) + P 2Û2
t − P (1− P )(F − 2Ût)F ]

σ2
X

Ŵt

.
(30)

Theorem 1. Using the proposed forced model, as t −→ ∞, the mean squared error of Rt(P )

may be obtained as:

MSE(Rt(P )) =
θ

P 2A2
(

c

2− c
)[σ4

Z(λ40 − 1) + 4B2C2
Z +D2(λ04 − 1)− 4Bσ2

Zλ12CZ

− 2Dσ2
Z(λ22 − 1) + 4BDλ30CZ ],

(31)

where, θ = 1
n , and

C2
Z = C2

Y σ
2
T +

σ2
S

Ȳ 2
, (32)



M. Azeem, A. Salam, S. Hussain / Eur. J. Pure Appl. Math, 18 (1) (2025), 5449 8 of 19

A = γ2 + (1− γ)2(σ2
T + σ2

S + 1) + 2γ(1− γ), (33)

B = γ2Z̄2+(1−γ)2PZ̄2(σ2
T +σ2

S+1)+γ(1−γ)P (Z̄σ2
S+2Z̄2)−P 2Z̄2−P (1−P )Z̄F, (34)

and,

D = σ2
Z − γ2P{(σ2

T + 1)σ2
S + Z̄2} − (1− γ)2P (σ2

T + σ2
S + 1)Z̄2

− 2γ(1− γ)P (Z̄2σ2
S + Z̄2) + P 2Z̄2 − P (1− P )(F − 2Z̄)F.

(35)

Proof. Define:

V̂t = σ2
Z(1 + dZt),

Ŵt = σ2
X(1 + dXt),

Ût = Z̄(1 + eZt),

(36)

so that,

dZt =
V̂t

σ2
Z
− 1, dXt =

Ŵt

σ2
X
− 1, and, eZt =

Ût

Z̄
− 1.

Applying expectation yields, E(dZt) = E(dXt) = E(eZt) = 0
Further,

E(d2Zt) = θ(
c

2− c
)[1− (1− c)2t](λ40 − 1),

E(d2Xt) = θ(
c

2− c
)[1− (1− c)2t](λ04 − 1),

E(e2Zt) = θ(
c

2− c
)[1− (1− c)2t]C2

Z ,

E(dZtdXt) = θ(
c

2− c
)[1− (1− c)2t](λ22 − 1),

E(dZteZt) = θ(
c

2− c
)[1− (1− c)2t]λ30CZ ,

E(dXteZt) = θ(
c

2− c
)[1− (1− c)2t]λ12CZ .

(37)

As t
∞−→, 1− (1− c)2t → 1 equation (37) tends to:

E(d2Zt) = θ(
c

2− c
)(λ40 − 1),

E(d2Xt) = θ(
c

2− c
)(λ04 − 1),

E(e2Zt) = θ(
c

2− c
)C2

Z ,

E(dZtdXt) = θ(
c

2− c
)(λ22 − 1),

E(dZteZt) = θ(
c

2− c
)λ30CZ ,

E(dXteZt) = θ(
c

2− c
)λ12CZ .

(38)



M. Azeem, A. Salam, S. Hussain / Eur. J. Pure Appl. Math, 18 (1) (2025), 5449 9 of 19

The ratio-type EWMA estimator may be written as follows:

Rt(P ) =
1

PA
[σ2

Z(1 + dZt)− γ2P{(σ2
T + 1)σ2

S + Z̄2(1 + eZt)
2}

− (1− γ)2P{(σ2
T + σ2

S + 1)Z̄2(1 + eZt)
2}

− 2γ(1− γ)P{Z̄(1 + eZt)σ
2
S + (Z̄2(1 + eZt)

2}+ P 2Z̄2(1 + eZt)
2

− P (1− P ){F − 2Z̄(1 + eZt)}F ]
σ2
X

σ2
X(1 + dXt)

.

(39)

Further simplification yields:

Rt(P ) − σ2
Y =

1

PA
[σ2

ZdZt − 2BeZt −DdXt]. (40)

Squaring equation (40), applying expectation, and after simplification, we get the required
result as:

MSE(Rt(P )) =
θ

P 2A2
(

c

2− c
)[σ4

Z(λ40 − 1) + 4B2C2
Z +D2(λ04 − 1)− 4Bσ2

Zλ12CZ

− 2Dσ2
Z(λ22 − 1) + 4BDλ30CZ ].

Remark 1: The mean square error of the ratio estimator using the proposed forced
model under traditional one-time surveys may be written as:

MSE(R(P )) =
θ

A2
[σ4

Z(λ40 − 1) + 4B2C2
Z +D2(λ04 − 1)− 4Bσ2

Zλ12CZ

− 2Dσ2
Z(λ22 − 1) + 4BDλ30CZ ].

(41)

Remark 2: Using equation (36) and equation (38), the mean square error of the ratio-
type estimator using the Saleem et al. [24] model under traditional one-time surveys can
be obtained as:

MSE(R(S)) =
θ

(1− γ)4σ4
T

[σ4
Z(λ40 − 1) +G2C2

Z +H2(λ04 − 1)− 2Gσ2
Zλ30CZ

− 2Hσ2
Z(λ22 − 1) + 2GHλ12CZ ],

(42)

where,
G = 2Z̄2{(1− γ)2σ2

T − 1}, (43)

and,
H = σ2

Z − (1− γ)2σ2
T (Z̄

2 + σ2
S) + Z̄2. (44)

A time-scaled variant of the mean square error given in equation (42) can be obtained as
follows:

MSE(Rt(S)) =
θ

(1− γ)4σ4
T

(
c

2− c
)[σ4

Z(λ40 − 1) +G2C2
Z +H2(λ04 − 1)− 2Gσ2

Zλ30CZ

− 2Hσ2
Z(λ22 − 1) + 2GHλ12CZ ].

(45)



M. Azeem, A. Salam, S. Hussain / Eur. J. Pure Appl. Math, 18 (1) (2025), 5449 10 of 19

Remark 3: Using equation (36) and equation (38), the mean square error of the ratio-
type estimator using the Gjestvang and Singh [7] forced model under traditional one-time
surveys can be obtained as:

MSE(RG(S)) =
θ

[P1 + P2(σ2
T + 1)]2

[σ4
Z(λ40 − 1) + J2C2

Z +K2(λ04 − 1)− 2Kσ2
Z(λ22 − 1)

− 2Jσ2
Zλ30CZ + 2JKλ12CZ ],

(46)

where,
J = 2{P2σ

2
T Z̄

2 + (P1 + P2)(1− P1 − P2)Z̄(F − Z̄)}, (47)

K = σ2
Z − P2σ

2
T Z̄

2 − (P1 + P2)(1− P1 − P2)(F − Z̄)2. (48)

A time-scaled variant of the mean square error given in equation (46) can be obtained as
follows:

MSE(Rt(GS)) =
θ

[P1 + P2(σ2
T + 1)]2

(
c

2− c
)[σ4

Z(λ40 − 1) + J2C2
Z +K2(λ04 − 1)

− 2Kσ2
Z(λ22 − 1)− 2Jσ2

Zλ30CZ + 2JKλ12CZ ].

(49)

Remark 4: The variance of the simple unbiased variance estimator can be obtained by
simply ignoring the terms containing λ04, λ22, and λ12 in equation (46). Thus, under
the proposed model, the variance of the simple unbiased variance estimator, V̂t, may be
derived as:

V AR(V̂t) =
θ

P 2A2
(

c

2− c
)[σ4

Z(λ40 − 1) + 4B2C2
Z + 4BDλ30CZ ]. (50)

Remark 5: The bias of Rt(P ) under the proposed model in time-scaled surveys can be
derived as:

Bias(Rt(P )) ≈
−θ

PA
(

c

2− c
)[P (1− P )Z̄2C2

Z + σ2
Z(λ22 − 1)− 2Gλ12CZ ],

where,

G = P [Z̄2{γ2 + (1− γ)2(σ2
T + σ2

S + 1)}+ {γ(1− γ)(Z̄σ2
S + 2Z̄2)− 2PZ̄2 − (1− P )FZ̄}].

6. A Real-World Survey using the Proposed Method

We applied the proposed time-based randomized survey method to obtain data from
a sample of 50 undergraduate students. Data was collected on the number of times the
students cheated in last examination. Using our suggested model, we collected the scram-
bled responses on three different days (three time-points). The interviewer guided the
respondents in calculating their scrambled responses. Using calculators, the respondents
calculated and reported their scrambled responses using the proposed model. The ob-
served responses at different time-points have been displayed in Table 1.



M. Azeem, A. Salam, S. Hussain / Eur. J. Pure Appl. Math, 18 (1) (2025), 5449 11 of 19

Table 1: Responses of Survey Participants at Different Time-Points

Responses on Day 1(t = 1) Responses on Day 2(t = 2) Responses on Day 3(t = 3)

1.4 9.6 -7.1 2.8 -0.6 2.1 -1.6 7.4 -2.0 -3.9 -7.3 -1.5 5.0 2.1 -6.3

7.3 -6.1 2.4 0.4 4.5 -3.4 3.1 -6.2 0.1 3.0 1.8 4.7 0.4 -4.0 4.1

-2.6 3.0 8.3 -6.9 3.0 -2.9 -7.3 3.0 -4.5 -2.6 -2.5 -0.4 -2.0 3.0 -2.5

0.3 -1.8 -4.2 -1.0 5.8 9.1 -1.1 4.8 -7.1 5.0 3.0 0.3 -0.2 -1.1 -1.6

-5.2 -7.4 5.8 3.0 -3.7 -3.3 5.4 -4.2 4.2 -2.9 -1.4 -3.5 1.3 8.1 3.0

0.8 0.3 1.5 4.2 -1.2 3.0 1.5 1.1 0.5 8.6 0.6 1.0 3.1 0.6 -4.7

4.9 -4.7 -0.6 7.1 4.3 2.7 3.0 -6.6 -3.5 2.4 -1.0 -0.5 -4.8 0.1 4.0

-3.5 8.4 5.9 3.7 8.0 -1.8 0.0 -0.9 1.4 -6.7 8.2 0.1 6.4 -3.8 7.8

-6.4 -3.1 3.4 -1.8 -5.1 1.4 -1.9 3.2 -4.2 0.2 -3.9 3.0 -2.5 3.0 -2.1

2.7 4.6 -6.1 2.1 0.1 -5.7 -4.2 -2.6 2.0 3.0 -0.4 -4.6 3.0 -0.9 -5.3

It should be noted that the number of times a candidate cheated in an examination is
a positive integer value, however, Table 1 shows that many of the observed responses are
negative real numbers. This is due to the scrambling process employed by the proposed
model which helps in protecting the privacy of the respondents.
Using the responses given in Table 1, we can calculate the estimator of population variance
as:
V̂t = Cs2Z + (1− t)V̂t−1.

The value of V̂t can be chosen by the researcher based on some prior knowledge about the
population variance. Alternatively, a pilot survey may also be conducted to get the value
of V̂t. In this example, we choose at time t = 0. Let c = 0.3, so the variance at time t = 1
may be calculated by the formula:

V̂1 = 0.3s2Z + 0.7(10).

At time-point t = 1 (Day 1), the sample variance of the 50 observed responses is s2Z =
21.75, therefore:

V̂1 = 0.3(21.75) + 0.7(10) = 13.525.

At time-point t = 2 (Day 1), the sample variance of the 50 observed responses is s2Z =
16.95, therefore:

V̂1 = 0.3(16.95) + 0.7(10) = 14.553.

At time-point t = 3 (Day 1), the sample variance of the 50 observed responses is s2Z = 13.69
, therefore:

V̂1 = 0.3(13.69) + 0.7(10) = 14.294.



M. Azeem, A. Salam, S. Hussain / Eur. J. Pure Appl. Math, 18 (1) (2025), 5449 12 of 19

In this example, we collected the data at three different time-points. Researchers may
collect the data at more than three time-points, depending on the availability of time and
other resources to conduct the survey.

7. Efficiency Comparison

The efficiency condition for the ratio-type EWMA variance estimator and traditional
ratio estimator using the proposed model may be derived as:

MSE(Rt(P )) < MSE(RP ).

After simplification, the efficiency condition reduces to the simple form:

c

2− c
< 1,

or,

c < 1.

Yan et al. [28] developed a model-evaluation metric as follows:

∆ = E(U − Y )2. (51)

Gupta et al. [8] suggested a better evaluation measure which can be expressed as:

δ =
V ar(µ̂)

∆
. (52)

Table 2 displays the empirical results of the efficiency comparison of the simple variance
estimator between traditional and time-scaled sample surveys, under different scrambling
models. Table 3 shows the results of the corresponding ∆ and δ values based on simple
variance estimator under different models. Likewise, Table 4 presents the efficiency com-
parison results for the ratio type variance estimators under various models. The ∆ and
δ values based on ratio estimator have been provided in Table 5 using different values of
the smoothing constant c. The superiority of the EWMA estimators over the traditional
survey-based estimators can be clearly observed from tables. Further, we see that the pro-
posed forced model provides the most efficient estimates of variance under both traditional
and time-scaled sample surveys.

8. Results and Discussion

Survey researchers employ one-time sample surveys for use with randomized response
models. This research study analyzed the efficiency of simple and ratio-type variance esti-
mators using time-scale surveys under different randomized response scrambling models.
Additionally, a new forced randomized response technique was also presented in Section



M. Azeem, A. Salam, S. Hussain / Eur. J. Pure Appl. Math, 18 (1) (2025), 5449 13 of 19

Table 2: Efficiency comparison of simple estimators for µY = 1.2,σ2
Y = 0.8, σ2

T = 0.5, σ2
S = 5, σ2

Z = 5,n = 500

EWMA Estimators One-Time based Estimators
c γ (Time-Scaled Surveys) (Traditional Surveys)

V ar(V̂t(GS)) V ar(V̂t(S)) V ar(V̂t(P )) s2Y (GS) s2Y (S) s2Y (P )

0.1 0.2 0.04 0.03 0.01 0.67 0.61 0.16
0.4 0.04 0.04 0.01 0.67 0.67 0.17
0.6 0.04 0.04 0.01 0.67 0.72 0.21

0.2 0.2 0.07 0.07 0.02 0.67 0.61 0.16
0.4 0.07 0.07 0.02 0.67 0.67 0.17
0.6 0.07 0.08 0.02 0.67 0.72 0.21

0.3 0.2 0.12 0.11 0.03 0.67 0.61 0.16
0.4 0.12 0.12 0.03 0.67 0.67 0.17
0.6 0.12 0.13 0.04 0.67 0.72 0.21

0.4 0.2 0.17 0.15 0.04 0.67 0.61 0.16
0.4 0.17 0.17 0.04 0.67 0.67 0.17
0.6 0.17 0.18 0.05 0.67 0.72 0.21

0.5 0.2 0.22 0.20 0.05 0.67 0.61 0.16
0.4 0.22 0.22 0.06 0.67 0.67 0.17
0.6 0.22 0.24 0.07 0.67 0.72 0.21

0.6 0.2 0.29 0.26 0.07 0.67 0.61 0.16
0.4 0.29 0.29 0.07 0.67 0.67 0.17
0.6 0.29 0.31 0.09 0.67 0.72 0.21

0.7 0.2 0.36 0.33 0.09 0.67 0.61 0.16
0.4 0.36 0.36 0.09 0.67 0.67 0.17
0.6 0.36 0.39 0.11 0.67 0.72 0.21

0.8 0.2 0.45 0.41 0.11 0.67 0.61 0.16
0.4 0.45 0.45 0.11 0.67 0.67 0.17
0.6 0.45 0.48 0.14 0.67 0.72 0.21

0.9 0.2 0.55 0.50 0.13 0.67 0.61 0.16
0.4 0.55 0.55 0.14 0.67 0.67 0.17
0.6 0.55 0.59 0.17 0.67 0.72 0.21

5. Results of the comparative analysis have been provided in Table 2-5 based on different
performance-evaluation measures.
In Table 2, the values of the sampling variance of the simple variance estimator under
various scrambling models have been presented. The results for both traditional surveys
and time-scaled surveys have been shown. Observing Table 2, we find that the EWMA
variance estimators are much more efficient than the traditional survey-based variance
estimators. Further, we also observe that the basic variance estimator under the pro-
posed forced model is more efficient than the estimator under the Gjestvang and Singh
[7] forced scrambling model and the Saleem et al. [24] model. This result holds not only
in traditional surveys but also in time-scaled surveys. Table 3 displays the values of pri-



M. Azeem, A. Salam, S. Hussain / Eur. J. Pure Appl. Math, 18 (1) (2025), 5449 14 of 19

Table 3: Comparison of ∆ and δ values based on simple estimators under different models for µY = 1.2,
σ2
Y = 0.8, σ2

T = 0.5, σ2
S = 5, σ2

Z = 5,n = 500

c γ Privacy Measure (∆ ) Unified Measure ( δ)
Gjestvang Saleem Proposed Gjestvang Saleem Proposed
& Singh et al. Model & Singh et al. Model

0.1 0.2 113.552 8.917 201.639 0.006 0.068 0.001
0.4 113.552 7.903 201.163 0.006 0.085 0.001
0.6 113.552 7.179 200.873 0.006 0.101 0.001

0.2 0.2 113.552 8.917 201.639 0.006 0.068 0.001
0.4 113.552 7.903 201.163 0.006 0.085 0.001
0.6 113.552 7.179 200.873 0.006 0.101 0.001

0.2 0.2 113.552 8.917 201.639 0.006 0.068 0.001
0.4 113.552 7.903 201.163 0.006 0.085 0.001
0.6 113.552 7.179 200.873 0.006 0.101 0.001

0.2 0.2 113.552 8.917 201.639 0.006 0.068 0.001
0.4 113.552 7.903 201.163 0.006 0.085 0.001
0.6 113.552 7.179 200.873 0.006 0.101 0.001

0.2 0.2 113.552 8.917 201.639 0.006 0.068 0.001
0.4 113.552 7.903 201.163 0.006 0.085 0.001
0.6 113.552 7.179 200.873 0.006 0.101 0.001

0.2 0.2 113.552 8.917 201.639 0.006 0.068 0.001
0.4 113.552 7.903 201.163 0.006 0.085 0.001
0.6 113.552 7.179 200.873 0.006 0.101 0.001

0.2 0.2 113.552 8.917 201.639 0.006 0.068 0.001
0.4 113.552 7.903 201.163 0.006 0.085 0.001
0.6 113.552 7.179 200.873 0.006 0.101 0.001

0.2 0.2 113.552 8.917 201.639 0.006 0.068 0.001
0.4 113.552 7.903 201.163 0.006 0.085 0.001
0.6 113.552 7.179 200.873 0.006 0.101 0.001

0.2 0.2 113.552 8.917 201.639 0.006 0.068 0.001
0.4 113.552 7.903 201.163 0.006 0.085 0.001
0.6 113.552 7.179 200.873 0.006 0.101 0.001

vacy measure ∆ and the unified measure δ under various models for different values of
the smoothing constant. The table shows that the ∆ values under the proposed forced
scrambling model are higher than those based on the existing scrambling models. This
implies that the proposed forced model offers the best level of privacy protection of all
three competitor models. Moreover, Table 3 also reveals that the δ values are the smallest
under the suggested forced scrambling model.
In Table 4, the values of mean squared errors of the ratio-type variance estimators have
been presented under various models. Results of mean square error have been provided
under both traditional and time-scaled sample surveys. From Table 4, we observe that



M. Azeem, A. Salam, S. Hussain / Eur. J. Pure Appl. Math, 18 (1) (2025), 5449 15 of 19

Table 4: Efficiency comparison of ratio-type estimators under different models for µY = 1.2,σ2
Y = 0.8, σ2

T = 0.5,
σ2
S = 5, σ2

Z = 5,n = 500

EWMA Estimators One-Time based Estimators
c γ (Time-Scaled Surveys) (Traditional Surveys)

MSE(Rt(DP )) MSE(Rt(S)) MSE(Rt(P )) MSE(RDP ) MSE(RS) MSE(RP )

0.1 0.2 1.18 0.02 0.02 22.33 0.47 0.29
0.4 1.18 0.03 0.01 22.33 0.51 0.18
0.6 1.18 0.03 0.01 22.33 0.55 0.22

0.2 0.2 2.48 0.05 0.03 22.33 0.47 0.29
0.4 2.48 0.06 0.02 22.33 0.51 0.18
0.6 2.48 0.06 0.02 22.33 0.55 0.22

03 0.2 3.94 0.08 0.05 22.33 0.47 0.29
0.4 3.94 0.09 0.03 22.33 0.51 0.18
0.6 3.94 0.10 0.04 22.33 0.55 0.22

0.4 0.2 5.58 0.12 0.07 22.33 0.47 0.29
0.4 5.58 0.13 0.05 22.33 0.51 0.18
0.6 5.58 0.14 0.05 22.33 0.55 0.22

0.5 0.2 7.44 0.16 0.10 22.33 0.47 0.29
0.4 7.44 0.17 0.06 22.33 0.51 0.18
0.6 7.44 0.18 0.07 22.33 0.55 0.22

0.6 0.2 9.57 0.20 0.13 22.33 0.47 0.29
0.4 9.57 0.22 0.08 22.33 0.51 0.18
0.6 9.57 0.23 0.09 22.33 0.55 0.22

0.7 0.2 12.02 0.25 0.16 22.33 0.47 0.29
0.4 12.02 0.28 0.10 22.33 0.51 0.18
0.6 12.02 0.29 0.12 22.33 0.55 0.22

0.8 0.2 14.89 0.31 0.19 22.33 0.47 0.29
0.4 14.89 0.34 0.12 22.33 0.51 0.18
0.6 14.89 0.36 0.14 22.33 0.55 0.22

0.9 0.2 18.27 0.39 0.24 22.33 0.47 0.29
0.4 18.27 0.42 0.15 22.33 0.51 0.18
0.6 18.27 0.45 0.18 22.33 0.55 0.22

the EWMA ratio estimators perform more efficiently than the traditional survey-based
estimators. Moreover, under both surveys, the ratio-type variance estimator under the
proposed forced model is observed to be more efficient than the available variance esti-
mators. Table 5 displays the computed values of the privacy measure ∆ and the unified
measure δ based on the ratio-type variance estimators. The results have been provided
under different scrambling models for different values of the constant ‘c’. We find that
the ∆ values under the proposed forced model are the highest of all three models. This
implies that the proposed forced response model offers the best level of privacy protection
of all three competitor models. Additionally, Table 5 also indicates that the δ values un-



M. Azeem, A. Salam, S. Hussain / Eur. J. Pure Appl. Math, 18 (1) (2025), 5449 16 of 19

Table 5: Comparison of ∆ and δ values based on ratio-type estimators under different models for µY = 1.2,
σ2
Y = 0.8, σ2

T = 0.5, σ2
S = 5, σ2

Z = 5,n = 500

c γ Privacy Measure (∆ ) Unified Measure ( δ)
Diana Saleem Proposed Diana Saleem Proposed
& Perri et al. Model & Perri et al. Model

0.1 0.2 113.552 8.917 201.639 0.197 0.053 0.001
0.4 113.552 7.903 201.163 0.197 0.065 0.001
0.6 113.552 7.179 200.873 0.197 0.076 0.001

0.2 0.2 113.552 8.917 201.639 0.197 0.053 0.001
0.4 113.552 7.903 201.163 0.197 0.065 0.001
0.6 113.552 7.179 200.873 0.197 0.076 0.001

0.3 0.2 113.552 8.917 201.639 0.197 0.053 0.001
0.4 113.552 7.903 201.163 0.197 0.065 0.001
0.6 113.552 7.179 200.873 0.197 0.076 0.001

0.4 0.2 113.552 8.917 201.639 0.197 0.053 0.001
0.4 113.552 7.903 201.163 0.197 0.065 0.001
0.6 113.552 7.179 200.873 0.197 0.076 0.001

0.5 0.2 113.552 8.917 201.639 0.197 0.053 0.001
0.4 113.552 7.903 201.163 0.197 0.065 0.001
0.6 113.552 7.179 200.873 0.197 0.076 0.001

0.6 0.2 113.552 8.917 201.639 0.197 0.053 0.001
0.4 113.552 7.903 201.163 0.197 0.065 0.001
0.6 113.552 7.179 200.873 0.197 0.076 0.001

0.7 0.2 113.552 8.917 201.639 0.197 0.053 0.001
0.4 113.552 7.903 201.163 0.197 0.065 0.001
0.6 113.552 7.179 200.873 0.197 0.076 0.001

0.8 0.2 113.552 8.917 201.639 0.197 0.053 0.001
0.4 113.552 7.903 201.163 0.197 0.065 0.001
0.6 113.552 7.179 200.873 0.197 0.076 0.001

0.9 0.2 113.552 8.917 201.639 0.197 0.053 0.001
0.4 113.552 7.903 201.163 0.197 0.065 0.001
0.6 113.552 7.179 200.873 0.197 0.076 0.001

der the proposed forced model are the smallest of all three models. These results make
the proposed forced model the most suitable model of all three models for application in
real-world sample surveys.
It may be noted that the mean square error presented in equation (31) depends on popula-
tion parameters. This makes it difficult to construct confidence intervals and to compare
the efficiency of estimators as the information about the values of parameters is rarely
available. However, using computer programs, Monte-Carlo simulations can be performed
in time-scaled surveys to estimate the sampling variance of the variance estimators. These
simulated sampling variances can be used to compare the efficiency of variance estimators.



M. Azeem, A. Salam, S. Hussain / Eur. J. Pure Appl. Math, 18 (1) (2025), 5449 17 of 19

Likewise, simulation-based analyses also help the researchers to find estimates of standard
errors for construction of confidence intervals.

9. Future Research Suggestions

For future researchers, we make the following recommendations:

• Future researchers may evaluate the performance of the variance estimators in the
case of response and non-response errors under time-scaled surveys.

• Besides simple unified measures, researchers can also evaluate the performance of
variance estimators using weighted unified measures, by using different weights for
efficiency and privacy.

• Complex sampling designs can be used with time-scaled surveys to develop new
forced response models and new variance estimators.

DECLARATIONS:

Funding Source
The authors didn’t receive any funding for this study.

Conflict of Interest
The authors declare no conflict of interest.

Data Availability Statement
The data associated with this study has not been deposited into a publicly available repos-
itory. All relevant data are available within the article and its references.

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