STRATIFIED RANDOM SAMPLING WITH RISK APPROACH Текст научной статьи по специальности «Математика»

STRATIFIED RANDOM SAMPLING WITH RISK

APPROACH

Astha Jain1, Diwakar Shukla2

1,2 Department of Mathematics and Statistics, Dr. Harisingh Gour Vishwavidyalaya,

Sagar, M.P., 470003 [email protected], 2 [email protected]

Abstract

In stratified random sampling, the sample size allocation is a problem which is tackled by many scientists and survey practitioners. Generally the proportional allocation, Neyman allocation and cost based allocation, are used to conduct sample surveys for gathering information from each strata. One can think of risk imposed on the life of investigators which is yet not considered while sample size allocation to risky strata. In this paper, the risk indicators stratum-wise are defined using police station records and hospital records. Such indicators are used for the determination of sample size allocation. For optimization, the Lagrange multiplier technique is used with two constants whose values need to be determined. An algorithm is proposed and analysed for such using simulation. The outcome of analysis provides that sample size allocation is directly proportional to the strata size and variability but inversely proportional to the square root of risk indicators of the stratum (with varying values of constants). This paper opens a new approach for the consideration of risk based sample size allocation and estimation in the setup of stratified sampling.

Keywords: Simple Random Sampling (SRSWOR), Stratified Random Sampling, Stratum, Sample survey, Lagrange Multiplier, Allocation to Strata, Risk data, Risk Indicators, Optimization, Variance of sample estimate.

1. Introduction

Sample surveys play important role in exploring the hidden characteristics of the whole population without complete enumeration. The sample survey methodologies are used in areas and regions where epidemic occurs. A survey is usually conducted to know about the patients health conditions, rate of spread of disease, estimation of average number of deaths due to disease etc. Where natural disaster happened, to collect facts about casualties, about root causes of natural disaster are further possible field based studies.

Stratification is a sampling technique used in surveys in order to improve upon the precision of the sample estimate. Several authors have developed optimal techniques to allocate sample sizes to stratum (for the single variable under study) using Lagrange multiplier optimization technique. There may be other situations such as war, naxalite movement, dense forest where issue risk is involved on the field officers, who are involved in data collection for the conduct of sample survey. The risk may be on life, infection due to disease or being unhealthy for short duration. In this study, the problem of risk occurrence during the data collection, if so exit, is considered in context to sample size allocation to each strata.

In literature, many studies exist, where the authors have considered the stratum size, variability and stratum cost of the data collection while surveying the population. Yadav and Verma

[5] studied the exponential ratio-type estimators under the linear cost function in the set up of stratified random sampling. Focus of study is on the estimation of population parameter with the help of collected data using proposed method. A linear function is used to determine the relation among sample sizes of each stratum. Yadav et al.[6] worked on the behaviour of ratio-product-cum-exponential-cum-logarithmic type estimators with one auxiliary variable in the stratified random sampling setup and analyzed such with the linear cost function using numerical illustrations. Ghosh[1] suggested a new method of allocation of sample sizes for stratum. In this, the author has taken the average of the optimum allocation for the different characters individually. Khan et al.[2] used compromised allocation in multivariate stratified sampling for an integral solution. Varshney [4] worked on an optimum allocation of sample sizes in the presence of non-response factor under the multivariate stratified double sampling setup. In such, authors have considered sample size allocation problem in stratified random sampling for single character as well as for multiple characters with varying cost functions.

Koyuncu and Kalidar[8] suggested a new family of estimators for stratified random sampling utilizing the information of the coefficient of kurtosis of the population and obtained efficient conditions between the adopted and proposed families. Theoretical finding are supported by the numerical examples with original data. Singh et al.[9] addressed the problem of various types of estimation of the main variable parameter in the presence of non-response and measurement error both incorporating the information of two auxiliary variables. In that, authors derived the optimum strata weights using the suitable calibration technique. Bhushan et al.[10] developed efficient classes of estimators in stratified sampling for combined ratio and separate ratio type estimators. Such estimators are theoretically justified and compared over the conventional estimator, classical ratio estimator and classical regression estimator using the simulation study. Tiwari et al.[11] proposed a general class of estimators for estimating the population mean of study variable using the support variable based correlated information. Members of such proposed class are identified and compared in terms of efficiency. Kadilar and Chingi [16] derived some ratio-type estimators and discussed their properties in the setup of stratified sampling. Aamir et al. [13] suggested a generalised class of exponential-type estimators for population mean by taking the two auxiliary variables for estimating the unknown means with the case of sub-sampling and non-response. In such, authors derived the conditions under which proposed estimators are more efficient as composed to other estimators. Cekim and Kalidar [12] suggested some estimators for estimating the population variances in stratified sampling, in the form of in-function type estimators. Ahnad et al.[17] proposed an improved family of estimators for estimating the population distribution function. The main aim of such contribution was to develop an enhanced family of log ratio-exponential based estimation procedure under stratified sampling. Zaman and Kalidar [15] suggested exponential ratio and product type estimators the mean by considering the two phase sampling setup in stratified sampling.

This paper considers the risk factor exposed on the life of survey workers over different strata. A sample size allocation keeping the method is discussed considering a risk function with computation of allocations variance optimal.

1.1. Risk in Survey Sampling

While data collection, using stratified sampling, some strata may have higher risk on the life of surveyor while others may have a little. For example, a strata of a population is affected by the nuxalite movement, next strata bears high rate of murders and killings, third one is affected by the dangerous epidemics(like malaria, dengue, COVID-19 ). The strata-wise risk on the life of investigators could be pre-estimated using police record and hospital records (for last one/five years) as below:

Strata I

Strata II

(1) Deaths due to murder and mass killing=an

(2) Deaths due to communal riots= a12

(3) Deaths due to epidemics and community diseases= a13

(1) Deaths due to murder and mass killing=«21

(2) Deaths due to communal riots= a22

(3) Deaths due to epidemics and community diseases= a23

The minimum risk may be assumed as ro which includes the normal risk of natural death during the survey work.

1.2. Symbols used for analysis

Let a population of finite size N, divided into L stratum. Each stratum is of size Ni where N1 + N2 + ... + Nl = N holds and samples are taken from each strata of size n such that n = (ni + n2 + ... + nL), where n denotes total size of sample. Notations used for population parameters are:

1 L Ni

Y = N E EY,

1

L Ni

i=1 j=1

Ni

N1

EE (Yn- Y)2

i=1j=1

Yi = N E Yi',

E(Yij- Y)2

i j=1

i Ni - 1 = i - j=1

(1.1) (1.2)

where Yij is the jth observation of the ith strata in a population of size N.

Let a sample of size n is drawn by the SRSWOR sampling scheme keeping ni from each stratum, then sample related notations are:

1

L ni

y = nE Eyii, i=1 j=1

1 ni

= n E yij,

s =

EE y - y)

l!=1

1 L ni

n - 1 fri 1 ..

i=1 j=1

1 ni

2

s2 = E (Vij- yi)2 i - j=1

(1.3)

(1.4)

Define risk indicators ri as:

1.3. Risk Indicators

where,

At i = 1, At i = 2,

(Ti-), for ith strata

Ni

T

r1(Risk) = —, for strata I T2

r2(Risk) = —, for strata II

Total : T1 Total strata I Population Total : T2 Total strata I Population

(an + «12 + U13)

N1

(a21 + U22 + «23) N2

(1.5)

(1.6)

(1.7)

(1.8)

(1.9) (1.10) (1.11) (1.12) (1.13)

These indicators are crude measures of the intensity of risk imposed on the life of field investigators who collect primary data through sample survey in a stratified population.

2

S

L

1

r

1.4. Motivation

The proportional allocation is stratum size based and Neyman allocation is size + variability based for ith stratum. There is one more method which is cost based allocation per stratum but involvement of stratum risk is yet not considered by any author. In order to utilize the information contained in the risk indicators r,, the problem of sample size determination is attempted in this paper.

2. Mean Estimation approach in Stratified Sampling The usual mean estimator under the stratified sampling is:

yst = E WiVi i=l

The variance for stratified random sampling is:

va.,) = E m- Wi\s

i=1

1 1

where Wi represents the weight of each stratum on the basis of its size i.e. Wi = (jf

(2.1)

(2.2)

Y,] i=l,2,...L, j=l,2,..,N-Population size=N

> f----Y

Stratum -11 Stratum -2| Stratum -3 Stratum-L

Stratum Siie=N1

Stratum Size=M:

Sample fronn

Stratum 2

Stratum Siie=N3

Sample from

Stratum3

H3

Stratum 5ize = NL

Sa rnple from

StratumL

"L

Figure 1: Sampling structure for L stratum based stratified sampling

3. Linear risk function Consider the linear risk function for the stratified sampling:

r =

ro + E

niri

i=1

(3.1)

L

L

where,

to: Minimum pre fixed risk exposed on life of investigators. ti: Risk per unit in a stratum.

r*: Total risk exposed on investigator while survey of entire population including natural death. The objective of this paper is to determine the sample size from each stratum using the linear risk function, keeping variance minimum. This objective can be achieved by optimizing following,

Minimize V(yst) (3.2)

subject to the conditions,

L

E tlnl = t* (3.3)

i=1 L

E ni = n (3.4)

i=1

For solution using the Lagrange multiplier technique defined and optimize the following function $

$ = V(yst) + n - n j + A2 ^E trnr - r*^ (3.5)

where A1, A2 are constants called Lagrange multipliers. Differentiating $ with respect to ni, A1,

A2 and equating to zero, one can get,

n WS (3 6)

ni = /1 , 1 (3.6) v Ai + Aiti

Summing (3.6) on both sides,

L L

(3.7)

n = ni = i=1 i=1

WiSi

_VAi + A 2 ti _

From (3.6) and (3.7), one get insights,

ni a Ni (3.8)

ni a Si (3.9)

1

ni a n ■ (3.10)

VA1 + Aiti

where ti is risk related to Ith strata.

4. Computational Algorithm for Optimal Variance along with choice

of A1 and A2

Step I : For given N, n calculate initial values Ni, Si, Yi and (NiSi) and ti of the population Step II : Find V(yst) using Neyman allocation, which is based on ni a Ni and ni a Si with expression ni = j I^Ws }. Find variance V(yst) using proportional allocation which is based on criteria ni a Ni only with expression ni = nWi

Step III : Find the risk ti and use risk function t* = tini. Step IV : Set

$ = V(yst) + A1 ^E ni - n j + Ai ^E tin - t*^ (4.1)

where A1, A2 are constants to determine under risk assumption.

Step V : For risk based allocation of sample size n,

ni « Ni

ni a Si

ni a

\Ml + ^2Ti

(4.2)

(4.3)

(4.4)

Step VI : Use simulation procedure to find values of A1 and A2 to optimize variance V (yst)

(a) Fix the values of A1,

(b) Vary A2 on x-axis of the graph and plot graph for variance, along with n1 and n1,

(c) Continue the process of creating graphs for different values of A1,

(d) When variance line becomes parallel to x-axis then stop the simulation process.

(i) Choose that input-data set n1, n2, A1, A2 (producing parallel line)

(ii) Use values to get optimal solution.

5. Empirical Study

Consider following data of size N= 244 from 6th Minor Irrigation Census - Village Schedule -Assam[7]. The crime data obtained from police station and hospitals as under(assumed data for a year):

Strata I

(a) Deaths due to bullet firing = 8

(b) Deaths due to riots = 11

(c) Deaths due to epidemic = 6 Total = 25

Total strata size= 127

Strata II :

(a) Deaths due to bullet firing = 11

(b) Deaths due to riots = 15

(c) Deaths due to epidemic = 10 Total = 36

Total strata size= 135 The basic data and basic computation is as under:

Table 1: Data for Strata (Source, please see [7])

i Ni Wt Y Si2 ri

1 127 0.5205 703.74 883.83 19%

2 135 0.48 413 644.922 26%

Table 2: The proportional allocation provides

n1 n2 n V(yst) prop

72 108 180 804.5

1

Figure 2: Variation when A\=1 fixed

Fig.(2) reveals that, for fixed value of A1 = 1, the variance of V(yst) has growing trend and under risk consideration . It is observed that A2 increases for fixed A1.Moreover, V(yst) fluctuates between between 800 to 1800. There is miled increase in n1 for increasing A2.

Figure 3: Variation when Ai=2 fixed

Fig(3) is an indicator of the analysis of V(yst), as the value of A2 increases for fixed value of A1 = 2, the value of V(yst) lies between 800 to 1000.

Figure 4: Variation when A\=3 fixed

Fig.(4) opens starting avenue for decrease in V(yst) as the value of A1 is increases, the V(yst) reduces.

Figure 5: Variation when A1=4 fixed Fig.(5) shows that the V(yst) line is tending to become parallel to the x-axis(on higher A2 values).

Figure 6: Variation when A1=10 fixed Fig.(6) represents the similar pattern as observed in Fig.(5) to get V(yst).

Figure 7: Variation when A1=100 fixed

Fig.(7) highlights that for higher values of A1, the relation between V(yst) over the incrementing values of A2 is almost parallel to x-axis. Such indicates for V(yst) being almost independent to the variation of A2.

Table 3: The Neyman allocation provides

ni n2 n V {]!st)Ney

71 119 180 803.5

Variation in Variance

30 5.5 30 5.4 305.3 305.2 30 5.1 SOS 304.5

0 2 4 5 S 10 12 14 16 18 JO 22 24 26 IS 30 32 34 36 3E « 12 44 46 4S £0

A,

Figure 8: Variation with respect to parameters

Fig.(8) depicts the relation between A1 and V(yst). The value of V(yst) is gradually decreasing as the values of A1 increases from 1 to 19. After A2 = 19 (approximately), there is no significant change in V(yst).

6. Comparison and Discussion

On comparing the different types of allocations (Table 5) it is evident that allocations are very close to each other and providing the optimal variance. The approach aimed at to utilize the

Table 4: The risk based allocation provides

n1 n2 n V (Vst)risk

72 118 180 805.26

Table 5: Different allocation methods provides

Proportional allocation provides n1 n2 n V (yst)prop

72 108 180 804.5

Neyman allocation provides n1 n2 n V (yst)prop

71 119 180 803.5

Risk based allocation n1 n2 n V (yst )risk

72 108 180 805.26

crime record information of police station and hospital records of the strata during sample survey. Such can be useful to determine the sample size allocation n from the ith strata (i = 1,2,3,..., L), so that n = Td=1 ni remains intact. Risk indicators are suggested and defined using the crime record and hospital records. The Lagrange multiplier technique provides two constants A1 and A2 whose values need to be computed using the available data. As evident in graphical pattern from [fig(2) to fig(8)], the increasing values of A1 provides the solution for best choice of n1 & n2 at the situation when variance remain stable (independent of increasing A2). An appendix added at the end provides choice of A1, A2 and n1 , n2. At A1=300, A2=0.01 one gets n1=108, n2=72 with lowest

variance 804.99 as displayed in the table(7) of appendix. When A1 increases then A2 decreases to attain same level of optimality. For any arbitrary choice of A1 the table(7) provides the value of A2 for quick selection. In general, 21 < A1 < 40 and 0.1 < A2 < 0.30 is the recommended rapid selection of A-values.

7. Conclusion

This paper presents a new idea of using the regional (strata) risk on the life of survey investigators with the help of risk indicators. In literature, when stratified sampling is used, the problem of sample size allocation appears that it could be resolved as per population strata size or as per population strata variability. The proportional allocation is based on population strata sizes while the Neyman allocation is based on size and variability both. Such allocations do not consider the risk factor imposed on the life of investigator. If risk is high for a particular strata then smaller sample size is required from that strata. The proposed risk based sample size allocation is like ni a Ni ,ni a Si and ni a + ^t incorporating two constants A1 & A2. An algorithm is proposed in this paper showing how to compute A1 and A2 constants with minimizing the population variability factor of the mean estimate. If 1 < A1 < 10 then it is suggested to choose A2 = 200 as per table 7. Similarly when 10 < A1 < 20 then recommended to choose A2 = 100 as per table 7, shown in appendix. Various graphs from (fig.(2) to fig.(8)) reveal that when variance line becomes parallel to x-axis for set of values (A1, A2, n1, n2), such provide the optimal solution for lowest variability due to the risk based sample size allocation. In general, one can work with risk based allocations choosing 21< A1 < 40 and 0.1< A2 < 0.3 (table 7) to get nearly optimal result. The crime data of all police stations and health data from hospitals can be utilized for risk computation and accordingly can be used in risk based sample size allocation. The table 7 attached in appendix helps in rapid selection of A2 for an arbitrary choice of A1

References

[1] Ghosh, S. P. (1958), A note on stratified random sampling with multiple characters, Calcutta Statistical Association Bulletin, 8(2-3):81-90.

[2] Khan, M.G.M., Ahsan, M.J. and Jahan, N. (1997): Compromise allocation in multivariate stratified sampling: An Integer Solution, Naval Reseatch Logistics, 44:69-79

[3] Varshney,R., and Mradula (2019), Optimum allocation in multivariate stratified sampling design in the presence of non-response with Gamma cost function, Joutnal of Statistical Computation and Simulation, 89(13):2454-2467.

[4] Varshney, R., Najmussehar and Ahsan, M.J. (2012), An optimum multivariate stratified double sampling design in presence of non-response, Optim. Lett, 6:993-1008.

[5] Yadav, S.K., Kumar Verma, M. and Varshney, R. (2024), Optimal strategy for elevated estimation of population mean in stratified random sampling under linear cost function, Ann. Data. Sci., https://doi.org/10.1007/s40745-024-00520-9

[6] Zaagan, Abdullah A., Kumar Verma, M., Mahnashi, Ali M., Yadav, S.K., Ahmadini, A.A.H., Meetei, M.Z., Varshney, R., (2024), An effective and economic estimation of population mean in stratified random sampling using a linear cost function, Heliyon, 10(10):e31291.

[7] https://data.gov.in/ Data source -6th Minor Irrigation Census - Village Schedule - Assam

[8] Koyunchu N. and Kadilar C. (2009), Ratio and product estimators in stratified random sampling, Journal of Statistical planning and Inference, 139:2552-2558.

[9] Singh, G., N., Bhattacharya, D. and Bandopadhyaya (2020), A general class of calibration estimators under stratified random sampling in presence of various kinds of non-sampling errors, Communication in Statistics (Simulation & Computation), Taylor and Francis, 52(2):320-333

[10] Bhushan, S., Kumar, A. and Singh, S. (2021), Some efficient class of estimators under stratified sampling, Communication in Statistics(Theory & Methods), Taylor & Francis, 52(6):1767-1796

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STRATIFIED RANDOM SAMPLING WITH RISK APPROACH_Volume 2°, March 2025

[11] Tiwari, K. K., Bhougal, S. and Kumar, S. (2020), A general class of estimators in stratified random sampling, Communication in Statistics (Simulation and Computation), Taylor & Francis, 52(2):1-16.

[12] Cekim, O. C., and Kalidar, C.(2019), In-type estimators for population variance in stratified random sampling, Communication in Statistics, (Taylor and Francis), 49(1):1-13

[13] Sohaib, A., Shabbir, J., Emam, W., Zahid, E., Aamir, M., Khalid, M., and Anas, M., M., (2024), An improved class of estimators for estimation of population distribution function under stratified random sampling, Heliyon, 10:e28272.

[14] Solanki, R. S., and Singh, H. P. (2015), Efficient class of estimators in stratified random sampling, Statistical Papers, 56:83-103.

[15] Zaman, T. and Kalidar, C. (2020), Exponential ratio and product type estimators of the mean in stratified two phase sampling, AIMS Mathematics, 6(5):4265-4279.

[16] Kadilar, C., and Cingi, H.(2003), Ratio estimator in Stratified Sampling, 45(2):218-225.

[17] Sanaullah, A., Amin, M. N., Hanif, M. and Koyuncu, N. (2018), Generalized exponential type estimators for population mean taking two auxiliary variables for unknown means in stratified sampling with sub-sampling the non-respondents, Int. Journal of Applied Computation and Mathematics, 4:56.

[18] Cochran, W.G.(2005), Sampling Technique, Wiley Eastern Publication.

Appendix

Table 6: A1 varies but X2 fixed

À1 A2 n1 n2 n V (yst) À1 a2 n1 n2 n V (yst)

1 0.01 109 71 180 805.469 30 0.01 108 72 180 804.998

2 0.01 108 72 180 805.134 32 0.01 108 72 180 804.997

3 0.01 108 72 180 805.063 33 0.01 108 72 180 804.997

4 0.01 108 72 180 805.037 34 0.01 108 72 180 804.997

5 0.01 108 72 180 805.024 35 0.01 108 72 180 804.997

6 0.01 108 72 180 805.017 36 0.01 108 72 180 804.997

7 0.01 108 72 180 805.012 37 0.01 108 72 180 804.997

8 0.01 108 72 180 805.009 38 0.01 108 72 180 804.997

9 0.01 108 72 180 805.007 39 0.01 108 72 180 804.997

10 0.01 108 72 180 805.005 40 0.01 108 72 180 804.997

11 0.01 108 72 180 805.004 41 0.01 108 72 180 804.997

12 0.01 108 72 180 805.003 42 0.01 108 72 180 804.997

13 0.01 108 72 180 805.002 43 0.01 108 72 180 804.997

14 0.01 108 72 180 805.001 44 0.01 108 72 180 804.997

15 0.01 108 72 180 805.001 45 0.01 108 72 180 804.997

16 0.01 108 72 180 805.000 46 0.01 108 72 180 804.997

17 0.01 108 72 180 805.000 47 0.01 108 72 180 804.997

18 0.01 108 72 180 805.000 48 0.01 108 72 180 804.997

19 0.01 108 72 180 804.999 49 0.01 108 72 180 804.997

20 0.01 108 72 180 804.999 50 0.01 108 72 180 804.997

21 0.01 108 72 180 804.999 20 0.01 108 72 180 804.999

22 0.01 108 72 180 804.999 25 0.01 108 72 180 804.998

23 0.01 108 72 180 804.998 50 0.01 108 72 180 804.997

24 0.01 108 72 180 804.998 100 0.01 108 72 180 804.996

25 0.01 108 72 180 804.998 150 0.01 108 72 180 804.996

26 0.01 108 72 180 804.998 200 0.01 108 72 180 804.996

27 0.01 108 72 180 804.998 250 0.01 108 72 180 804.996

28 0.01 108 72 180 804.998 300 0.01 108 72 180 804.996

29 0.01 108 72 180 804.998 1000 0.01 108 72 180 804.996

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Table 7: Aj and A2 are varying

Ai A2 ni U2 n V (yst)

I.00 0.50 118.94 61.06 180.00 863.32 2.00 0.49 116.89 63.11 180.00 843.62 3.00 0.48 115.43 64.57 180.00 832.17 4.00 0.47 114.34 65.66 180.00 824.95 5.00 0.46 113.48 66.52 180.00 820.12 6.00 0.45 112.79 67.21 180.00 816.75 7.00 0.44 112.23 67.77 180.00 814.30 8.00 0.43 111.76 68.24 180.00 812.49 9.00 0.42 111.36 68.64 180.00 811.11 10.00 0.41 111.01 68.99 180.00 810.03

II.00 0.40 110.71 69.29 180.00 809.19 12.00 0.39 110.45 69.55 180.00 808.51 13.00 0.38 110.22 69.78 180.00 807.96 14.00 0.37 110.02 69.98 180.00 807.51 15.00 0.36 109.83 70.17 180.00 807.14 16.00 0.35 109.67 70.33 180.00 806.84 17.00 0.34 109.51 70.49 180.00 806.58 18.00 0.33 109.38 70.62 180.00 806.36 19.00 0.32 109.25 70.75 180.00 806.17 20.00 0.31 109.14 70.86 180.00 806.02 21.00 0.30 109.03 70.97 180.00 805.88 22.00 0.29 108.93 71.07 180.00 805.77 23.00 0.28 108.84 71.16 180.00 805.67 24.00 0.27 108.76 71.24 180.00 805.58 25.00 0.26 108.68 71.32 180.00 805.50 26.00 0.25 108.61 71.39 180.00 805.44 27.00 0.24 108.54 71.46 180.00 805.38 28.00 0.23 108.48 71.52 180.00 805.33 29.00 0.22 108.42 71.58 180.00 805.29 30.00 0.21 108.36 71.64 180.00 805.25 31.00 0.20 108.31 71.69 180.00 805.21 32.00 0.19 108.25 71.75 180.00 805.18 33.00 0.18 108.21 71.79 180.00 805.16 34.00 0.17 108.16 71.84 180.00 805.13 35.00 0.16 108.12 71.88 180.00 805.11 36.00 0.15 108.08 71.92 180.00 805.10 37.00 0.14 108.04 71.96 180.00 805.08 38.00 0.13 108.00 72.00 180.00 805.07 39.00 0.12 107.97 72.03 180.00 805.05 40.00 0.11 107.93 72.07 180.00 805.04