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Volume 18, Issue 4, December 2019, Pages 361 - 366

An Application of Hermite Distribution in Sensitive Surveys

Authors
Said Farooq Shah1, Zawar Hussain2, *
1Department of Statistics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan
2Department of Social & Allied Sciences, Cholistan University of Veterinary & Animal Sciences, Bahawalpur, Pakistan
*Corresponding author. Email: zhlangah@yahoo.com
Corresponding Author
Zawar Hussain
Received 7 April 2016, Accepted 10 September 2017, Available Online 21 November 2019.
DOI
10.2991/jsta.d.191112.002How to use a DOI?
Keywords
Hermite distribution; Randomized response techniques; Sensitive surveys
Abstract

In this article, we proposed an efficient estimator for estimating population proportion of individuals possessing sensitive attribute in a finite dichotomous population. We used the Hermite distribution to randomize the responses in the randomization design of Kuk [1]. The relative efficiency results depicted that the proposed technique is relatively better than those of Kuk [1], Singh and Grewal [2] and Hussain et al. [3] and Hussain et al. [4].

Copyright
© 2019 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

A random variable Z is said to have a Hermite distribution if its probability mass function (pmf) is given by

PZ=Pr(Z=z)=e−(a+b)∑i=1[z2]az−2ibi(z−2i)!i!,z=0,1,2,…,∞.
where a and b are the two parameters taking positive numbers. Furthermore, the mean and variance of Z are given by μZ=a+2b, and σZ2=a+4b, respectively. The distribution with pmf (1) is denoted by Hera,b.

For estimation of population proportion π of the sensitive group, Warner [5] introduced the randomized response technique (RRT) in order to reduce non-response and misreporting in sensitive surveys. Several modifications of his RRT and new RRTs have been suggested. For better understanding of RRTs we refer to Blair et al. [6] and the references therein.

Kuk [1] modified Warner [5] model and argued that respondents feel insecure to answer a sensitive question, even when it is generated by a randomization device. He suggested using two decks each containing cards of two different colors, say C1 and C2. A respondent belonging to sensitive (non-sensitive) group is directed to use first (second) deck with proportion θ1θ2 of C1 (C2) cards. The ith respondent, using either the first or second deck, is asked to randomly draw a card and report the color of the card drawn without disclosing the deck he/she have used. Let XiYi be the color of the card drawn from the first (second) deck. According to this design XiYi follows Bernoulli distribution with parameter θ1θ2. The reported response, Zi, can be written as

Zi=αiXi+1−αiYi,
here αi is an indicator variable which taking value 1 if the respondent possesses sensitive attribute and 0, otherwise. Evidently, Eαi=π.

Recently, Singh and Grewal [2] argued that the respondent have to report XiYi the number of cards he/she have drawn for getting first C1 card from first (second) deck according to their status. Here XiYi follows geometric dtribution, Gθ1Gθ2. By changing the distribution of Xi and Yi Singh and Grewal [2] improved the efficiency of estimator of π. Following Singh and Grewal [2] work, Hussain et al. [3] and Hussain et al. [4], respectively, used negative binomial and geometric distribution of order k, as randomization device [in their notation X and Y are distributed as NBr,θ1 and NBr,θ2 and Gk1θ1 and Gk2θ2, respectively] and provided efficient RRTs. We intend to use the Hermite distribution, in Kuk [1] set up. The rest of the article is arranged as follows: In Section 2, we present briefly the proposed survey method and give an unbiased estimator of π and its variance. Relative efficiency comparisons are made in Section 3. Section 4 is about discussion of the results and giving a conclusive statement.

2. PROPOSED RRT

Suppose, in a sensitive survey, we provide two decks of cards to the respondents. Random number, X1,X2,X3,…,XT, generated from Hera1,b1, are written on cards placed in deck 1, and random numbers Y1,Y2,Y3,…,YT, generated from Hera2,b2, are written on cards placed in deck 2. Each respondent in the sample is directed to use one of the two provided decks depending on his/her own status on sensitive attribute A. If a respondent possesses (does not possesses) sensitive attribute A, he/she is asked to draw a card from deck 1 (deck 2) and report the number XiYi written on the drawn card. Evidently, Xi and Yi follows Her a1,b1, and Her a2,b2, respectively. The expected randomized response from the ith respondent may be written as

EZi=πμX+1−πμY.

On solving (3) for π and estimating EZi by the sample mean, z¯=1n∑i=1nzi, of reported responses, we get

π^P=z¯−μYμX−μY,
where μX=a1+2b1 and μY=a2+2b2.

By (4), the variance of π^P is given by

Varπ^P=Varz¯μX−μY2=π1−πn+σX2π+σY21−πnμX−μY2,
where σX2=a1+4b1, and σY2=a2+4b2.

3. EFFICIENCY COMPARISONS

It is difficult to conclude from analytical comparison, here, so numerical comparisons are made between the proposed RRT and those proposed by Kuk [1], Singh and Grewal [2], Hussain et al. [3] and Hussain et al. [4]. Let π^Kuk, π^SG, π^H1 and π^H2 denote the estimators proposed by Kuk [1], Singh and Grewal [2], Hussain et al. [3] and Hussain et al. [4], respectively. The estimators π^SG, π^H1 and π^H2 and their variances can be readily obtained by (4) and (5) by setting different parameters as mentioned earlier. The estimator π^Kuk may also be obtained from (4) but its variance is given by

Varπ^Kuk=θKuk1−θKuknkθ1−θ22+π1−πn1−1k,
where θKuk=πθ1+1−πθ2, and k is the number of repetitions.

Now, we define the relative efficiency of the proposed estimator π^P with respect to π^Kuk, π^SG, π^H1 and π^H2 as

REJ=Varπ^JVarπ^P,   J=Kuk, SG, H1, H2.

To know the extent of relative efficiency we have computed the REJ for different values of the design parameters, and the results are reported in the Tables 1–5 given below.

θ1 θ2 REKuk RESG REH1 REH2 θ1 θ2 REKuk RESG REH1 REH2
π=0.1 π=0.2
0.1 0.2 50.116 11.084 4.263 1.805 0.1 0.2 27.497 8.393 3.387 1.993
0.3 17.432 4.263 1.989 1.802 0.3 9.184 3.893 1.887 1.99
0.4 9.695 2.937 1.547 1.801 0.4 4.92 2.945 1.571 1.989
0.5 6.418 2.451 1.386 1.800 0.5 3.133 2.574 1.447 1.988
0.6 4.642 2.217 1.307 1.800 0.6 2.172 2.385 1.384 1.988
0.7 3.537 2.084 1.263 1.800 0.7 1.577 2.273 1.347 1.988
0.8 2.786 2.001 1.235 1.800 0.8 1.174 2.2 1.322 1.988
0.9 2.244 1.945 1.217 1.800 0.9 0.885 2.148 1.305 1.988
0.2 0.3 66.537 31.547 11.084 1.830 0.2 0.3 37.436 21.202 7.656 2.022
0.4 20.116 9.000 3.568 1.814 0.4 10.933 7.067 2.945 2.004
0.5 10.256 4.853 2.186 1.807 0.5 5.37 4.319 2.029 1.996
0.6 6.379 3.411 1.705 1.803 0.6 3.202 3.313 1.693 1.991
0.7 4.389 2.747 1.484 1.801 0.7 2.098 2.827 1.531 1.988
0.8 3.204 2.389 1.365 1.799 0.8 1.444 2.552 1.44 1.986
0.9 2.425 2.175 1.293 1.798 0.9 1.016 2.38 1.382 1.985
0.3 0.4 76.642 57.505 19.737 1.882 0.3 0.4 43.693 37.104 12.957 2.072
0.5 21.221 14.589 5.432 1.839 0.5 11.761 10.684 4.15 2.029
0.6 10.116 6.916 2.874 1.818 0.6 5.411 5.742 2.503 2.006
0.7 5.945 4.322 2.009 1.805 0.7 3.041 3.996 1.921 1.993
0.8 3.884 3.164 1.623 1.797 0.8 1.877 3.18 1.649 1.984
0.9 2.695 2.558 1.421 1.792 0.9 1.209 2.733 1.5 1.978
0.4 0.5 80.432 83.274 28.326 1.958 0.4 0.5 46.27 52.785 18.184 2.135
0.6 20.747 19.611 7.105 1.871 0.6 11.669 13.914 5.227 2.054
0.7 9.274 8.495 3.4 1.825 0.7 5.043 6.847 2.871 2.01
0.8 5.116 4.832 2.179 1.798 0.8 2.65 4.417 2.061 1.983
0.9 3.126 3.24 1.648 1.781 0.9 1.509 3.313 1.693 1.965
0.5 0.6 77.905 103.168 34.958 2.045 0.5 0.6 45.166 64.933 22.233 2.191
0.7 18.695 22.642 8.116 1.895 0.7 10.656 15.929 5.899 2.063
0.8 7.73 8.958 3.554 1.816 0.8 4.266 7.264 3.01 1.992
0.9 3.892 4.583 2.096 1.77 0.9 2.029 4.369 2.045 1.947
0.6 0.7 69.063 111.505 37.737 2.108 0.6 0.7 40.38 70.233 24 2.207
0.8 15.063 22.263 7.989 1.883 0.8 8.724 15.902 5.89 2.029
0.9 5.484 7.674 3.126 1.765 0.9 3.08 6.626 2.798 1.929
0.7 0.8 53.905 102.6 34.768 2.083 0.7 0.8 31.914 65.374 22.38 2.136
0.9 9.853 17.053 6.253 1.784 0.9 5.871 13.003 4.923 1.913
0.8 0.9 32.432 70.768 24.158 1.868 0.8 0.9 19.767 47.043 16.27 1.907
Table 1

Relative efficiency of π^P for r=k=3, k1=4, k2=2, a1=11, b1=12, a2=1 and b2=2.

θ1 θ2 REKuk RESG REH1 REH2 θ1 θ2 REKuk RESG REH1 REH2
π=0.3 π=0.4
0.1 0.2 20.324 7.817 3.197 2.161 0.1 0.2 16.971 7.935 3.233 2.358
0.3 6.873 3.972 1.915 2.157 0.3 5.767 4.243 2.002 2.354
0.4 3.798 3.113 1.629 2.156 0.4 3.257 3.38 1.714 2.352
0.5 2.525 2.762 1.512 2.156 0.5 2.232 3.017 1.593 2.352
0.6 1.845 2.577 1.451 2.155 0.6 1.69 2.821 1.528 2.351
0.7 1.427 2.465 1.413 2.155 0.7 1.359 2.7 1.488 2.351
0.8 1.146 2.389 1.388 2.155 0.8 1.138 2.618 1.46 2.351
0.9 0.945 2.336 1.37 2.155 0.9 0.98 2.559 1.441 2.351
0.2 0.3 28.211 18.296 6.69 2.193 0.2 0.3 24.073 17.633 6.465 2.395
0.4 8.352 6.718 2.831 2.173 0.4 7.176 6.906 2.89 2.372
0.5 4.236 4.362 2.045 2.164 0.5 3.72 4.637 2.133 2.362
0.6 2.648 3.465 1.746 2.159 0.6 2.4 3.747 1.837 2.356
0.7 1.845 3.017 1.597 2.156 0.7 1.739 3.291 1.685 2.352
0.8 1.372 2.756 1.51 2.153 0.8 1.352 3.02 1.595 2.349
0.9 1.066 2.588 1.454 2.152 0.9 1.103 2.843 1.535 2.348
0.3 0.4 33.282 31.056 10.944 2.248 0.3 0.4 28.727 29.241 10.335 2.456
0.5 9.127 9.761 3.845 2.201 0.5 7.971 9.79 3.851 2.403
0.6 4.362 5.62 2.465 2.176 0.6 3.91 5.878 2.547 2.374
0.7 2.595 4.104 1.96 2.16 0.7 2.415 4.404 2.056 2.357
0.8 1.732 3.372 1.715 2.15 0.8 1.69 3.673 1.812 2.346
0.9 1.239 2.958 1.577 2.143 0.9 1.278 3.251 1.671 2.338
0.4 0.5 35.535 43.563 15.113 2.31 0.4 0.5 30.931 40.555 14.106 2.523
0.6 9.197 12.465 4.746 2.226 0.6 8.155 12.343 4.702 2.43
0.7 4.174 6.606 2.793 2.178 0.7 3.829 6.857 2.873 2.376
0.8 2.366 4.521 2.099 2.148 0.8 2.278 4.849 2.204 2.343
0.9 1.507 3.541 1.772 2.128 0.9 1.543 3.879 1.881 2.32
0.5 0.6 34.972 53.282 18.352 2.356 0.5 0.6 30.686 49.371 17.045 2.566
0.7 8.563 14.197 5.324 2.228 0.7 7.727 14.014 5.259 2.428
0.8 3.673 7.038 2.937 2.154 0.8 3.475 7.331 3.031 2.347
0.9 1.961 4.558 2.111 2.107 0.9 1.987 4.945 2.236 2.294
0.6 0.7 31.592 57.676 19.817 2.353 0.6 0.7 27.992 53.486 18.416 2.55
0.8 7.225 14.324 5.366 2.181 0.8 6.686 14.253 5.339 2.369
0.9 2.859 6.634 2.803 2.08 0.9 2.849 7.053 2.939 2.26
0.7 0.8 25.394 54.211 18.662 2.256 0.7 0.8 22.849 50.694 17.486 2.427
0.9 5.183 12.211 4.662 2.048 0.9 5.033 12.508 4.757 2.214
0.8 0.9 16.38 40.352 14.042 2.000 0.8 0.9 15.257 38.792 13.518 2.136
Table 2

Relative efficiency of π^P for r=k=3, k1=4, k2=2, a1=11, b1=12, a2=1 and b2=2.

θ1 θ2 RE(Kuk) RE(SG) RE(H1) RE(H2) θ1 θ2 RE(Kuk) RE(SG) RE(H1) RE(H2)
π=0.5 π=0.6
0.1 0.2 15.324 8.514 3.417 2.615 0.1 0.2 14.753 9.6 3.765 2.975
0.3 5.189 4.691 2.143 2.61 0.3 4.929 5.382 2.359 2.969
0.4 2.969 3.764 1.834 2.608 0.4 2.831 4.329 2.008 2.967
0.5 2.076 3.366 1.701 2.607 0.5 2.003 3.869 1.854 2.966
0.6 1.61 3.148 1.629 2.607 0.6 1.576 3.614 1.769 2.965
0.7 1.328 3.012 1.583 2.606 0.7 1.322 3.453 1.716 2.965
0.8 1.141 2.918 1.552 2.606 0.8 1.154 3.342 1.679 2.964
0.9 1.008 2.851 1.529 2.606 0.9 1.036 3.261 1.652 2.964
0.2 0.3 22.274 18.243 6.66 2.658 0.2 0.3 22.047 20.047 7.247 3.027
0.4 6.637 7.471 3.069 2.631 0.4 6.518 8.471 3.388 2.995
0.5 3.484 5.116 2.284 2.619 0.5 3.433 5.867 2.52 2.98
0.6 2.293 4.17 1.969 2.612 0.6 2.282 4.8 2.165 2.971
0.7 1.703 3.676 1.805 2.607 0.7 1.718 4.235 1.976 2.966
0.8 1.36 3.378 1.705 2.604 0.8 1.393 3.89 1.861 2.962
0.9 1.141 3.181 1.639 2.602 0.9 1.188 3.659 1.784 2.96
0.3 0.4 26.907 29.71 10.483 2.729 0.3 0.4 26.988 32.188 11.294 3.113
0.5 7.506 10.425 4.054 2.667 0.5 7.518 11.7 4.465 3.038
0.6 3.741 6.429 2.722 2.634 0.6 3.773 7.341 3.012 2.998
0.7 2.366 4.887 2.208 2.613 0.7 2.415 5.625 2.44 2.973
0.8 1.703 4.107 1.948 2.6 0.8 1.765 4.744 2.146 2.957
0.9 1.328 3.649 1.795 2.59 0.9 1.4 4.218 1.971 2.945
0.4 0.5 29.224 40.83 14.189 2.803 0.4 0.5 29.576 43.906 15.2 3.2
0.6 7.795 13.031 4.923 2.697 0.6 7.929 14.541 5.412 3.073
0.7 3.741 7.471 3.069 2.635 0.7 3.851 8.518 3.404 2.999
0.8 2.293 5.386 2.375 2.596 0.8 2.4 6.212 2.635 2.952
0.9 1.61 4.358 2.032 2.569 0.9 1.718 5.054 2.249 2.919
0.5 0.6 29.224 49.517 17.085 2.847 0.5 0.6 29.812 53.082 18.259 3.248
0.7 7.506 14.768 5.502 2.692 0.7 7.753 16.465 6.053 3.064
0.8 3.484 8.012 3.25 2.599 0.8 3.668 9.161 3.618 2.954
0.9 2.076 5.538 2.425 2.538 0.9 2.238 6.428 2.707 2.881
0.6 0.7 26.907 53.687 18.475 2.82 0.6 0.7 27.694 57.6 19.765 3.207
0.8 6.637 15.116 5.618 2.618 0.8 6.988 16.941 6.212 2.972
0.9 2.969 7.819 3.185 2.496 0.9 3.224 9.035 3.576 2.828
0.7 0.8 22.274 51.255 17.664 2.668 0.7 0.8 23.224 55.341 19.012 3.017
0.9 5.189 13.552 5.097 2.436 0.9 5.635 15.441 5.712 2.75
0.8 0.9 15.324 40.135 13.958 2.327 0.8 0.9 16.4 44.188 15.294 2.604
Table 3

Relative efficiency of π^P for r=k=3, k1=4, k2=2, a1=11, b1=12, a2=1 and b2=2.

θ1 θ2 REKuk RESG REH1 REH2 θ1 θ2 REKuk RESG REH1 REH2
π=0.7 π=0.8
0.1 0.2 15.18 11.472 4.365 3.528 0.1 0.2 17.067 14.922 5.472 4.495
0.3 4.944 6.489 2.704 3.52 0.3 5.332 8.464 3.319 4.484
0.4 2.82 5.215 2.279 3.518 0.4 2.974 6.777 2.756 4.48
0.5 1.999 4.65 2.091 3.516 0.5 2.087 6.021 2.505 4.478
0.6 1.584 4.334 1.985 3.516 0.6 1.648 5.596 2.363 4.477
0.7 1.339 4.133 1.918 3.515 0.7 1.394 5.324 2.272 4.477
0.8 1.18 3.994 1.872 3.515 0.8 1.231 5.135 2.209 4.476
0.9 1.069 3.893 1.838 3.515 0.9 1.12 4.997 2.163 4.476
0.2 0.3 23.421 23.524 8.382 3.595 0.2 0.3 27.326 30.218 10.57 4.587
0.4 6.811 10.159 3.927 3.554 0.4 7.741 13.244 4.912 4.53
0.5 3.564 7.077 2.9 3.534 0.5 3.976 9.244 3.579 4.503
0.6 2.369 5.794 2.472 3.523 0.6 2.611 7.554 3.016 4.488
0.7 1.79 5.106 2.243 3.516 0.7 1.959 6.64 2.711 4.479
0.8 1.461 4.682 2.102 3.512 0.8 1.592 6.073 2.522 4.472
0.9 1.254 4.396 2.006 3.509 0.9 1.365 5.687 2.393 4.468
0.3 0.4 29.086 37.352 12.991 3.703 0.3 0.4 34.477 47.565 16.352 4.735
0.5 8.034 13.944 5.189 3.608 0.5 9.373 18.117 6.536 4.605
0.6 4.021 8.845 3.489 3.557 0.6 4.632 11.565 4.352 4.535
0.7 2.578 6.803 2.808 3.526 0.7 2.942 8.901 3.464 4.492
0.8 1.893 5.74 2.454 3.505 0.8 2.145 7.498 2.997 4.463
0.9 1.511 5.099 2.24 3.49 0.9 1.705 6.645 2.712 4.443
0.4 0.5 32.176 50.639 17.421 3.812 0.4 0.5 38.518 64.166 21.886 4.881
0.6 8.614 17.266 6.296 3.651 0.6 10.228 22.383 7.959 4.663
0.7 4.193 10.262 3.961 3.558 0.7 4.943 13.43 4.974 4.535
0.8 2.627 7.532 3.052 3.498 0.8 3.078 9.886 3.793 4.453
0.9 1.893 6.142 2.588 3.457 0.9 2.207 8.058 3.183 4.396
0.5 0.6 32.691 61.069 20.897 3.867 0.5 0.6 39.451 77.223 26.238 4.953
0.7 8.549 19.545 7.056 3.638 0.7 10.306 25.345 8.946 4.642
0.8 4.079 11.069 4.23 3.499 0.8 4.908 14.528 5.34 4.453
0.9 2.514 7.836 3.153 3.407 0.9 3.019 10.335 3.942 4.327
0.6 0.7 30.631 66.322 22.648 3.808 0.6 0.7 37.275 83.938 28.477 4.864
0.8 7.841 20.202 7.275 3.518 0.8 9.606 26.301 9.264 4.474
0.9 3.678 11.009 4.21 3.339 0.9 4.528 14.549 5.347 4.231
0.7 0.8 25.996 64.082 21.901 3.56 0.7 0.8 31.99 81.513 27.668 4.517
0.9 6.489 18.657 6.76 3.234 0.9 8.13 24.552 8.681 4.082
0.8 0.9 18.785 52.03 17.884 3.036 0.8 0.9 23.596 67.15 22.881 3.796
Table 4

Relative efficiency of π^P for r=k=3, k1=4, k2=2, a1=11, b1=12, a2=1 and b2=2.

θ1 θ2 REKuk RESG REH1 REH2 θ1 θ2 REKuk RESG REH1 REH2
π=0.9 π=0.9
0.1 0.2 22.467 22.733 7.978 6.629 0.3 0.7 3.828 13.575 4.925 6.624
0.3 6.578 12.867 4.689 6.612 0.8 2.733 11.4 4.2 6.578
0.4 3.504 10.244 3.815 6.606 0.9 2.133 10.067 3.756 6.545
0.5 2.383 9.058 3.419 6.603 0.4 0.5 54.467 95.933 32.378 7.245
0.6 1.844 8.387 3.196 6.601 0.6 14.244 34.067 11.756 6.897
0.7 1.541 7.956 3.052 6.6 0.7 6.763 20.556 7.252 6.693
0.8 1.351 7.656 2.952 6.599 0.8 4.133 15.133 5.444 6.561
0.9 1.224 7.435 2.878 6.599 0.9 2.911 12.307 4.502 6.47
0.2 0.3 37.578 45.667 15.622 6.777 0.5 0.6 56.244 115.267 38.822 7.353
0.4 10.244 20.2 7.133 6.686 0.7 14.578 38.6 13.267 6.861
0.5 5.084 14.081 5.094 6.643 0.8 6.862 22.304 7.835 6.56
0.6 3.244 11.467 4.222 6.618 0.9 4.161 15.892 5.697 6.359
0.7 2.378 10.04 3.747 6.603 0.6 0.7 53.578 125.4 42.2 7.202
0.8 1.899 9.148 3.449 6.593 0.8 13.8 40.2 13.8 6.588
0.9 1.605 8.54 3.247 6.586 0.9 6.467 22.467 7.889 6.203
0.3 0.4 48.244 71.4 24.2 7.014 0.7 0.8 46.467 122.333 41.178 6.637
0.5 12.8 27.6 9.6 6.806 0.9 11.911 37.867 13.022 5.957
0.6 6.17 17.667 6.289 6.693 0.8 0.9 34.911 102.067 34.422 5.481
Table 5

Relative efficiency of π^P for r=k=3, k1=4, k2=2, a1=11, b1=12, a2=1 and b2=2.

4. DISCUSSIONS OF RESULTS

From Tables 1–5, it is observed that the proposed estimator performs better than all the considered estimators over the whole range of π. It is also worth mentioning that the proposed RRT is simple to apply as respondents have to just draw a card and report the number written on the card, while in the RRTs proposed by Singh and Grewal [2], Hussain et al. [3] and Hussain et al. [4] respondents might have consumed a lot of time for observing a specific type of card and sequences of specific cards. Therefore, it is concluded that the proposed strategy of randomizing the response using Hermite distribution performs well without incurring any additional sampling and administrative cost.

CONFLICT OF INTEREST

There are no conflicts of interest among the authors.

AUTHORS' CONTRIBUTIONS

Zawar Hussain and Said Farooq Shah conceived the idea and contributed in calculation of results and writing of the manuscript.

Funding Statement

There is no funding for this work.

REFERENCES

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Journal
Journal of Statistical Theory and Applications
Volume-Issue
18 - 4
Pages
361 - 366
Publication Date
2019/11/21
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
DOI
10.2991/jsta.d.191112.002How to use a DOI?
Copyright
© 2019 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

risenwbib
TY  - JOUR
AU  - Said Farooq Shah
AU  - Zawar Hussain
PY  - 2019
DA  - 2019/11/21
TI  - An Application of Hermite Distribution in Sensitive Surveys
JO  - Journal of Statistical Theory and Applications
SP  - 361
EP  - 366
VL  - 18
IS  - 4
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.191112.002
DO  - 10.2991/jsta.d.191112.002
ID  - Shah2019
ER  -