Data Gathering, Analysis and Protection of Privacy Through by Arijit Chaudhuri, Tasos C. Christofides, C.R. Rao PDF

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I¼1 We have the following theorem. Theorem 5. sZ1 Z2 is an unbiased estimator of sZ1 Z2 . Proof. We have " # n n X 1 X 1     ðZ1i À Z1 ÞðZ2i À Z 2 Þ ¼ Z1i Z2i À nZ1 Z 2 sZ1 Z2 ¼ ðn À 1Þ i¼1 ðn À 1Þ i¼1 Thus " # n X 1 EðZ1i Z2i Þ À nEðZ1 Z2 Þ EðsZ1 Z2 Þ ¼ ðn À 1Þ i¼1 (14) Since À Á EðZ1i Z2i Þ ¼ E½XS1 YS2 Š ¼ EðXY ÞEðS1 S2 Þ ¼ sxy + m1 m2 ðt11 + y1 y2 Þ and CovðZ1 , Z2 Þ CovðZ1 , Z2 Þ ¼ ¼ EðZ1 Z2 Þ À EðZ1 ÞEðZ2 Þ n (15) 50 Handbook of Statistics We find that EðZ1 Z2 Þ ¼ sxy ðt11 + y1 y2 Þ + t11 m1 m2 CovðZ1 , Z2 Þ + m1 m2 y1 y2 + EðZ1 ÞEðZ2 Þ ¼ n n (16) Using (15) and (16) in (14), we get E ð sZ 1 Z 2 Þ ¼ & '!

For example, X may stand for the respondents’ income and Y may stand for the respondents’ expenditure. The problem of estimating the correlation coefficient both between two sensitive variables, and between a sensitive and a nonsensitive variable is considered. Asymptotic properties of the proposed estimators are investigated through analytical expressions. 2 TWO SCRAMBLING VARIABLE RANDOMIZED RESPONSE TECHNIQUE Suppose X denotes the response to the first sensitive question (eg, income), and Y denotes the response to the second sensitive question (eg, expenditure).

We have Eðd1 d0 Þ ¼ nÀ1 G1 (32) where " s2Z s2Z t20 s2Z s2Z t02 s2Z s2Z ðA22 À 1Þ + 1 2 ðA22 À 1Þ À 2t20 m21 s2Z2 CZ1 A12 + 1 2 ðA22 À 1Þ 1 2 ny21 ny22 o É s2Z t02 t20 m21 n 2 1 + nÀ1 ðA22 À 1Þ À 2 CZ1 + 2CZ1 A12 À 2s2Z1 t02 m22 CZ2 A21 ny21 y22 ny22 # o s2Z t20 t02 m22 n À 1 C2Z2 + 2CZ2 A21 + 4t20 t02 m21 m22 rZ1 Z2 CZ1 CZ2 ny21 n   o G1 ¼ s2x s2y t20 + y21 t02 + y22 + s2Z s2Z t20 t02 È 1 2 Theorem 8. The bias of the estimator rxy defined at (22) of rxy is seen to be approximately ! 3 1 1 (33) BðrXY Þ ¼ nÀ1 rXY ðA1 + B1 Þ À ðD1 + F1 Þ + G1 8 2 4 where A1, B1, D1, F1, and G1 have been defined in Lemmas 4–9.

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