A Test for Mean with Additional Observations

Abstract

X₁ and X₂ are two random variables following a bivariate normal distribution, $N_2\hspace{0.17em}\left({\mu }_1,\hspace{0.17em}{\mu }_2\hspace{0.17em};\hspace{0.17em}\hspace{0.17em}{\sigma }_{\hspace{0.17em}1}^2,\hspace{0.17em}{\sigma }_{\hspace{0.17em}2}^2,\hspace{0.17em}\rho \right)$ , with all parameters unknown. Given n₁ indepen­ dent observations on (X₁ , X₂) and n₂ additional independent observations on X₁ , we propose a procedure for inferring about µ₂ utilizing all available information. The proposed procedure is compared with that based on student's t­statistic utilizing only the observations on X₂. The generalization to the multivariate case is also done.