LetY have ann-variate normal distribution with covariance matrixσ2I and mean vectorXβ, whereX is a knownn×p matrix. The problem of estimatingθ=σ2+β′X′CXβ is studied. The admissibility and inadmissibility of the estimators of the form\(b\hat S^2 + \hat \beta 'X'CX\hat \beta \), where\(\hat \beta = (X'X)^ - X'Y\) and\(S^2 = (Y - X\hat \beta )'(Y - X\hat \beta )\), are established. Another class of admissible quadratic estimators ofθ is derived.