Let X1,X2, . . . ,Xn be a random sample from N(0, ?),

QUESTION:

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from \(N(0, \theta)\), where \(\sigma^{2}=\theta>0\) is unknown. Argue that the sufficient statistic \(Y=\sum_{i=1}^{n} X_{i}^{2}\) for \(\theta\) and \(Z=\sum_{i=1}^{n} a_{i} X_{i} / \sum_{i=1}^{n} X_{i}\) are independent. HINT: Let \(x_{i}=\theta w_{i}, i=1,2, \ldots, n\), in the multivariate integral representing \(E\left[e^{t Z}\right]\).

Equation Transcription:

Text Transcription:

X_1,X_2,...,X_n

N(0,theta)

sigma^2=theta>0

Y=sum_i=1^n X_i^2

Z=sum_i=1^n a_iX_i/sum_i=1^n  X_i

x_i= theta w_i,i=1,2,...,n

E[e^tZ]