Let X1,X2, . . . ,Xn be a random sample from N(?1, ?2).

QUESTION:

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from \(N\left(\theta_{1}, \theta_{2}\right)\). Show that the sufficient statistics \(Y_{1}=\bar{X}\) and \(Y_{2}=S^{2}\) are independent of the statistic.

\(Z=\sum_{i=1}^{n-1} \frac{\left(X_{i+1}-X_{i}\right)^{2}}{S^{2}}\)

because \(Z\) has a distribution that is free of \(\theta_{1}\) and \(\theta_{2}\).

 HINT: Let \(w_{i}=\left(x_{i}-\theta_{1}\right) / \sqrt{\theta_{2}}, i=1,2, \ldots, n\), in the multivariate integral representing \(E\left[e^{z^{z}}\right]\).

Equation Transcription:

Text Transcription:

X_1,X_2,...,X_n

N(theta_1,theta_2)

Y_1=bar X

Y_2=S^2

Z=sum_i=1^n-1 (X_i+1-X_i)^2/S^2

Z

theta_1

theta_2

w_i=(x_i-theta_1)/sqrt theta_2,i=1,2,...,n

E[e^tZ]