a question on negative mean square error for simple random sampling,
i have calculated some mean square errors of many types of ratio estimator.
 
Well, i obtained negative mean square error. Is there a mistake? negative MSE is a normal thing?
 A: You must distinguish between estimates of a parameter and the parameter itself. Thus, some estimates of a positive variance component can be negative, because one random term is subtracted from another.  Here, as @Sjoerd stated, the MSE is always positive and the formula would be positive  if true values of the component  parameters were input. But you have entered estimates of $\beta_y$ and $\theta$, better designated as $\hat{\beta_y}$ and $\hat{\theta}$. It is unpleasant, but not contrary to estimation theory, that $\hat{\beta_y} -2\hat{\theta}$, the difference of two random variables, is negative enough to make $\widehat{MSE}(t_2)$ negative. This does not mean that  $MSE(t_2)$ itself is negative. 
A: It definitely seems like you have made a mistake, as the mean squared error can not be negative. I did look at the picture that you added, but I find it hard to determine what is going on. Nevertheless let me state the MSE for you here in case you ended up with the wrong definition. 
The MSE of a set of predicted (calculated) values $Y' = (y'_{0}, .., y'_{n})$ with respect to their targets $Y = (y_{0}, .., y_{n})$ is given by 
MSE = $\frac{1}{n}\sum_{i=1}^{n} (y'_{i} - y_{i})^{2}$
I realize I am stating the obvious here, but not the square in the above formula which prevents the outcome from becoming negative. 
