Estimate average percentage error based on another gaussian measurement I have a model where the error is proportional to the throughput. This is, the observations I got come from a measurement instrument that has some error and it measures material going through in little buckets. I am approximating the model as follows
$$
x_i = x~(1 + \epsilon_X) \\ \epsilon_X \sim N(\mu_X,\sigma_X)
$$
I am also able to determine via another instrument, the total material before and after, so I am able to state (after simplifying)
$$
\delta_i = \delta + \epsilon_D' \approx x_i \\ \epsilon_D' = \epsilon_D^1 - \epsilon_D^2 \\ \epsilon_D^{\{1,2\}} \sim N(\mu_D, \sigma_D)
$$
The mean $\mu_D$ does not matter, as it cancels out in $\epsilon_D'$, but the mean $\epsilon_X$, if not zero, is what I am after. Things get complicated because the variance of $x_i$ depends on the actual throughput, so it seems I have heteroscedasticity in the model.
Put other way, $\epsilon_X$ is given as a random percentage of the measured quantity.
So, I have a number of $\delta_i$ readings, with random noise; and another set of $x_i$ readings with noise as above; they should match, but they don't due to noise. Actually, I am interested in the problem of finding out if the noise in $x_i$ is not zero-mean.
Any pointers would be appreciated.
Update
For what is worth, in addition to Alecos' answer, I tried also what seanv507 suggested: by averaging $(1/n)\sum_{i=1}^n(Ln(x_i)-Ln(\delta_i))$, and then reverting with $Exp(.)$. It is very close to the ratio model (as said $1+x \approx e^x$ for small x) with the additional advantage that $x_i$ is is never negative under this model (if this is what one wants).
 A: I will change notation to simplify.  Using information provided by the OP in the comments,
We have a latent, unobservable random variable $Y$. We measure this random variable through two different methods, each with measurement error. These methods are represented as
$$X_i = Y_i(1+u_i), \qquad u_i \sim N(\mu_{u}, \sigma^2_{u}),\;\; i=1,...,n$$
with $Y_i$ independent of $u_i$
and
$$Z_i = Y_i+w_i, \qquad w_i \sim N(0, \sigma^2_{w}),\;\; i=1,...,n$$
with $w_i$ independent of $Y_i$
Our interest lies in whether $\mu_{u} =0$ or not.
Now, it is a crucial assumption that $Y$ is a random variable (and not deterministic), and also that it is an ergodic-stationary variable. If $Y$ was deterministic, then the random variable $X$ would not have a constant variance for each $i$. If $Y$ was a random variable but not ergodic/stationary, then again $X$ would not have a constant variance. $u$ is ergodic/stationary by construction, and the same holds for $Z$. 
Since both $Y$ and $u$ are ergodic/stationary, then their product is also ergodic/stationary. So
$$\operatorname{Var}(X_i) = Var(Y+Yu) $$
and constant $\forall i$, and consistently estimated by the sample variance.
Then:  
Combine the two functional equations of the measurements to obtain (solving for $Y_i$ and equating)
$$X_i = (Z_i -w_i)+Y_iu_i \Rightarrow E(X_i) = E(Z_i) - E(w_i) + E(Y_iu_i)$$
$$\Rightarrow E(X_i) = E(Z_i) + E(Y_i)E(u_i)$$
where we have used the assumption of independence between $Y_i$ and $u_i$ to separate the expected value.
Under a null assumption
$$H_o: E(u_i) = \mu_{u}=0 \Rightarrow E(X_i) = E(Z_i)$$
and we can apply a Welch's test for the equality of means, using sample means and sample variances of $X$ and $Z$.
Again, it is critical that we can assume that the $X$ series has a constant variance.
ADDENDUM
We have that $E(Z) = E(Y)$. Applying a Method-of Moments approach
We have 
$$E(X) = E(Z) + E(Y)E(u)= E(Z)\cdot [1+E(u)] \Rightarrow E(u)=\mu_u = \frac {E(X)}{E(Z)} -1$$
So
$$\hat \mu_u = \frac {(1/n)\sum_{i=1}^nx_i}{(1/n)\sum_{i=1}^nz_i} -1$$
The (additional) critical assumption for the Welch's test and for this estimator, is that $Y_i$ is independent of $u_i$, and so we can separate the expected value $E(Yu) = E(Y)E(u)$.
