I have a simulation in which I specify certain parameters, say $a$, $b$, and $c$. I have a model that I fit to the simulation results and find best fit parameters $a'$, $b'$, and $c'$. I do this many times for a given set of $a$, $b$, and $c$, and I get best fit parameters for each trial $a_i'$, $b_i'$, and $c_i'$. I want to check if the $a_i'$, $b_i'$, and $c_i'$ are statistically different from the known true values $a$, $b$, and $c$.
One way I had thought of doing this was just to check if the $a_i'$, $b_i'$, and $c_i'$ are normally distributed about their respective true values. But in this case what test do I use? I am not comparing two distributions, I am comparing one distribution to a fixed value (i.e. the standard deviation of the true values is identically zero).
Here's how it could work for a single parameter. I assume you would use the mean of the $a_i$s to get your estimate $\hat{a}$ of the true parameter $a$. For a hypothesis test, your null hypothesis would be that $\hat{a}=\bar{a_i}=a$. Making the assumption that the $a_i$s are normally distributed, you could use the one-sample t test. The test statistic is $$ t = \frac{\hat{a}-a}{s/\sqrt{n}}$$ where $n$ is the number of trials in your simulation, and $s$ is the sample standard deviation of your $a_i$. Then $t$ is a sample from the t-distribution with $n-1$ degrees of freedom.
The test gives you a p-value, which is the probability of seeing values at least as extreme as your estimate under the assumption that the null hypothesis is true. Low p-values are inconsistent with the null-hypothesis; typically you might select some significance level $\alpha$ in advance, e.g. $\alpha=0.05$ and reject the null-hypothesis when $p<\alpha$ .
Note that $\alpha$ is also the probability of wrongly rejecting the null hypothesis, or type 1 error. This means that if you run $S$ simulations, then you would expect to wrongly reject the null hypothesis $S\alpha$% of the time. A more insidious problem is that once you start doing hypothesis tests for each parameter you get more and more likely to reject at least one of the null hypotheses. If you have 100 parameter estimates, then you're extremely likely to find that one of your estimates is too extreme compared to the corresponding true parameter! There are several ways round this, e.g. coming up with a single test that simultaneously looks at all of your parameters at once if you have some insight from your problem domain. Alternatively, there are various ways to address combine multiple hypothesis tests, e.g. search for the Bonferroni correction, the Sidak correction, or the Benjamini-Hochberg procedure.
Personally, I would be tempted to simply plot the error bars (mean $\pm$ standard error) of each of your estimates as the number of trials increases from 1 to $n$ and check that they look sensible. Looking at the p-values for the total number of trials $n$ would be a useful further check, bearing in mind that if you run enough simulations you're bound to see some high p-values occasionally.