The maximum likelihood estimator for the parameter of the exponential distribution under type II censoring can be derived as follows. I assume the sample size is $m$, of which the $n < m$ smallest are observed and the $m - n$ largest are unobserved (but known to exist.)
Let us assume (for notational simplicity) that the observed $x_i$ are ordered: $0 \leq x_1 \leq x_2 \leq \cdots \leq x_n$. Then the joint probability density of $x_1, \dots, x_n$ is:
$f(x_1, \dots, x_n) = {m!\lambda^n \over {(m-n)!}}\exp\left\{-\lambda\sum_{i=1}^nx_i\right\}\exp\left\{-\lambda(m-n)x_n\right\}$
where the first exponential relates to the probabilities of the $n$ observed $x_i$ and the second to the probabilities of the $m-n$ unobserved $x_i$ that are greater than $x_n$ (which is just 1 - the CDF at $x_n$.) Rearranging terms leads to:
$f(x_1, \dots, x_n) = {m!\lambda^n \over {(m-n)!}}\exp\left\{-\lambda\left[\sum_{i=1}^{n-1}x_i+(m-n+1)x_n\right]\right\}$
(Note the sum runs to $n-1$ as there is a "$+1$" in the coefficient of $x_n$.) Taking the log, then the derivative w.r.t. $\lambda$ and so on leads to the maximum likelihood estimator:
$\hat{\lambda} = n / \left[\sum_{i=1}^{n-1}x_i+(m-n+1)x_n\right]$
Since you are a tutor, any knowledge is always for a good cause. So I will provide some bounds for the MLE.
We have arrived at
$$(1-\lambda x_{(n)})e^{\lambda x_{(n)} } + \lambda n x_{(n)} - 1 = 0$$
with $x_{(n)}\equiv M_n$. So
$$(1-\hat \lambda x_{(n)})e^{\hat \lambda x_{(n)}} = 1-\hat \lambda x_{(n)}n $$
Assume first that $1-\hat \lambda x_{(n)} >0$. Then we must also have $1-\hat \lambda x_{(n)}n>0$ since the exponential is always positive. Moreover since $x_{(n)}, \hat \lambda > 0\Rightarrow e^{\hat \lambda x_{(n)}}>1$. Therefore we should have
$$\frac {1-\hat \lambda x_{(n)}n}{1-\hat \lambda x_{(n)}}>1 \Rightarrow \hat \lambda x_{(n)}>\hat \lambda x_{(n)}n$$
which is impossible. Therefore we conclude that
$$\hat \lambda >\frac 1{x_{(n)}},\;\; \hat \lambda = \frac c{x_{(n)}}, \;\; c>1$$
Inserting into the log-likelihood we get
$$\ell(\hat\lambda(c)\mid x_{(n)}) = \log \frac c{x_{(n)}} + \log n - \frac c{x_{(n)}} x_{(n)} + (n-1) \log (1 - e^{-\frac c{x_{(n)}} x_{(n)}})$$
$$= \log \frac n{x_{(n)}} + \log c - c + (n-1) \log (1 - e^{-c})$$
We want to maximize this likelihood with respect to $c$. Its 1st derivative is
$$\frac{d\ell}{dc}=\frac 1c -1 +(n-1)\frac 1{e^{c}-1}$$
Setting this equal to zero, we require that
$$e^{c}-1 - c\left(e^{c}-1\right)+(n-1)c =0$$
$$\Rightarrow \left(n-e^c\right)c = 1-e^c$$
Since $c>1$ the RHS is negative. Therefore we must also have $n-e^c <0 \Rightarrow c > \ln n$. For $n\ge 3$ this provides a tighter lower bound for the MLE, but it doesn't cover the $n=2$ case, so
$$\hat \lambda > \max \left\{\frac 1{x_{(n)}}, \frac {\ln n}{x_{(n)}}\right\}$$
Moreover (for $n\ge 3$) rearranging the 1st-order condition we have that
$$c= \frac{e^c-1}{e^c-n} > \ln n \Rightarrow e^c -1 > e^c\ln n -n\ln n $$
$$\Rightarrow n\ln n-1>e^c(\ln n -1) \Rightarrow c< \ln{\left[\frac{n\ln n-1}{\ln n -1}\right]}$$
So for $n\ge 3$ we have that
$$\frac 1{x_{(n)}}\ln n < \hat \lambda < \frac 1{x_{(n)}}\ln{\left[\frac{n\ln n-1}{\ln n -1}\right]}$$
This is a narrow interval, especially if $x_{(n)}\ge 1$. For example (truncated at 3d digit )
$$\begin{align}
n=10 & &\frac 1{x_{(n)}}2.302 < \hat \lambda < \frac 1{x_{(n)}}2.827\\
n=100 & & \frac 1{x_{(n)}}4.605 < \hat \lambda < \frac 1{x_{(n)}}4.847\\
n=1000 & & \frac 1{x_{(n)}}6.907 < \hat \lambda < \frac 1{x_{(n)}}7.063\\
n=10000 & & \frac 1{x_{(n)}}9.210< \hat \lambda < \frac 1{x_{(n)}}9.325\\
\end{align}$$
Numerical examples indicate that the MLE tends to be equal to the upper bound, up to second decimal digit.
ADDENDUM: A CLOSED FORM EXPRESSION
This is just an approximate solution (it only approximately maximizes the likelihood), but here it is:
manipulating the 1st-order condition we want to have
$$\lambda = \frac 1{x_{(n)}}\ln \left[\frac {\lambda x_{(n)}n -1}{\lambda x_{(n)} -1}\right]$$
Now, one can show (see for example here) that
$$E[X_{(n)}] = \frac {H_n}{\lambda},\;\; H_n = \sum_{k=1}^n\frac 1k$$
Solving for $\lambda$ and inserting into the RHS of the implicit 1st-order condition, we obtain
$$\lambda = \frac 1{x_{(n)}}\ln \left[\frac {nH_n\frac {x_{(n)}}{E[X_{(n)}]} -1}{ H_n\frac {x_{(n)}}{E[X_{(n)}]} -1}\right]$$
We want an estimate of $\lambda$, given that $X_{(n)}=x_{(n)}$, $\hat \lambda \mid \{X_{(n)}=x_{(n)}\}$. But in such a case, we also have $E[X_{(n)}\mid \{X_{(n)}=x_{(n)}\}] =x_{(n)}$. this simplifies the expression and we obtain
$$\hat \lambda = \frac 1{x_{(n)}}\ln \left[\frac {nH_n -1}{ H_n -1}\right]$$
One can verify that this closed form expression stays close to the upper bound derived previously, but a bit less than the actual (numerically obtained) MLE.
Best Answer
I cannot speak as to the use of these symbols but let me show you instead the traditional way, why the mle is biased.
Recall that the exponential distribution is a special case of the General Gamma distribution with two parameters, shape $a$ and rate $b$. The pdf of a Gamma Random Variable is:
$$f_Y (y)= \frac{1}{\Gamma(a) b^a} y^{a-1} e^{-y/b}, \ 0<y<\infty$$
where $\Gamma (.)$ is the gamma function. Alternative parameterisations exist, see for example the wikipedia page.
If you put $a=1$ and $b=1/\lambda$ you arrive at the pdf of the exponential distribution:
$$f_Y(y)=\lambda e^{-\lambda y},0<y<\infty$$
One of the most important properties of a gamma RV is the additivity property, simply put that means that if $X$ is a $\Gamma(a,b)$ RV, $\sum_{i=1}^n X_i$ is also a Gamma RV with $a^{*}=\sum a_i$ and $b^{*}=b$ as before.
Define $Y=\sum X_i$ and as noted above $Y$ is also a Gamma RV with shape parameter equal to $n$, $\sum_{i=1}^n 1 $, that is and rate parameter $1/\lambda$ as $X$ above. Now take the expectation $E[Y^{-1}]$
$$ E\left [ Y^{-1} \right]=\int_0^{\infty}\frac{y^{-1}y^{n-1}\lambda^n}{\Gamma(n)}\times e^{-\lambda y}dy=\int_0^{\infty}\frac{y^{n-2}\lambda^n}{\Gamma(n)}\times e^{-\lambda y}dy$$
Comparing the latter integral with an integral of a Gamma distribution with shape parameter $n-1$ and rate one $1/\lambda$ and using the fact that $\Gamma(n)=(n-1) \times \Gamma(n-1)$ we see that it equals $\frac{\lambda}{n-1}$. Thus
$$E\left[ \hat{\theta} \right]=E\left[ \frac{n}{Y} \right]=n \times E\left[Y^{-1}\right]=\frac{n}{n-1} \lambda$$
which clearly shows that the mle is biased. Note, however, that the mle is consistent. We also know that under some regularity conditions, the mle is asymptotically efficient and normally distributed, with mean the true parameter $\theta$ and variance $\{nI(\theta) \}^{-1} $. It is therefore an optimal estimator.
Does that help?