Let independent random samples, each of size $n$, be taken from the $k$ normal distributions with means $$\mu_j = c + d[j-(k+1)/2], j = 1, 2,..., k,$$ respectively and common variance $\sigma^2$, Find the maximum likelihood estimators of $c$ and $d$.
Is this the correct likelihood function?
$$L(\mu) = (\sqrt{2\pi}\sigma)^{nk}\exp\left[\frac{\sum_{j=1}^n\sum_{j=1}^k(X_j-\mu_j)^2}{2\sigma^2}\right]$$
No. Let $X_{ji}$ be the $i$-th observation from $j$-th sample. Then the likelihood will be $$L(c,d)=\dfrac{1}{(2\pi \sigma^2)^{nk/2}}\exp\left[-\dfrac{1}{2\sigma^2}\sum_{j=1}^k\sum_{i=1}^n(x_{ji}-\mu_j)^2\right]$$
Setting $\dfrac{\partial}{\partial c} L(c,d)=0$ yields $c_{mle}=\dfrac{1}{nk}\sum_{j=1}^k\sum_{i=1}^n x_{ji}$
Setting $\dfrac{\partial}{\partial d} L(c,d)=0$ yields $d_{mle}=\dfrac{\sum_{j=1}^k j\left(\sum_{i=1}^n x_{ji}-\dfrac{1}{k}\sum_{j=1}^k\sum_{i=1}^n x_{ji}\right)}{n\sum_{j=1}^k \left(j-\dfrac{k+1}{2}\right)^2}$