I have the following problem:
The distribution of random variable $X$ (which depends on parameters $\lambda <0.5$ and $\mu <0.5$) is given by:
$ X: 1 \quad 2 \qquad 3 \\ P: \lambda \quad \mu \quad 1-\lambda- \mu$
I should find the values of estimators $\lambda$, $\mu$ using the method of moments and say if they are consistent or not (here I have problems).
My solution: I find actual values of $\lambda$ and $\mu$ by solving the system: $\begin{cases}E(X)=\bar{X}\\ E(X^2)=\bar{X^2} \end{cases}$
and then, in order to show consistency, I need to find variance for n $\to\infty$. I'm totally lost here.
I think the method of moments estimation here should be something like this:
$\hat \lambda = \frac{1}{n}\sum\limits_{i=1}^n [x_{i} = 1], \ \hat \mu = \frac{1}{n}\sum\limits_{i=1}^n [x_{i} = 2], \ $ where $n$ - number of observations, $[z = b] = \left\{\begin{array}{rcl} 1, z = b \\ 0, z \not = b \end{array}\right.$
Then, let's introduce new r.v's: $x_{\lambda} = \left\{\begin{array}{rcl} 1, x = 1 \\ 0, x \not = 1 \end{array}\right., x_{\mu} = \left\{\begin{array}{rcl} 1, x = 2 \\ 0, z \not = 2 \end{array}\right.$.
$\mathbb{E}(x_{\lambda}) = \mathbb{P}(x_{\lambda} = 1) = \lambda, \ \mathbb{E}(x_{\mu}) = \mathbb{P}(x_{\mu} = 1) = \mu$.
$\mathbb{D}x_{i, \lambda} = \mathbb{P}(x_{\lambda} = 1) - \mathbb{P}(x_{\lambda} = 1)^2 = \lambda - \lambda^2$
$\mathbb{D}x_{i, \mu} = \mathbb{P}(x_{\mu} = 1) - \mathbb{P}(x_{\mu} = 1)^2 = \mu - \mu^2$
$\hat \lambda = \frac{1}{n}\sum\limits_{i=1}^n x_{i, \lambda} \ \Rightarrow \mathbb{E} \hat \lambda = \frac{1}{n}\sum\limits_{i=1}^n \mathbb{E} x_{i, \lambda} = \frac{1}{n}\sum\limits_{i=1}^n \lambda = \lambda \Rightarrow \hat \lambda $ is unbiased estimator of $\lambda$ (so is $\hat \mu$).
Now compute it's variance: $\mathbb{D}\hat \lambda = \frac{1}{n^2} \sum\limits_{i=1}^n \mathbb{D} x_{i, \lambda} = \frac{\lambda - \lambda^2}{n} \to 0 $ when $n \to \infty$. So $\hat \lambda$ is consistent (so is $\hat \mu$).