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Consider $X_1,X_2,...,X_n$ i.i.d $U(-\theta,0)$. I want to find the maximum likelihood estimator of $\theta$. I know that $f(x,\theta)=\frac{1}{\theta}$ for $-\theta < x < 0$ and that $L_n(\theta, x …

Let $S \sim Binomial(N,\theta)$, where $S$ can be seen as $S_n = B_1+...+B_N$ and $B_i$'s are i.d.d $Bernoulli(\theta)$. I have to show that $\frac{\hat{N}_{MLE}}{N}$ converges to $1$ in probability …

Let $X_1,X_2,\ldots,X_n$ be i.i.d $\operatorname{Uniform}(-\theta,0)$. Now consider all of the estimates of the form $S_\rho=\rho \hat{\theta}_\text{MLE}$. I have to find which of these estimates has …

Let $Y_i=\alpha_0+\beta_0 X_i + \epsilon_0$, where $\epsilon_i \sim N(0, \sigma_0^2)$ and $X_i \sim N(\mu_x,\tau_0^2)$ are independent. The data $(X_i, Y_i)$ are generated from $Y_i=\alpha_0+\beta_0 …

Let $Y_i=\alpha_0+\beta_0 X_i + \epsilon_0$, where $\epsilon_i \sim N(0, \sigma_0^2)$ and $X_i \sim N(\mu_x,\tau_0^2)$ are independent. The data $(X_i, Y_i)$ are generated from $Y_i=\alpha_0+\beta_0 …

Consider $X_1, X_2, \ldots,X_n$ iid $\operatorname{Poisson}(\lambda)$ random variables, where $\lambda \in [a,b]$, $0<a<b$. How do you find the maximum likelihood estimator of the restricted $\lambda …