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Let $x_1, \cdots, x_n \sim f(x;\theta) = \frac{1}{2\theta}, -\theta < x < \theta$ and $\theta > 0$. I'm tasked with finding $\hat{\theta}$ using the moment method. It's clear that $f(x;\theta) \sim \m …

I apologize for asking so many questions. My last question, I am stuck with evaluating $\mathrm{Var}\left(\frac{1}{n}\sum_{i=1}^{n}{(\ln(x)-\mu)^2}\right)$, where $x \sim \mathrm{LogNorm}(\mu, \sigma …

I am working on an academic paper proposing a new estimator for a population mean. It works quite well in simulations across various superpopulation models (linear, quadratic, and strictly nonlinear). …

I'm trying to compare the results of a by hand, one-way between-groups ANOVA with those given from R. Weirdly, I get completely different results. I used formulas given by my professor: \begin{align} …

I'm tasked with finding $\mathbb{E}\left(\hat{\sigma}^2\right)$ where $\hat{\sigma}^2 = \frac{\sum_{i=1}^{n}{\left(\ln(x)-\mu\right)^2}}{n}$, $x \sim \mathrm{Lognorm}(\mu, \sigma^{2}$) . So far I get, …

I am trying to derive the bias and variance of $\hat{\mu}_{MLE} = \frac{-\sum_{i=1}^{n}{\ln(x)}}{2\sigma^2n}$, where $x \sim \mathrm{LogNorm}(\mu, \sigma^2)$. So far I have $$ \mathrm{Bias} = \mu\lef …