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MLE of $\theta$ from $N(\theta+2, \theta^2)$

Your last derivation can be rearranged into a Quadratic Eunction form (See Quadratic Form, Quadratic Function): $$ \underbrace{-n}_{a} {\theta}^{2} \underbrace{-\sum_{i} \left( {x}_{i} - 2 \right)}_{b}...
Royi's user avatar
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1 vote
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Prove that the MLE for variance is consistent

From the expectation you computed, you can see that $\widehat{\sigma}^{2}$ is asymptotically unbiased for $\sigma^{2}$. If you compute its variance, and show that it goes to zero as $n \rightarrow \...
aprobabilityspace's user avatar
2 votes

Computing the likelihood function

In general, likelihood is only defined up to proportionality. Here the likelihood for $\theta$ is proportional to both your expressions. Since $\left(x_i+\frac{1}{2}\right)^{n_1}$ and $x_j^{n_2}$ do ...
Henry's user avatar
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3 votes
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Computing the likelihood function

$L(x;\theta)=\prod_{i=1}^{n}f(x_i)$ is the correct formula for the likelihood when the underlying distribution has a density. The $n_1$ and $n_2$ are defined as $n_1=\operatorname{Card}\left(\left\{i\...
Davide Giraudo's user avatar
3 votes

Finding MLE given dependent observations from uniform distribution $U(0,\theta)$

By definition, when $X_1,\dots, X_n$ are dependent, the MLE is $X_{(n)}=\max\limits_i X_i$ if A: and $X_1,\dots, X_n$ have a joint pdf $f$ B: The parameter $\theta$ only affects the univariate ...
Amir's user avatar
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5 votes
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Finding MLE given dependent observations from uniform distribution $U(0,\theta)$

The hypothesis that the sample maximum is the MLE, even if there are dependencies, is false. The MLE really depends on how the $X_i$ depend on each other. Consider this example: $$ X_1 \sim U(0, \...
nicola's user avatar
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1 vote
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Check MLE estimator is asymptotically normal

In this particular case, you should recognize that if $$X \sim \operatorname{Beta}(1/\theta, 1),$$ then $$Y = -\log X \sim \operatorname{Exponential}(\lambda = 1/\theta),$$ hence $$\hat \theta_n \sim \...
heropup's user avatar
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1 vote
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Min-max optimization and prediction of a parameter in a mathematical model

Let $x$ be a vector in $\mathbb{R}^{11}$ denoting the 11 parameters, i.e., the 10 parameters plus $R$. Let $f$ be some function (which you are trying to find) so that $f(x)$ is a reasonable estimate ...
D.W.'s user avatar
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1 vote

Efficient and unbiased estimation of the location ($\mu$) of truncated normal distribution with known scale ($\sigma^2$) and truncation points

One observation is not enough to infer $\mu$ with much accuracy at all. I think you are inherently going to be limited by your lack of data. I suspect the best you can do is take a Bayesian approach. ...
D.W.'s user avatar
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6 votes
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Given the maximum likelihood function- estimate the value of the parameter

$\beta$ is the minimum of this Pareto distribution: the question says the density is $0$ below $\beta$ so there is zero probability of observing data below $\beta$. Suppose for example you knew $\...
Henry's user avatar
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4 votes

Given the maximum likelihood function- estimate the value of the parameter

As stated in the solution, the maximum likelihood function, for a fixed $\alpha$, is increasing in $\beta$. Thus, for some fixed sample $(x_1,\ldots,x_n)$ we want to make $\beta$ as big as possible. ...
Julio Puerta's user avatar
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