All Questions
Tagged with expected-value maximum-likelihood
35
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How to prove that the MLE of a uniform distribution is biased using the formula given below? [duplicate]
I've calculated the MLE of the uniform distribution on [0,theta] as maxi{Xi} but don't know how to prove it is biased. The formula I have learned to prove it is unbiased is E(θ^)-θ=0.
Was stuck on how ...
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1
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68
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Definition of cost function and likelihood: how one appears from the other
I was reading this paper and the book "Introduction to Quantum State Estimation" by Yong Siah Teo and I am facing some issues trying to understand how the definition of the cost function ...
- 15
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1
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145
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What do you think of this proof for Fisher information?
I want to prove This formula:
The score function is basically the derivative of the maximum likelihood's log, so to get the information I make another derivative of that:
$$ -E[∂/∂θ s(X;θ)] = -E[∂/∂θ ...
- 699
1
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88
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Finding the maximum likelihood solution corresponds to finding the root of a regression function. How?
Given a pair of RVs $z,\theta$ governed by a joint distribution $p(z,\theta)$. Conditional expectation of $z$ given $\theta$ defines a deterministic function (called as regression functions) $f(\theta)...
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Issue with Casella&Berger derivation of EM likelihood equality
In the explanation of the EM (Expectation maximization) algorithm p.328 in the book "Statistical inference" by G. Casella and R. Berger, 2nd edition, they present the following:
$\mathbf{Y} =...
- 387
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2
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Variance of MLE poisson distribution
I am working on problems related to finding MLE from Mathematical Statistics with Applications, 7th Edition - Wackerly. Below is the exercise 9.80 that I'm a bit confused over. My concern is mostly ...
- 337
2
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1
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88
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Bias of MLE scales with $1/N$?
I was reading this paper (link) and it gave me some confusion.
$P(r|\theta)$ is a distribution that generates sample $r$ based on some Poisson distribution, whose mean and variance are defined as some ...
- 281
2
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343
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What does "Expectation with respect to true unknown parameter" mean?
I am trying to study the asymptotic properties of MLE, but I am having trouble understanding an expression that seems to be consistently used in all lecture notes available online (page 93,page 18,...
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3
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1
answer
711
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Expectation for the MLE for a Uniform Discrete Random Variable
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Problem Statement: Suppose that $n$ integers are drawn at random and with replacement from the integers $1,2,\dots,N....
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431
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Information Matrix for Conditional Likelihood
I am studying the MLE theory on my own and I am confused by the difference between the fisher information matrix for the full sample and for one observation, when it comes to conditional likelihood. ...
- 1,389
10
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Why is the observed Fisher information defined as the Hessian of the log-likelihood?
In an MLE setting with probability density function $f(X, \theta)$, the (expected) Fisher information is usually defined as the covariance matrix of the fisher score, i.e.
$$
I(\theta) = E_\theta \...
- 101
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47
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Expected value of unbiased estimator of $\sigma$ in binomial sum
Suppose that $Y_1, Y_2, \dots, Y_r$ are random independent variable such as $Y_i \sim B(m_i, \pi)$, the idea first is to find $\hat{\pi}$ which is the maximum likelihood estimator an use it to find ...
- 121
1
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126
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Unbiased estimator for a parameter from a transformed distribution
I am solving an exercise in which I have to show that a certain estimator is unbiased for a given parameter. However, after a couple lines of computation I got stuck in the following scenario:
$$
\...
- 37
5
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1
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6k
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Difference between the expectation of x bar squared and the expectation of x squared
I am trying to understand the derivation of the expectation of the maximum likelihood (MLE) of variance, however I am confused as to what the difference is between $\bar{x}$ and $x$. Below you find ...
- 341
2
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958
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Why is the expected value of variance different than the expected value of Maximum Likelihood variance?
The expected value of variance is:
The expected value of the maximum likelihood of variance is:
$E\left[\frac{1}{N}\sum_{n=1}^{N}(X-\mu_{ml})^{2})\right] = \frac{N-1}{N}\sigma ^{2} $
Why does ...
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