Questions tagged [estimators]
A rule for calculating an estimate of a given quantity based on observed data [Wikipedia].
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Using Rao-Blackwell to improve the estimator of P(X/Y < t)
X and Y are independent N (0, 1) random variables, we want to approximate P (X/Y ≤ t), for a fixed number t.
The first part of the problem was to describe a naive Monte Carlo estimate. I described ...
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What is the difference between unbiasedness, consistency and efficiency of estimators? How are these interrelated among themselves? [duplicate]
!Efficiency(https://stackoverflow.com/20240427_193105.jpg). Given snapshot of the book states that among the class of consistent estimators, in general, more than one consistent estimator of a ...
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Terminology clarification about sample moments
According to MathWorld (link): "The sample raw moments are unbiased estimators of the population raw moments".
While in Wikipedia (link) it is said:
...the $k$-th raw moment of a population ...
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Why can we get better asymptotic global estimators even for IID random variables?
Let $X_1,...,X_N$ be IID random variables sampled from a parametrised distribution $p_\theta$, and suppose my goal is to retrieve $\theta$ from these samples.
We know that the MLE provides an ...
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Standard practice to show Biased CRBs
I have a problem with four-parameter estimation. I have derived the variances for the estimated parameters using Monte Carlo simulations (numerical ones) and theoretical ones using the inverse of the ...
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What is the distribution of the unbiased estimator of variance for normally distributed variables?
I must be making some mistake in my derivation of the distribution of the unbiased variance estimator for i.i.d. $X_{i} \sim \mathcal{N}\left(\mu, \sigma^{2}\right)$.
We have $\bar{X} =\frac{1}{n}\sum\...
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Demonstrating $SU=U(\sigma^2 I+D^2)$ as a Sufficient Condition in Maximum Likelihood Estimation
I am working on an exercise related to maximum likelihood estimation (in the context of principal component analysis) for the distribution
$$p(x) = Gauss(b, WW^T+\sigma^2I)$$
In particular, I want to ...
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Degrees of freedom for estimation
In the context of estimators, why is it that in general dividing by the degrees of freedom(instead of the sample size) leads to unbiasedness? I see the value in substituting degrees of freedom for ...
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Assumptions needed for consistency of plug-in estimator
Assume $X,Z$ are random variables and let $x_0$ be a fixed number. I want to estimate $A =\mathbb{E}_{X,Z}[\frac{X}{P(X=x_0|Z)}]$.
If $P(X=x_0|Z=z)$ is known for all $z$ we can apply the LLN and ...
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When are mean and variance estimates uncorrelated or independent
I know that in the case of the normal distribution, the MLE estimates of the mean and the variance are independent. My impression is that this is a rare property for a distribution to have. Are there ...
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Sufficient conditions for asymptotic efficiency of MLE
Maximum-likelihood estimators are, according to Wikipedia, asymptotically efficient, that is they achieve the Cramér-Rao bound when sample size tends to infinity. But this seems to require some ...
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Is there a good review on complete class theorems?
I'm trying to get an overview of the various results called "complete class theorems" and their relatives, especially the ones that say things along the lines of "every admissible ...
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Covariance of Best Linear Unbiased Estimators and arbitrary LUE
I'm working on a problem involving two linear unbiased estimators $T$ and $T'$ of a parameter $\theta$, defined from a sample $\{X_1, \dots, X_n\}$ with mean $\theta$ and finite variance. I aim to ...
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Distribution of $F_n^{-1}(3/4)-F_n^{-1}(1/4)$ [closed]
Given $X_1,X_2,...X_n\overset{\text{iid}}{\sim}F$, find the distribution of the sample inter quartile range, $F_n^{-1}(3/4)-F_n^{-1}(1/4)$ in terms of $F$ where, $F_n$ is the emperical distribution ...
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Probability mass function of sample median (Bootstrap)
Consider a sample $X_1,X_2,...X_n\overset{\text{iid}}{\sim}F$. Let $T_n=F_n^{-1}(1/2)$ be the sample median where, $F^{-1}(x)=\inf\{t:F(t)\ge x\}$ and $F_n(y)=\frac{1}{n}\sum_{i=1}^n\mathbb{I}(X_i\le ...
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