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Questions tagged [estimators]

A rule for calculating an estimate of a given quantity based on observed data [Wikipedia].

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Sum of asymptotically independent random variables - Convergence

Let $\theta_N=\frac{1}{N}\sum_{i=1}^N \pi_i\cdot g_i$ where $0<\pi_i<1$ and $0<g_i<1/\pi_i$ such that $\theta_N\overset{N\rightarrow \infty}{\rightarrow}\theta$. If $X_i\sim Ber(\pi_i)$, I ...
Pierfrancesco Alaimo Di Loro's user avatar
3 votes
2 answers
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Must maximum likelihood method be applied on a simple random sample or on a realisation?

I guess my trouble is not a big one but here it is: when one applies maximum likelihood, he considers the realization $(x_1, \dots, x_n)$ of a simple random sample (SRS), leading to ML Estimates. But ...
MysteryGuy's user avatar
5 votes
2 answers
497 views

Asymptotic unbiasedness + asymptotic zero variance = consistency?

Here, Ben shows that an unbiased estimator $\hat\theta$ of a parameter $\theta$ that has an asymptotic variance of zero converges in probability to $\theta$. That is, $\hat\theta$ is a consistent ...
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Is Coefficient of Variation a valid measure of relative efficiency?

I'm wondering if it is always valid to use Coefficient of Variation (CV) to determine relative efficiency of parameter estimators, and to compute statistically equivalent sample sizes based on that ...
feetwet's user avatar
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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 ...
stat_student123's user avatar
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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 ...
Parth's user avatar
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6 votes
1 answer
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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 ...
user1420303's user avatar
1 vote
1 answer
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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 ...
glS's user avatar
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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 ...
CfourPiO's user avatar
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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 ...
Andrea's user avatar
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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 ...
secretrevaler's user avatar
0 votes
1 answer
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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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2 votes
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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 ...
Snildt's user avatar
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comparing estimators for variance, cofusion about biases and mean square errors

I wanted to compare biases and mean square errors for three estimators of variance. I took one unbiased (first) estimator and two biased. ...
romperextremeabuser's user avatar
5 votes
2 answers
124 views

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 ...
N. Virgo's user avatar
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1 vote
2 answers
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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 ...
Taha Rhaouti's user avatar
1 vote
0 answers
119 views

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 ...
reyna's user avatar
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1 answer
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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 ...
reyna's user avatar
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Intuition behind between-group covariance matrix from MANOVA?

Suppose that we have samples from $m$ different $p$-dimensional normal multivariate distributions, where they share a common covariance matrix $\Sigma$ but the mean vectors may be different for each ...
Bergson's user avatar
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Why does not this underlying hypergeometric distribution lead to unbiased estimators?

This example is take from Lippman's "Elements of probability and statistics". Let N be the number of fish in a lake the warden wants to estimate. He catches 100 fish, tags them and releases ...
Tryer's user avatar
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1 answer
108 views

Verifying mean and covariance estimators of a two-dimensional normal distribution

Here I try to verify estimators of the mean and covariance matrix of the two-dimensional normal distribution $N(\mu, A)$ with $\mu=[-2,3]^T$ and $A=\begin{pmatrix} 5 & 11\\ 11 & 25 \end{...
H.Y Duan's user avatar
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9 votes
1 answer
96 views

Adjusted R2 and bias

Consider the population $R^2$: \begin{equation} \rho^2 = 1- \frac{\sigma^{2}_u}{\sigma^{2}_y} \end{equation} This equation describes the proportion of the variation in $y$ in the population explained ...
Dimitru's user avatar
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115 views

What's the difference and relationship between theta, theta star and theta hat?

I understand that $\theta$ is the true distribution parameter (great explanation here). I also know that $\hat\theta$ is an estimator of the true $\theta$ (so for example, MLE is an example of $\hat\...
HeyJude's user avatar
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2 votes
1 answer
75 views

Is an estimator that always have a value of zero is a linear estimator?

Consider a simple linear regression model: $$Y=\beta_0+\beta_1 X +u$$ Here, we can consider an estimator that does not use any data: $$\hat{\beta}_1=0$$ That is, regardless of the observed data, the ...
MinChul Park's user avatar
3 votes
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57 views

What is a measure of hardness-of-approximation by samples?

Suppose there is a large vector $\mathbf{x}$ of real numbers, and I want to estimate a certain aggregate function $f(\mathbf{x})$ by taking a small sample of the population $\mathbf{x}$. I would like ...
Erel Segal-Halevi's user avatar
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Optimality criterion for mean estimators

Assume a sample size of $n>5$, a given variance $\sigma^2 > 0$ and a $\delta \in (2e^{-n/4}, 1/2)$. Proof that there exists a distribution with variance $\sigma^2$ such that for any mean ...
MrLCh's user avatar
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1 vote
1 answer
40 views

For a biased estimator, how does one call the point for which the expected value of the estimator is equal to the observed sample estimate? [closed]

Let $\hat{\theta}$ be a biased estimator whose bias depends on the true value $\theta_0$, such that $E[\hat\theta|\theta_0]= f(\theta_0)\neq \theta_0$. Let $t_{sample}$ be a sample realization of $\...
Matifou's user avatar
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1 vote
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42 views

Showing that the estimator of the log posterior used in stochastic gradient MCMC is unbiased

In the SGLD paper as well as in this paper it is claimed (paraphrasing) that the following estimator: $$\widetilde{U}(\theta) = -\dfrac{|\mathcal{S}|}{|\widetilde{\mathcal{S}}|} \sum_{{x}\in \...
Tan's user avatar
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3 votes
1 answer
107 views

Confidence interval on ratio of estimates for exponential random variables

Given exponential random variable X, the MLE for the scale parameter is $\hat{\beta_x} = \bar{x}$, and the confidence interval for that estimate is: $$\frac{2n\bar{x}}{\chi^2_{\frac{\alpha}{2},2n}} &...
feetwet's user avatar
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1 answer
80 views

What is the variance decomposition method?

For $i = 1, \ldots, m$ and $j = 1, \ldots , n$ we have observations $x_{ij}$. We can assume that $$ x_{ij} = y_{i} + z_{ij}, \qquad y_{i} \sim \mathcal{N}(\mu_{y},\sigma_{y}^{2}), \quad z_{ij} \sim \...
math_space's user avatar
2 votes
1 answer
78 views

Maximum Likelihood Estimation for a Unique Probability Density Function

In the context of estimating parameters for a uniquely distributed set of independent and identically distributed random variables, I am examining the following probability density function $ f(x|\...
Occhima's user avatar
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0 answers
21 views

How to show that the influence function of minimum density power divergence estimator with positive tuning parameter is bounded?

In the linked paper, in the influence function section, the term ${u_{\theta}(y)}{f_{\theta}(y)}^\alpha$ is directly called bounded which i do not get the explanation of? Here $\alpha > 0$ is the ...
Amlan Dey's user avatar
2 votes
1 answer
91 views

Mathematical Step for consistency

Let me state my problem from the beginning: Let $i$ be an index representing countries ($i = {1,2,\ldots,N }$), and $t$ represent time, denoted as available data for country $i$ ($t = {1,2,\ldots,T_i }...
Maximilian's user avatar
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1 answer
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Unable to estimate AR(p) coefficients and $\sigma^2$

I am currently trying to solve this problem pertaining to the Yule-Walker equations: Let $\{X_t\}_{t\in Z}$ be a causal autoregressive process given by $$X_t = \varphi X_{t−2} +W_t$$ with $\{W_t\}_{t\...
Patrick O'Rourke's user avatar
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0 answers
26 views

One off the advantages of the bootstrap is that i don't have to worry about having a good estimator?

I'm learning about the theory of estimators and saw that sometimes the analytical formula of the estimator has to be diferent off the formula for the parameter, for example the standart deviation, and ...
Roger Danilo Figlie's user avatar
1 vote
0 answers
25 views

Large samples property of bayes procedures

I was reading through Wasserman's All of Statistics and I came across this property in the Bayesian statistics chapter: I think I don't really get what is supposed to be the intuition behind it, and ...
DeadKarlMarx's user avatar
3 votes
1 answer
121 views

Is the sample mean an unbiased estimator of population mean in the presence of autocorrelation?

I've seen previous questions here that the sample mean can be considered an unbiased estimator of the population mean. e.g.1, 2. While the examples seem to refer to independent sample points, it seems ...
JMenezes's user avatar
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1 vote
1 answer
66 views

How to estimate how heavy a tail is?

Suppose I have data coming from a single variate distribution. I want to estimate how heavy the tail of the distribution is. For example, if the data comes from the Zipf distribution, I would want the ...
user2316602's user avatar
1 vote
1 answer
108 views

Difference between consistent and unbiased estimator [duplicate]

I have a problem where I have to think of an example to explain a practical example of consistency and unbiased. The example I thought of is the sample mean. Consistency is when the estimator (sample ...
stats_noob's user avatar
0 votes
0 answers
26 views

How to derive the (partial) maximum likelihood estimator for a simple autoregressive model

I am trying to derive two maximum likelihood estimators which I have seen in a statistics book, but I am unable to derive them and would really like some help. It goes like this: Consider the simple ...
Rstrobaek's user avatar
1 vote
1 answer
170 views

Is convergence in probability implied by consistency of an estimator?

Every definition of consistency I see mentions something convergence in probability-like in its explanation. From Wikipedia's definition of consistent estimators: having the property that as the ...
Estimate the estimators's user avatar
0 votes
0 answers
9 views

Variations of Correlation Coefficient of Simple Linear Regression with Estimators [duplicate]

Suppose we are using an Ordinary Least Squares (OLS) estimator of $\alpha_{0}$ and $\alpha_{1}$ for the simple linear regression below: $$ H_{i} = \alpha_{0} + \alpha_{1}X_{i} + \epsilon_{i} $$ How ...
Plesiozaurus's user avatar
4 votes
1 answer
182 views

Is it more correct to say "bias of the standard error of the estimator" or "bias of the standard error of the estimate"

I understand an estimator is a "rule" (e.g., a function, say $g$) that produces an estimate ($\hat\theta$) of an estimand (say, a population parameter, $\theta$). My question: is it ...
moses.rivera100's user avatar
1 vote
1 answer
67 views

Does increasing number of observations lead to the decreasing of Mean Square Error of consistent estimators?

I know that not all weakly consistent estimators exhibit MSE-consistency : https://stats.stackexchange.com/a/610835/397467. Anyway, does increasing the sample size leads to a reduction in their mean ...
whn's user avatar
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0 answers
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How to compute the FGLS estimator for simulated data in Matlab (or any other language really)?

first time posting here so please let me know if I can improve my question. As an exercise, I have to simulate 1000 iterations of a sample of 500 observations from a linear model: $$ y_i = \beta_0 + \...
Bernie's user avatar
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3 votes
1 answer
51 views

Properties of statistical estimators when data is a collection of estimates

Assume I have a statistical estimator $\theta$ that has nice properties (say, unbiased and consistent) when the data $Y=\{y_1,y_2,\dots,y_n\}$ is i.i.d. (possibly with additional assumptions). But now,...
12345's user avatar
  • 213
3 votes
1 answer
132 views

How does Huber compute the $\operatorname{var}(s_n)/E[s_n]^2$ and $\operatorname{var}(d_n)/E[d_n]^2$?

(N.B. I am cross posting this question from math stackexchange since after x days I have still not received any responses.) How does Huber in book 'Robust statistical procedures' in chapter 1 compute ...
peter's user avatar
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0 answers
26 views

How to know errorbars on accuracy to nonlinear fit: $A(1-\cos(x+\phi))$ with poisson noise?

I am trying to write some code that accurately estimates the parameters for the following function: $$ Y = A(1-V \cos(X+\phi)) $$, where this output data is poisson distributed. To do this, I first ...
Steven Sagona's user avatar

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