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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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How do I find bias and variance of estimators of a binomial distribution?

A product-lot arrives in two containers with respectively 300 and 700 units in each container. We examine 30 units in the first container and find that 𝑋1 of them is defective. We check 70 units in ...
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How I can sketch the proof of consistency of only one beta in multiple regression?

Now assume you additionally obtained data on average parental incomes (PI) and the ethnic composition (EC) of the pupils in school. You regress the score on STR PI EC and a constant. State the ...
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Expectation on estimator for Poisson distribution

I'm reading through the textbook "All of Statistics" and one of the problems gives the following estimator for the lambda parameter of the Poisson distribution: $\hat{\lambda} = \frac{\sum_{i=1}^n ...
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Least square estimate for post-stratification sampling

I figured out via the normal linear regression method that Beta0 hat = ybar - Beta1 hat xbar. But I am not sure how to find out the least square estimate for Chat. Is anyone able to help me? Thanks!! ...
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Definition of curvature

Kay (Fundamentals of Statistical Signal Processing) defines the curvature of a log-likelihood function to be the "negative of the second derivative of the logarithm of the likelihood function at its ...
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What is the maximum likelihood estimator for $e^{-\theta}( (P(X_i = 0))$?

Suppose $X_i$ is $iid$ $Poisson(\theta)$ What is the maximum likelihood estimator for $e^{-\theta}(= P(Xi = 0))$? I already found the MLE for the $\theta$. how do you then find the MLE of $e^{-\...
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Can a biased but consistent estimator have a non zero convergent bias?

I understand that an estimator can be biased and yet consistent, and for me intuitivly in these cases the bias converge to zero as n goes to infinity, however can it be the case that the bias won't ...
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Estimator for $\frac{1}{\lambda}$ using $\min_i X_i$ when $X_i$ are i.i.d $\mathsf{Exp}(\lambda)$

Let $X_1,\ldots,X_n$ be i.i.d. $\mathsf{Exp}(\lambda)$ random variables, where $\lambda$ is unknown. Consider $f_{\min}(x) = \min_{i}(X_i)=$ $ n \lambda $ Exp$(n\lambda x)$. I am told that $\hat \...
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Spherical error variance in OLS estimation of AR($p$)

Consider the linear model $\boldsymbol y=\boldsymbol X\beta+\boldsymbol\varepsilon$. One of the assumptions for the OLS estimator is the spherical error variance assumption which states that $\...
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Partitioned regression model: estimator of beta 1

below is an exercise that is really giving me a hard time, I believe that there is a simple way around it but I can not find it: Assume the correct regression model is Y = X$\beta$ + $\epsilon$ for E(...
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Maximum likelihood estimate of Gaussian given rounded observations

Suppose there is a hidden gaussian with mean $\mu$ and variance $\sigma^2$, and that $X_i \sim \mathcal{N}(\mu,\sigma^2)$ where the $X_i$ are i.i.d. If I can only oberve the rounded value of $X_i$, i....
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Invariance property

I am a bit confused regarding what exactly is the invariance property of sufficient estimators, consistent estimators and maximum likelihood estimators. As far as I know, Invariance property of ...
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Existence of Huber M-estimators

I am working on a paper about optimization using the Huber's Loss function, which is defined as: \begin{equation} \psi(x)=\begin{cases} \frac{x^2}{2\gamma},& \text{if } \lvert x\rvert\leq\gamma\\ ...
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Based on the ideas of Parameter Estimation and Fitting Probability Distributions, what stops us from making any function be a PDF(PMF)?

Currently I am doing an introduction to parameter estimation and fitting probability distributions to sets of data. So in a small synopsis what I understand the whole process to be like is the ...
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1answer
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OLS estimator for regression without intercept [duplicate]

Consider a linear regression model: $Y_i = \beta_1 A_i + \beta_2 B_i + u_i$ where all variables are assumed to have mean 0, and $A_{i}$ is distributed independently of both $B_{i}$ and $u_{i}$, but $...
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Distribution for average of multiple binomial proportions

Assume we have a population $N$ and a proportion $p$ of that population with a characteristic of interest. Both $N$ and $p$ are unknown. Furthermore, assume that we have $k$ random samples $(n_i, x_i)$...
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Consistency vs. Asymptotic Efficiency of estimator

I'm thinking about the relationship between an asymptotically efficient estimator and a consistent estimator, and I'd like to make sure that my thinking is correct. An estimator is asymptotically ...
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Asymptotically unbiased estimator vs consistent estimator [duplicate]

I'm wondering if there is a difference between an asymptotically unbiased estimator and a consistent estimator. For asymptotically unbiased estimators, the expected value of the estimator converges ...
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2answers
55 views

Maximum Likelihood Estimator (MLE) for $2 \theta^2 x^{-3}$

I'm having a bit of trouble solving this. $$ f(x_i; \theta) = 2 \theta^2 x_i^{-3}, 0 \le \theta \le x_i \lt \infty $$ I start by finding $f(\textbf{x}; \theta)$: $$ f(\textbf{x}; \theta) = \prod{f(...
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In linear regression, are the noise terms independent of the coefficient estimators?

In the Wikipedia article on the bias-variance tradeoff, the independence of the estimator $\hat f(x)$ and the noise term $\epsilon$ is used in a crucial way in the proof of the decomposition of the ...
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Is this the only way to determine if a parameter can be estimated efficiently?

I am tasked with determining if a particular parameter can be estimated efficiently. Given that an efficient estimator is an unbiased estimator which achieves the Cramer-Rao lower-bound, is the only ...
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Can't find much online about “Linear Regression Estimators” — Looking for help making sense of notes on the topic

I have recently been lectured on how to implement linear regression estimators for a project I have going - I was walked through it works but I couldn't make sense of what was going on. See below for ...
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107 views

How to prove variance of OLS estimator in matrix form?

I am reading Wooldridge's Introductory Econometrics (2000), don't judge me, old version = cheap second hand book, and in the page P94 Theorem 3.2 of Multiple Regression Analysis, it says that: $$ Var(...
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1answer
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Covariance of An Empirical Distribution Function Evaluated at Different Points

The problem is extracted from All of Statistics (Exercise 7.5), Larry Wasserman. I don't have a solution manual to the book so I post here the problem together with my attempted answer: Let $x$ and $...
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1answer
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Is a Kalman filter ever the optimal way to estimate a dynamic value given a full history of measurements?

I'm trying to get some intuition for Kalman filtering, and I conceived this toy example: Say that I have a sensor that tracks a moving 1-dimensional target. Say that the measurements from the sensor ...
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Finding Chernoff bounds maximum estimators

I am currently trying to resolve the following exercise about Chernoff bounds: Let $X_{1}, X_{2}, \dots, X_{n}$ be independent, identically distributed (i.i.d) random variables with distribution $N(0,...
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Distribution of coefficient on the error correction term in ECM and VECM

According to statistic academic literature, the cointegration test on coefficient $\alpha$ of the error term included in ECM or VECM does not follow a standard distribution. My question is: If so, ...
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consistency of an estimator not based on total sample size

How do I show the consistency of an estimator of a parameter, say $\mu$, that is not based on the sample size $n$ but a function of $n_{i}$'s where $\sum_{i=1}^{K}n_{i}=n$ ? Consider for example the ...
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1answer
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Estimator of ratio of central moments

In the context of Control Variates one has to estimate, for example, the following ratios of central moments: $$\frac{\mu_{1,1}}{\mu_{0,2}} \quad \text{and} \quad \frac{\mu_{1,1}^2}{\mu_{0,2}}$$ ...
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Confusion in terminologies for simple linear regression model [closed]

Please go through my draft summary below and let me know if my conventions are correct, comprehensible, and non ambiguous. Simple Linear Regression Model Let given observed sample set be $\{(x_1,...
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Posterior mean estimator with MCMC (Metropolis Hastings Algorithm) - Concrete example

I have a little project for which I have to estimate parameters on a PSF (Point Spread Function = response of the system to a dirac, i.e a star in my case). I have the 6 parameters to estimate : $p=(\...
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142 views

Why do we divide by the degree of freedom?

This might be trivial and vague question, but I still don't understand why when creating test statistics or estimators we always divide by the degree of freedom. Just to give examples of what I'm ...
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Why is it important that estimators are unbiased and consistent?

I am clear on the definition of unbiasedness and consistency. But why are these the criteria we use to judge whether an estimator is a good one? There are other criteria, of course, like the variance ...
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Admissibility does not imply minimax

The answer to minimax estimator explains why minimax does not imply admissibility. The relevant statement is from https://www.stat.berkeley.edu/~yuekai/201b/lec6.pdf which says, minimaxity does not ...
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Existence of Minimum Risk equivariant estimator

$X\sim f(x-\theta)$ . Let $y_i=x_i-x_n$ for $i =1,\cdots,n-1$. Let loss $L(\theta,t)=\rho(t-\theta)$ . Let there exist an equivariant estimator $T_0$ with finite risk. If $\rho$ is convex and ...
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Efficient Estimator from Insufficient Statistic

Suppose that I have a statistic $T(X)$, and I know for sure that it is not sufficient to estimate a parameter $\theta$. Is it still possible to have an estimator $\hat\theta(T(X))$ that is efficient (...
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Heteroskedasticity-consistent standard errors

See https://en.wikipedia.org/wiki/Heteroscedasticity-consistent_standard_errors. Assume the model of interest is the linear regression model. If the errors are heteroskedastic, $\hat{\sigma}^2_i = \...
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Dominating Positive Part James-Stein

Is dominating Positive Part James Stein estimator when estimating the mean of a multivariate normal of dimension 3 with known variance(all equal) an open problem? If not, what is this estimator ...
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Interpretation of MINE Mutual Estimator function evaluated on individual samples

This paper proposes an estimator for MI over two channels with finite samples. The estimator (eq. 10 in the paper) uses an expression obtained from a parametric NN evaluated over a mixture of joint ...
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Consistent unbiased estimator for the location parameter of Cauchy (theta, 1)

Given Cauchy distribution with pdf $p(x) = \frac{1}{\pi ((x - \theta)^2 + 1)}$ how can I find a consistent unbiased estimator for $\theta$? My reasoning so far Tried MLE, but there seems to be no ...
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Dependence of estimator covariance on sample count

Say that $X$ is a set $\{X_1, X_2, ..., X_N\}$ of (non-independent) random variables, and that $\hat{\mu}$ is a set $\{\hat{\mu}_1, \hat{\mu}_2, ..., \hat{\mu}_N\}$ of estimators. Each $\hat{\mu}_i$ ...
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Show that if $E\psi(x-\theta)= 0 $ then $P(X< \theta) \leq p \leq P(X \leq \theta)$

Define $$\psi(x)=\begin{cases} 1-p & x < 0 \\ 0 & x=0 \\ -p & x> 0 \end{cases}$$. I have to show that if $$E\psi(x-\theta)= 0 $$ then $$P(X< \theta) \leq p \leq P(X \leq \theta)$$...
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The definitions of estimator and estimate

This example demonstrates the difference between a theoretical observation and a realized observation. A theoretical observation is a random variable with a probability distribution, while its ...
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Bayes Estimator

Wikipedia has a section on the Bayes Estimator. https://en.wikipedia.org/wiki/Bayes_estimator Isn't Bayes Estimator simply the value of the parameter that minimizes the expected loss of a loss ...
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How to estimate contribution of noise sources

We're trying to estimate the contribution of a device on a performance indicator on the quality of transmission of some signal. The value for performance indicator is assumed to be normally ...
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Estimator for repeated sampling and fitting

Say I have a Normal distribution $\mathcal{N_1}(\mu_1,\sigma_1)$. Now I will sample $N$ samples $X_1$ from this distribution, and use estimators for $\hat\mu_2$ and $\hat\sigma_2$ to fit a new ...
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Robust estimator of a geometric average?

Is the geometric average of a sample the best robust estimator of the geometric average of a population, or is it something else like the median or trimmed mean or even the mode? Would it be correct ...
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1answer
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Trouble understanding some r code

I have this code here (it's out of a book called "An Introduction to Bootstrap Methods with Applications to R") We are working with the estimator: $S_n^2=\frac{\sum_{i=1}^{n}(X_i-X_b)^2}{n}$ where $...
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What is the difference between a true estimation and an estimator?

In a machine learning class, the instructor was explaining Maximum Likelihood Estimation. He mentioned that $\hat{\theta}_{MLE}=argmax_\theta P(D;\theta)$ Where D is the data observed, $\theta$ is ...
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Validating new estimators

Can anyone please tell me whether there it be a problem if I validate an estimator by taking samples from one population. This is how most of the estimators for respondent driven sampling has been ...