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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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Can any unbiased estimator be changed into a consistent estimator when estimating functions of the mean [closed]

For an i.i.d sequence of Random Variables $X_1, \dots, X_n$, each with mean $\mu = \mathbb E[X]$, the goal is to estimate some continuous function $f$ evaluated at the mean, $f[\mathbb E[X]]$. If ...
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39 views

Kaplan-Meier method or estimator?

This is a really basic question. I say Kaplan-Meier estimator and always tell people to say estimator and not method or methodology. However, I've seen a few places where statisticians also say Kaplan-...
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25 views

Hopefully a quick semantics question (Maximum Likelihood Estimator)

I was working through a portion of this paper, when I came across something that seemed odd to me. In Appendix E (pg24), equation (E4), the following line pops up: $\widehat{D} = \frac{\widehat{\...
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“Appropriate conditions” for method of moments estimator to exist, be consistent, and asymptotically normal?

My statistics text has the following theorem, and alludes to "appropriate conditions on the model", but never specifies what those conditions are. What conditions are necessary? Let $\hat{\theta}_n$...
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Sample mean lognormal variables

Suppose you've got $x_1, ..., x_n$ independant realisation drawn from a $LogNormal(\mu, \sigma^2)$. Could someone explain me why $exp(\mu + 0.5*\sigma^2)$ $\neq$ $\frac{1}{n}(x_1 + ... + x_n)$ ? Here ...
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How does the TraMiner Package Calculate Standard Error Using Weighted Data?

The TraMiner Package includes an option to include sampling weights in the analysis. However, I haven't found any discussion in the package documentation (or associated user manual) of how standard ...
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15 views

Corroborating a differnce in differences identification strategy

I read in Mostly harmless econometrics that a good way of testing whether a difference in differences is a good identification strategy is running this equation: where the first sums are post-...
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Rao-Blackwell for Minimum-Variance Unbiased Estimator

Let $X$ be an observation from a distribution with probability mass function:$f(x;\theta) = \left(\frac{\theta}{2}\right)^{|x|}(1-\theta)^{1-|x|}I_{\{-1,0,1\}}(x), \, \theta \in (0,1).$ Use Rao-...
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Within estimator and between estimator?

I have understood that the within estimator is for fixed effects model. Can I say that the between estimator can only be used for a random effects model? (in reference to panel data)
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156 views

Sample mean is always an optimal estimator of the mean?

Suppose we have $T_i,i=1..n$ i.i.d. with unknown distribution and we want to estimate $E[T]$. Note that in this setting we are not estimating E[T] as a parameter of a parameter-dependent family of ...
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Standard error of sample variance

We know that an unbiased estimator of the variance is: $$ \hat{\sigma}^2_{unbiased} = \frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2$$ I was wondering, does it have the smallest possible standard error? ...
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Normal distributed random sample: find the least variance from the set of all unbiased estimators of $\theta$

Let $X_{1},X_{2},\ldots,X_{n}$ be a random sample from $X\sim\mathcal{N}(0,\sigma^{2})$. (a) Find the least variance from the set of all unbiased estimators of $\sigma^{2}$. (b) Find a sufficient ...
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Parameter Estimation using hazard function

Following the notation in this paper [ref], assume we have a random sample of $x_1, \dots, x_n$ of a distribution with PDF $f(x)=f(x,\theta)$ and CDF $F(X)=F(x,\theta)$ and we wish to estimate $\...
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Mean Squared Error as quantifier of the Bias-Variance tradeoff

I have acquired the impression that many of the people doing statistical work, will prefer a biased estimator $\hat b$ to an unbiased one $\hat \beta$, if the former has lower Mean Squared Error. This ...
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Manipulation of asymptotic bounds for distance between estimators

Suppose I know some asymptotic bounds: $$\mathbb{E}(|D(a,\hat{a})|) \lesssim O(n^{-1/2}),$$ where $D$ is some distance between probability measures, and $a$ is a probability measure while $\hat{a}$ ...
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Prove that bias of MLE for Weibull process

In the case of fixed observation interval $[0,T]$ of a Weibull process, I learned that the MLE of the shape parameter $\beta$ and MLE of the scale parameter $\alpha$ are as follow: $$\frac{1}{\hat{\...
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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 $X_1, X_2,...,X_n$ is a random sample from a $\text{Poisson} (\theta)$ distribution with probability mass function: $$P(X=x)=\frac{\theta^ {x}e^{-\theta}}{x!}, x=1,2,...; 0<\theta$$ What ...
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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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2answers
42 views

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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125 views

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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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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92 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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227 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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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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47 views

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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52 views

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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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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236 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 ...