Questions tagged [estimators]

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

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Given that some statistics are estimators, are test-statistics consistent, efficient, complete and unbiased, estimators?

Given that some statistics are estimators, are test-statistics consistent, efficient, complete and unbiased, estimators? Are sufficient statistics consistent, efficient, complete and unbiased, ...
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What is the definition of a non-linear estimator? I heard that ratio of estimators is non-linear

Why don't we consider nonlinear estimators for the parameters of linear regression models? says that LASSO is a non-linear estimator. I think LASSO has a solution via matrix multiplication. I don'...
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+50

Are "Moments" More Robust Then "MLE"?

I am an MBA Student taking courses in Statistics. We are learning about different ways to estimate the parameters (i.e. coefficients) of a Regression Model. Our professor indicated that there are two ...
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What is Bayes estimator of $\theta$ when loss function is $L(\theta,a)=I(|\theta -a|>\delta)$?

Suppose $X$ given $\theta$ has pdf $f(x\mid \theta)=e^{-(x-\theta)}I(x>\theta)$ and there is a standard Cauchy prior on $\theta$. As part of an exercise, I am trying to find a Bayes estimator of $\...
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Is asymptotic unbiasedness different from unbiasedness in practice?

Given some estimator T for a parameter θ, by definition T is unbiased if its bias B(T) is 0. It is asymptotically unbiased if B(T) is not 0, but some value that tends to 0 as n goes to infinity. My ...
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Is the variance of an estimator a random-variable? [closed]

Is the variance of an estimator a random-variable? If so, the mean of the variance and the variance of the variance exist. An estimator of the variance of this variance of an estimator also exists.
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What can be said about the distribution of estimators obtained by using bootstrapping? [duplicate]

A common technique to estimate the uncertainty—for example the variance—in an estimate α (this could the the mean, for example) produced by some estimator applied to a small dataset with n examples, ...
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Theory behind this natural sequential estimator of the mean

Suppose we have i.i.d. real random variables $X_1, \cdots$. We want to estimate the expectation $E[X_1]$ to within some desired error $\epsilon$, but we do not know the variance. We want to use as few ...
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Which metric to use for estimating accuracy of a climate model?

Let's assume I have 3 different climate models for a specific region that project the temperature. I also have the observations of that region for the same time frame(the real temperatures). My ...
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Sample vs Population variance

I understand that sample variance is $$\bar X= \frac{1}{N} \Sigma _i^N X_i$$ whereas population variance (for a continuous variable) is $$E[X]= \int f(X)\cdot X \:dx$$ I do not see how these relate. ...
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Does a Heteroskedasticity and Autocorrelation Consistent Estimator for generalized linear (mixed/non-mixed) models exist?

Does a Heteroskedasticity and Autocorrelation Consistent Estimator for generalized linear models exist? That would make GEEs outdated unless no-free lunch theorem suggests otherwise. I am only aware ...
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Are the classical moments consistently estimated from a single realization drawn from a given PSD?

Given a sequence $\{x_k\}_{k=-N}^{N}$ having power spectral density $S(f)$, we know that that "single realization PSD" $$ \frac{\Delta t^2}{T} \left| \sum_{k=-N}^{N} x_n \exp(-2\pi i f n \...
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Confusion regarding proof that the variance estimator is unbiased for finite population

Going through Sharon L. Lohr's Sampling design book (2nd Edition), I have no issues with the content all the way until it goes into the proof in chapter 2 on SRSWOR that $E[s^2] = S^2$, where $S^2$ is ...
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Rescaling logistic regression coefficients such that variance remains constant

I'm reading "A Modern Maximum-Likelihood Theory for High-dimensional Logistic Regression" by Pragya Sur, and trying to recreate Figure 2, for my own edification. The covariates, $X$, are i.i....
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Nonlinear regression with derivative dependence

I am trying to perform a functional approximation on some experimental data. I have a set of measurements of a d-dimensional state $\mathrm x \in \mathbb{R}^d$, together with velocities $\dot{\mathrm ...
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What does substituting a population total for an estimate mean for the standard error?

I have to calculate the minimum sample size for a proportion estimate required in order to meet a criterium on the standard error of this kind: $$ \text{SE}_P = \sqrt{\frac{P(1-P)}{n}} \sqrt\frac{N-n}...
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Why we do not define the reciprocal variance of the Minimum Variance Unbiased Estimators as the FIsher information?

If I give you data on death rate of rats in China and ask you to estimate the population of Cuba based on that, you'll surely say that the data contains no information about the quantity to be ...
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Is there an analog of Lehman-Scheffe theorem for Bayes/MAP/biased estimators?

In statistics, the Lehmann-Scheffé theorem is a prominent statement, tying together the ideas of completeness, sufficiency, uniqueness, and best unbiased estimation. ${ }^{[1]}$ The theorem states ...
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Unbiasedness properties of Ratio/Proportion-type estimator

I have a ratio estimator, $\hat{a} = n_1/(n_0+n_1)$, where $n_x$ refers to the frequency of $x$-valued data. Note that, $E(n_0)$ and $E(n_1)$ exists and strictly positive. Usually, to show an ...
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How would you justify to a non-statistician why we should use an unbiased estimator instead of a maximum likelihood estimator?

Say we have the maximum likelihood estimator (which is usually biased) and an unbiased estimator and the sample size is small enough that these estimator are substantially different in magnitude. We'...
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Bias of the Variance Estimator in autocorrelated samples

Given samples $X_1, \cdots , X_n$ identically distributed with mean $\mu$ and variance $\sigma^2$, but not "linearly" independent (e.g. in the sense that $\mathrm{Cov}(X_i, X_{i-1}) \neq 0$ ...
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Which fit indices in lavaan are appropriate for WLSMV estimator?

I'm conducting a few CFAs with Likert-type data (7 and 5 categories), and hoping to compare my results with the MLR (and MLM) estimators to those with WLSMV in lavaan. I haven't been able to find any ...
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Parameter estimation of state-space models with hidden variables

I have a time-series analysis problem, that I am having trouble finding a suitable regression technique for. I have a coupled linear three dimensional system \begin{align*} X_{t} & =\left(1+J\...
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Does a linear regression assume that the (unconditional) predictor data is i.i.d?

Say I have a linear, cross sectional relationship - $y_{i}=x_{i}b+e_{i}$. Where $E(e_{i}|X_{j})=0$ for all relevant $i,j$. Given this, one can prove that the OLS estimator is unbiased. However, ...
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Why not use exact probability in 0.632 or 0.632+ method with small sample size?

The .632 estimator (and extensions like .632+) developed by Bradley Efron are founded on the following premise. Suppose we have a data set with $n$ observations, and we draw $B$ nonparametric ...
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"Median" version of L-moments

The L-moments are useful as robust summary statistics for various probability distributions, similar to the moments but only requiring the mean of the distribution to exist. Each L-moment is a linear ...
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What does asymptotic efficiency mean in statistic?

I reads some comparison articles, and always find " asymptotic efficiency", "asympototically less efficient", and "asympotoically normal". I really confused about the ...
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Estimating subtotals from a simple random sample

Suppose, for illustration, that we have a population of $N=10$ enterprises $E_1,\cdots,E_{10}$. We extract a sample of $n=4$ enterprises by a simple random sampling method. The sampled enterprises ...
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Is collinearity really not a problem for GLM?

I have read that collinearity is not a problem for GLM. Is it really? I here estimate two models. The dependent variable is default, a dummy equals to 1 if someone ...
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When can we get unbiased estimate given biased data?

There was a recent "hot take" tweet by Andrej Karpathy (without any comment or clarification from the author): real-world data distribution is ~N(0,1) good dataset is ~U(-2,2) It provoked ...
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Do confidence intervals make sense for win rates in sport?

Imagine we have 2 teams play 10 matches against each other with team A winning 6 of them I.e. 60%. In this setting do confidence intervals for the probability of winning make sense? On one hand I ...
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Efficient ways to measure the degree of independence of a moderately large number of variables

I have a process that generates values for variables $x_{1}, x_{2}, \dotsc, x_{n}$ where $n \approx 40$, and the value of each $x_{i}$ lies between $0$ and $1$. The process generates these in batches ...
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Distribution estimation from interval times

In Formula 1 races, an interval time is the lag behind the leader at a split. If first place completes the first lap at 1:30, second place completes the first lap at 1:32, and third place completes ...
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How does perfect multicollinearity affect $R^2$ and $R_{\text{adj}}^2$?

I'd like to know how does perfect collinearity affect measures of fit (R squared and R squared adjusted). A mathematical approach is not necessary, just the general intuition is fine.
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OLS estimator question: using a subset versus using a dummy-interacted variables

Suppose that we are interested in the following model: $$y_i=\beta_1+\beta_2x_{i2}+\beta_3x_{i3}+u_i$$ Here, there is a dummy variable $d_i$. I am wondering whether the following estimators are ...
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An estimator $\hat f$ such that $\hat f(x) : =y_{i^*}$ where $i^* := \operatorname{argmin}_i |x-x_i|$

I'm think of such an estimator $\hat f$ defined as below. It is inspired from the continuity of $f$. Let $f:\mathbb R \to \mathbb R$ be continuous and $(X_0, y_0)$ a square-integrable random vector ...
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Estimating conditional mutual informations from 2D histograms

I have binned marginal and joint distributions of two event features X and Y, i.e. p(X), p(Y) and p(X,Y) where the marginal distributions in X and Y are obtained by summing p(X,Y) over the bins of the ...
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Fisher matrix for a discrete distribution

Let $\mathbf{X} = \{X_1, \ldots, X_n\}$ be a sample of i.i.d. variables following a discrete distribution with parameters $\mathbf{p}^T = (p_1, p_2, p_3)$. How can I find the Fisher information matrix ...
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Construction of statistics of a discrete distribution

I have the following problem: we consider an i.i.d sample $\mathbf{X} = (X_1,...,X_n)$ of the discrete set $\{1,...,N\}$. An agent has to infer the probability distribution of $X_i$. I wanted to use ...
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In estimating $X + Y$, is it helpful if I know random variables $X$ and $Y$ are identically and independently distributed?

Suppose I have $$X \sim Dist_1$$ $$Y \sim Dist_2$$ and I want to estimate $X + Y$. I can sample from $Dist_1$ and $Dist_2$ and generate samples for $X + Y$. So far so good. Now suppose I discover that ...
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sufficient statistics for bernoulli distribution

Let Y1, . . . , Yn be a random sample of size n where each Yi ~ Bernoulli(p), and let Y = $\sum$ Yi for i = 1, . . . , n. The estimator is W= (Y+1)/(n+2) Is the estimator a sufficient statistics for ...
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Parameter estimator and its variance estimator covary

In classic linear regression, estimators of the coefficients of the mean model and the estimator for the residual variance are uncorrelated. However, what to do when this is not the case, for instance ...
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Robust distance weighted mean

Given a data sample $\{x_i\}_1^n$, instead of hard omitting outliers by e.g. trimming, one can form a weighted average where we soft penalize observations out in the tails. \begin{align} \mu = \frac{...
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Statistical Inference: Definition of contrast function

Reading a paper recently regarding results on parameter estimation and I came across the terminology "contrast function" which was a function constructed out of a sample. If I compare it to ...
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Inverse transform sampling : comparing bias, variance and mse for an estimator

Starting from the PDF of the Pareto distribution, \begin{equation} f_{\theta_1, \theta_2}(x) = \begin{cases} \frac{\theta_1 \theta_2^{\theta_1}}{x^{\theta_1 + 1}}, &\quad x \geq \theta_2 \...
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What order of power mean best estimates the median of a gamma distribution?

Suppose we have a gamma-distributed random variable $X$ whose shape/scale parameters are known to be $\alpha$ and $\beta$. What order $p$ for the sample power mean $\hat M_p[X]$ minimizes $$ (\mathcal{...
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Which type of quantiles are safest to report in R?

With this topic in mind: If there is no censoring, can be the naive 3rd quantile different from the one calculated with from the Kaplan-Meier? I'm wondering which one is the safest option. Of course I ...
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Is it possible that the correlation between $\hat{b}$ and $\hat{c}$ can be negative multiple linear regression? [duplicate]

Given the following linear regression model as following, with two explanatory variables $x_1$ and $x_2$ and response $y$ $$y_i=a+bx_{i1}+cx_{i2}+\epsilon_{i}$$ We say that $\hat{a}, \hat{b}, \hat{c}$ ...
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What is this type of data called?

An event occurs once per period, such as once per year. Time is measured in discrete units, such as days of the year. Let $A_y$ be the day in year $y$ on which this event occurs. However, we do not ...
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On the naming of two different median estimators

Assume that $X \sim \mathcal{E}(\lambda)$ is, for example, exponential with $\lambda > 0$. Given a data sample $X_1, \ldots, X_n$, assume that I want to estimate the median of $X$. Consider these ...
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