Questions tagged [estimators]
A rule for calculating an estimate of a given quantity based on observed data [Wikipedia].
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Estimators for capability indices
The capability indices $C_{pk}$ and $P_{pk}$ are defined for a normally distributed random variable $X$ with mean $\mu$ and standard deviation $\sigma$ and specification limits $-\infty <LSL < ...
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Deriving Properties of Estimators (Bias and Variance)
I have the following probability distribution function given by:
\begin{equation}
\label{eq:function}
f(x) = \frac{4a}{x^5} \exp \left[ {- \frac{a}{x^4}} \right] \quad \quad 0 \leq x \leq \infty \...
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Relationship between POLS, FE, and BE estimators in panel data
In some old lecture notes I came across this relationship between all three estimators:
$$
\hat{\beta}_{POLS} = W_{BE} \hat{\beta}_{BE} + W_{FE} \hat{\beta}_{FE},
$$
where the weights $W_{BE}$ and $W_{...
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How to find asymptotically normal estimator if I know probability density function [closed]
I have $X_1, X_2,\ldots,X_n$ be a random sample of size n from a distribution with probability density function:
$$p(x) = \theta^2xe^{-\theta x}I (x > 0).$$
How can I find an asymptotically normal ...
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Show that $T=\sum_{i=1}^n X_i$ is a sufficient statistic for $p$ [duplicate]
I try to use the definition of sufficient statistic to prove that
Suppose that $X_1,\dots, X_n$ is an iid random sample from $X\sim \mathrm{Bernoulli}(p)$. Show that $T=\sum_{i=1}^n X_i$ is a ...
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Compare variance estimators in complex sampling designs by simulations
I need to find the best variance estimator of my parameter $\theta$ using complex sampling data. My survey data with dimension N are drawn with a two-stages stratified sampling.
I thus began from my ...
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1
answer
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Understanding Wilson confidence interval for estimating precision of ML model
Two related questions from statistics noob.
Can someone recommend a good statistics textbook that covers Wilson confidence interval? The reason why I'm looking for Wilson CI specifically is that I am ...
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Let $X_1,\dots, X_n$ be random sample from $Bernoulli(p)$. Which estimator is better?
Let $X_1,\dots, X_n$ be random sample from $Bernoulli(p)$. Compare the risks of the squared loss of two estimators of $p$:
$$
\hat{p}_1=\bar{X}, \, \hat{p}_2=\frac{n\bar{X}+\alpha}{\alpha+\beta+n}
$$...
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Estimating a light field from diffracted images
I am a physicist, and the following problem has arisen in my research:
Given functions $G_j:\mathbb{R}^2\rightarrow\mathbb{R}$ and parameters $\alpha_j\in\mathbb{R}$, for $j=1,2,\dots, N$, find two ...
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Consistent or inconsistent estimator
If $\hat{\theta}_n$ is an estimator for the parameter $\theta$, then the two sufficient conditions to ensure consistency of $\hat{\theta}_n$ are:
Bias($\hat{\theta}_n)\to 0$ and Var$(\hat{\theta}_n)\...
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Are bias and variance used as metrics to evaluate estimators in Bayesian inference?
Consider the parameter $\theta$, which is a deterministic unknown in the frequentist paradigm. Given a random variable $X \sim p_X(x ; \theta)$, consider the estimator $\Theta(X)$ of $\theta$, and the ...
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Kernelization vs pre-defined basis functions: which one is better and why?
I am learning about kernels and how linear models can use them to model nonlinear data. Consider, for example, linear regression for nonlinear function $y(\textbf{x})$. The idea is to project the ...
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In some sense, is linear regression an estimate of an estimate of an estimate?
Consider the problem of estimating a random variable $Y$ using another random variable $X$.
The best estimator of $Y$ by a function of $X$ is the conditional expectation $E[Y|X]$. It minimizes the ...
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Asymptotic property of estimators
I'm studying the asymptotic properties of estimators.
Let $\{ \hat{\theta}_T : T=1,2,3... \}$ be a sequence of estimators of the $p \times1$ vector $\theta \in \Theta $, and $T$ is the sample size. ...
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Minimizing variance of sequence of independent but not identically distributed random variable
I tried to work on the problem
Let $(X_n)$ be a sequence of independent random variables with $E[X_n]=\mu$ and $Var[X_n]=n$ for every $n \in \mathbb{N}$. Find the statistic of the form $\sum_{i=1}^...
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Philosophical insight of Bias Variance Decomposition
As we know that we can perform a Bias Variance decomposition of an Estimator with MSE as loss function and it will look like below:
$$\operatorname{MSE}(\hat{\theta}) = \operatorname{tr}(\operatorname{...
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Extending Minimal sufficient statistics to arbitrary dimension
I am wondering if the following reasoning is correct regarding minimal sufficiency and dimension. Given $X_1,\dots,X_n$ i.i.d. $N(\mu,1)$, we know that the sample mean $S = \bar{X}$ is a minimal ...
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Can a “reverse unbiased” estimator be created?
Suppose we have a parameter $\theta$ that we want to estimate. We sample an observation (random variable) $X$ from a known distribution $D_{X|\theta}$. Then, we can compute an estimator $\hat\theta(X)$...
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Estimator for the propensity for consumption c = C/Y
I've an exercise where it asks to propose an estimator for the propensity for consumption: $c = C/Y$ where $C$ is the consume and $Y$ is the income.
Since the consumption function $C = c_0 + c_1 Y$ is ...
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In OLS, does the uncorrelatedness between regressors and residuals require a constant?
I'm reading this PDF.
It shows how to obtain the OLS estimator and its properties.
It is said that from the normal equations we obtain $X' e = 0$.
Where $X$ is the design matrix and $e$ is the vector ...
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Variance estimation for small sample size
The following variance estimator of a set of data points $x = (x_1, ..., x_N)$
$$
\text{Var}\,(x) = \frac{1}{N-1} \sum_{i=1}^N (x_i - \bar{x})^2
$$
has itself a large variance when $N$ is small (in my ...
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How did we derive the least square estimator using OLS?
How does multiplying a matrix with its transpose equal "minimizing" it?
When calculating the partial derivative, where does the X' come from?
Why setting the value of third equation to 0 is ...
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Should use DWLS for a SEM with binary exogenous predictors and latent variable outcome?
I need to analize a SEM with 4 binary predictors (yes/no), 3 mediators (continous but non-normal variables) with a latent variable that is reflected of 4 ordinal and 1 continuos variable. I have look ...
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How can we compare biases of two estimators with no parametric form?
I was reading in my textbook that the bias of a statistical estimator $\hat{\theta}_n$ can be quantified as $B(\hat{\theta}_n,\theta)=E[\hat{\theta}_n-\theta]$. This expectation seems to be w.r.t. to ...
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MLE estimator in null hypothesis?
Suppose i have some data $D$ from a distribution with a unknown parameter $\mu$. Now suppose i construct an estimator $\hat{\mu}$ based on mle and note what value the estimator takes on the given data ...
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Sampling distribution, bias and variance of cross-validation methods (particularly LOOCV)
(TL;DR version below) If my understanding is correct, bias/variance are measures of goodness of fit of a statistical estimator w.r.t. the sampling distribution. So if I have a statistic $t(X)$ that ...
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Maximum likelihood vs generalized method of moments
I am trying to understand how maximum likelihood (MLE) and generalized method of moments (GMM) are related to each other. In particular, I often see people saying that MLE can be written in terms of ...
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Is there an analogue for standard error of the mean based on mean absolute deviation?
We can estimate the standard error (SE) of the sample mean as the sample standard deviation divided by the square root of the number of samples, cf. https://en.wikipedia.org/wiki/Standard_error#...
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1
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Covariance Between $x_{it}$ and $\alpha_i=\frac{1}{T}\sum_{t=1}^T x_{it}$ in Panel Data
I have unit $\times$ time panel data, $x_{it}$. I have taken the time average for each unit: $$\alpha_i=\frac{1}{T}\sum_{t=1}^T x_{it}.$$ How would you calculate $\mathbf{Cov}(x_{it}, \alpha_i)$?
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Gasser Müller estimator for estimating the derivative $m'(x)$ of a nonparametric regression function
I would like to compare the performance of the Gasser Müller estimator with other estimators for estimating the the derivative $m'(x)$ of the regression function $m(x)$.
Let's say we have the ...
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Is $\sum(x^2 - \bar{x}^2)/(n-1)$ an unbiased estimator of variance?
We know that sample variance $\sum(x- \bar{x})^2/(n-1) = \sum(x^2- 2x\bar{x}+\bar{x}^2)/(n-1)$ is an unbiasedd estimator of variance
Is $\sum(x^2 - \bar{x}^2)/(n-1)$ also an unbiased estimator of ...
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How do we know the true value of a parameter, in order to check estimator properties?
For example, we say that an estimator is unbiased if the expected value of the estimator is the true value of the parameter we're trying to estimate. However, if we already know the true value of the ...
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Estimators for gaussian distribution including label for data point (Bayes classification - ML)
I am struggling to understand how I am supposed to derive two estimators for a standard gaussian distribution in bayes classification. I have a data set $ \chi = [x^t,r^t]_{i=1}^N$, and two classes. I ...
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What is the expression for covariance in the context of Monte-Carlo estimator? [duplicate]
I am trying to calculate the variance:
$$
\langle(\bar{O}-<O>)^2\rangle
$$
of the Monte-Carlo estimator
$$
\bar{O}=\frac{1}{M}\sum_{m=1}^M{O_m}
$$
For uncorrelated samples.
In order to do so, I ...
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Maximum likelihood estimation normal
I am looking for the maximum likelihood estimators of the random sample with normal distribution $N(\mu,\gamma\mu)$. My question is whether from the conventional estimators, $\bar{X} $ and $S^2$, it ...
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Confusing usage of Central Limit Theorem
The CLT defined in Introduction to Mathematical Statistics (Hogg) 8th ed., states that given the samples $\mathbf X\sim\mathcal N(\mu,\sigma) $ with the mean and variance estimator $\bar X,S^2$, the ...
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Minimax estimator for geometric distribution
I'm trying to solve this problem:
Let $X$ be a single sample from Geo($p$) where $p ∈ (0, 1)$. Find a minimax estimator for $p$ under the loss $L(p, δ(x)) = (p−δ(x))^2/
p(1−p)$ .
I'm trying to put ...
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Estimate multivariate normal distribution of a random vector by sampling it using N biased observers
Let be $\mathbf{X}$ $r$-dimentional random vector, following multivariate Gaussian distribution with mean $\mathbf{0}$ and covariance matrix $\mathbf{F}$:
$$
\mathbf{X} \sim G(0, \mathbf{F})
$$
The ...
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1
answer
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Consistency of a simple estimator for $y_i = \beta_1 x_i + u_i$
Let $y_i = \beta_1 x_i + u_i$ for $i=1,2,..,n$. If I define $$\hat \beta_1 = \frac{y_1 + y_n}{x_1 + x_n}$$ then whether my $\hat \beta_1$ will be consistent or not in this setup?
For my estimator to ...
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Variance of an estimator [duplicate]
I have tried to proof the variance of this estimator.
Is this right?
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Unbiased estimators for uniform distribution
I have a given question that I just cannot figure out. From what I can understand, both estimators would be unbiased (since e_1 is the sample mean and e_2 is an unbiased estimator for uniform ...
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Endogenous sample selection vs exogenous sample selection
I am trying to understand how bias differs between exogenous sample selection and endogenous sample selection.
To give a concrete example, I am trying to understand the relationship between income, ...
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Confidence Intervals for Normalized Random Variables
I think I have a pretty simple question about constructing confidence intervals for normalized random variables.
If I have i.i.d random variables $X_1, X_2, X_3, ..., X_n \sim F$ for some distribution ...
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Closeness of two estimators of median under non parametric setup in a large sample situation
Median Regression under non-parametric set-up (Nadaraya Watson Estimate)
Data: $\{(Y_i,X_i):1\le i\le n\}$
Interested in estimating $\phi(x)=\text{median}(Y|X=x).$
Possible estimates are
Minimize the ...
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Show asymptotic consistency of Gauss test
The Gauss test is defined as: $$\varphi_N(X_1,...,X_N) = 1_{\{\sqrt N \frac{|\overline x - \mu_0|}{\sigma} > \phi^{-1}(1-\alpha/2)\}}(x_1,...,x_n)$$
where we reject $H_0: \mu = \mu_0$ when $\...
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What is the joint distribution between the sample mean and sample mediant of rounded normal variables?
I am curious about the relationship between the arithmetic mean and the (generalized) mediant. I took $10^4$ samples (each of size $n=3$) of $\operatorname{Round}(X_i,\text{decimals}=3)$ where $X_i \...
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Bias and variance of estimators - Normal Sample
If we consider the two following estimators $$\hat{\mu_1} = \frac{\bar{X_1}+\bar{X_2}}{2}$$ $$\hat{\mu_2} = \frac{n_1\bar{X_1}+n_2\bar{X_2}}{n_1+n_2}$$ where $X_1, X_2$ are samples from a normal ...
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Aproximate maximum of two multivariate Gaussians with multivariate Gaussian
Given two multivariate Gaussians $G_1(\mathbf{x}), G_2(\mathbf{x})$ (not PDFs!) with the same center at the coordinate origin and different covariance matrix $\mathbf{F}_1, \mathbf{F}_2$, where $\...
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Parameter estimation on exponential distribution from a bounded subset of that distribution
I have a random variable that is exponentially distributed with some $\lambda$. I'm sampling observations from this variable, but I'm limited to observing only those that are less than some maximum ...
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Substitution principle - Sample mean estimator
We can read in Keith Knight - Mathematical Statistics Example 4.22 the following.
EXAMPLE 4.22: Suppose that $\theta(F)=\int_{-\infty}^{\infty} h(x) d F(x)$. Substituting $\widehat{F}$ for $F$, we get
...
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