Questions tagged [estimators]

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

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56 views

Rao-Blackwell and unbiased estimators of zero

In Casella & Berger we are trying to prove that any estimator $\phi$ based on a complete sufficient statistic T is the unique best unbiased estimator of its expectation. However, in the preceding ...
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Bootstrapping and Estimators

I have a textbook that states something to the effect of: The mean of a bootstrap distribution is not an accurate approximation of of the mean of the sampling distribution. But, the spread of the ...
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Finding the MLE of Uniform distribution [duplicate]

Let $x_{1} = 2.4$ , $x_{2} = 9.2$ , $x_{3} = 5.2$ , $x_{4} = 4.1$ , $x_{5} = 2.1$, $x_{6} = 3.1$ be the observed values of a random variable of size 6 from the uniform distribution with parameters $(\...
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Consistency of Posterior Distribution [closed]

I'm reading the book $\textit{Bayesian Nonparametrics}$ and on pg 17 it tackles the consistency of the following posterior distribution $X_{1}, X_{2},..., X_{n}\sim Bern(\theta)$ $\theta \sim Beta(a,b)...
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Mean and covariance estimators for indirect (transformed) observations of Gaussian

Suppose we have a multivariate Gaussian random variable $\mathbf{x}\sim \mathcal{N}(\mathbf{\mu}, \mathbf{\Sigma})$, and $n$ measurements $\mathbf{y}_1 = H_1 \mathbf{x}_1, \mathbf{y}_2 = H_2 \mathbf{x}...
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Coming up with a better estimator for this quantity

I have data from the following generative process: $$ Z \sim F(z)\\ p = g(Z)\\ X \sim \text{Bernoulli}(p) $$ Where $F(z)$ is an unknown distribution on $[-10^5, 10^5] \cap \mathbb{Z}$, and $g$ is an ...
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Method of moments estimate of Pareto Distribution

The Pareto distribution has the following $cumulative \ distribution \ function$ : $$F(x;\alpha ,\Theta ) = \left\{\begin{matrix} 1 - (\frac{\alpha}{x})^{\theta}\ \ if \ \alpha \leq x\ & \\ 0 \ ...
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Panel Data Short-Run & Long-Run estimators

Hello kind folk of crossvalidated, It would seem that I need your help once again. I have some data in panel format, which I really shouldn't get too specific about. The aim is to find the short and ...
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How many samples does one need to perform polynomial regression of degree $m$?

Suppose $(X_i, Y_i)$, $i = 1,\dots, n$ are random variables such that $$X_i\sim N(0,1)$$ $$Y_i = f(X_i) + \epsilon_i$$ where the $\epsilon_i$ are i.i.d. standard Gaussian and $f(x)=\sum_{k = 0}^\infty ...
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Meaning of Invariance of Maximum Likelihood Estimator

In Casella-Berger, the invariance of MLE is defined as: Assuming that $\hat{\theta}$ is MLE of $\theta$, then for any function $\tau$, $\tau(\hat{\theta})$ is MLE of $\tau(\theta)$. In the case of a ...
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The law of large numbers when using the midrange estimator for expectation

I have a growing sequence $A_1 \subseteq A_2 \cdots$ of finite sets that contain samples of $X \sim U([a, b])$. From this sequence I construct a sequence of estimates of $\mathbb{E}[X]$ using the ...
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Should a Test Statistic Consist of a Consistent Estimator for the Parameter of Interest?

Suppose that we want to test the following hypothesis: $$H_0: \theta \in \Theta_0\quad vs \quad H_1: \theta\in \Theta_0^c.$$ Suppose that our test statistic is $T_n$. Then, should the test statistic $...
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NN for resource management of an VM

Are there any papers/projects that deal with neural networks learning/adaptation for resource management (learning of system behavior and resource adaptation such as memory, CPU for an VM)? e.g. some ...
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Given a mean, what is the range of variance values that make for possible Beta distribution parameters

The beta distribution can have its parameter estimated via method of moments, which I will be doing. $$\hat\alpha = \bigg(\dfrac{\bar x (1-\bar x)}{var(X)} - 1\bigg)\bar{x}\\ \hat\beta = \bigg(\dfrac{\...
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Testing bias and consistency for a parameter given variance less than infinity

I proceeded to find the expectation of the estimator to check for bias. Since Therefore and hence biased. An estimator is consistent if MSE tends to 0 as n tends to infinity but I do not know how ...
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Maximum likelihood estimate for multivariate sum of normal distributions

For each $j = 1,\dots,N$, let $\mu_j \in \mathbb{R}^N$ denote a known column vector, $\Sigma_j \in \mathbb{R}^{N\times N}$ a known covariance matrix, and $\theta_j \in \mathbb{R}$ an unknown parameter,...
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Problem in Reproducing the MLEs for a Given Distribution

I am testing a code I am using in R to compute estimates for the MLEs of the parameters for a given distribution. As an example, to check if the code works, I have chosen the paper A New Two-parameter ...
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Estimating conditional probability when events are sampled

Suppose I have many people who eat different fruits (apples, oranges, bananas &c): ...
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Are loss functions only used to evaluate estimators?

Given the likelihood $p(x;\theta)$, we want to construct an estimator $\hat\theta(X)$ that takes in the observation $x$ and returns an estimate of $\theta$. There are many different ways to evaluate ...
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Definition of the bias of an estimator

I'm quite confused about the definition of the bias of an estimator. Suppose we have unknown distribution $P(x, \theta)$, and construct the estimator $\hat{\theta}$ that maps the observed data sample ...
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How to calculate Hodges-Lehmann estimator of slope in rank regression?

Suppose we have $n$ paired observations $(x_1,y_1),(x_2,y_2),\ldots,(x_n,y_n)$, where $y$ is the response variable and $x$ is the covariate. Consider a simple linear regression model $$y_i=\alpha+\...
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Estimator of failure by cause j at time t

I am reading Dirk Moore's Applied Survival Analysis Using R page 124. Let $S(t)$ be cumulative survival curve of population facing cause of death due to a set of cause $\{1,2,\dots,n\}$ where $1,2,\...
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Estimating the $\chi^2$-divergence with Monte Carlo: which distribution to sample from?

Notation: let the $\chi^2$-divergence between $p, q$ be defined as $$\chi^2 (p||q) := \int \left ( \frac{p(x)}{q(x)} \right )^2 q(x)\mathrm{d}x -1 = \int \frac{p(x)}{q(x)} p(x)\mathrm{d}x - 1. $$ ...
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Types of consistency: practical difference

Consistency is usually a desired property for an estimator. We have the definition of consistency for an estimator $T_n$ for $\theta$, stating that it converges in probability to $\theta$, and the ...
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What low bias has to do with "good fit"

Look at the bias-variance decomposition below: In pratice we often consider low bias as good fit to train data but i dont understand the why this by bias-variance decomposition.Why if i got "...
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Is it wrong to say that a Riemann sum is an unbiased estimate of an integral?

Would it be wrong to say that a Riemann sum approximation of an integral \begin{align} \int_a^b f(t) \mathrm{d}t \approx \sum_{k=1}^{n_\text{samples}} f(t^{\ast}_k)\Delta t, \end{align} where $\...
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Estimating failure rate in a changing population of identical widgets

We wish to estimate the failure rate of a changing population of indistinguishable widgets. Starting with zero widgets, at the start of week $w$ we observe the number of new widgets $n_w$ added to ...
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Bias of MLE scales with $1/N$?

I was reading this paper (link) and it gave me some confusion. $P(r|\theta)$ is a distribution that generates sample $r$ based on some Poisson distribution, whose mean and variance are defined as some ...
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Finding the Maximum Likelihood Estimator for a random variable of mixed type

I am having issues solving the follow problem from my textbook: Suppose we have $x_1,...,x_n$ i.d.d. data from a r.v. $X$ with unknown distribution function $F_\theta$, for $\theta = (\theta_1,\...
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When is subsetting survey data prior to analysis NOT a problem?

Survey researchers are typically advised to not subset their data prior to analysis because it will produce incorrect variance estimates. My understanding of the inferential issue is that removing ...
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James-Stein-style estimator when we place greater importance on some components

The James-Stein estimator allows us to get a better overall estimate of a mean vector (length $\ge 3$) than we would be able to get by estimating the components independently. My intuition is that, ...
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Is the Hodges-Lehmann estimator 'optimal' for estimating the location parameter of Logistic distribution?

Is the Hodges-Lehmann estimator $\hat\theta_{HL}=\operatorname{median}\limits_{1\le i\le j\le n}\left\{\frac{X_i+X_j}{2}\right\}$ in some sense 'optimal' for estimating the location parameter $\theta$ ...
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Where does the denominator vanish to in the MAP derivation?

According to MAP estimator: $$\hat\theta_\text{MAP}=\arg\max_\theta P(\theta|D) = \arg\max_\theta \frac{P(D|\theta)P(\theta)}{P(D)}=\arg\max_\theta {P(D|\theta)P(\theta)} $$ The denominator $P(D)$ ...
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Demonstration: Integral (discrete summing) of $C_\ell$ has a better variance than one single $C_\ell$ [closed]

I am faced to a issue to prove the gain in term of variance (that is to say a variance smaller) that we get by computing the integral of $\hat{C}_\ell$ over $\ell$ (actually a discrete summing since I ...
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Is there an estimator for the existence of Lyapunov-motivated stability?

Preface: This question is now asking about dynamical stability in a particular sense, and whether its existence can be inferred from data. It is motivated by commentary below the question "Is ...
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Reconciling the concept of Total Mean Squared Error

I'm trying to understand the idea of Total Mean Squared Error in a text that I am working through and wanted to verify if my understanding of it is correct. The following passage comes from Applied ...
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Are there guidelines to the 'Alphabet Soup' of Statistics? [closed]

This is a somewhat 'meta-statistical' question that pertains to the discipline of Statistics, rather than to Statistics itself. I will not be accepting any answers due to the subject being open-ended. ...
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Variance of an estimator

How does one obtain the variance of the estimator phat2?
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Bias of standard errors as estimates

I'm reading on the standard errors used in various hypothesis tests. For example, in tests of one population proportion, we use $\sqrt{p(1-p)/n}$. For comparison of two population proportions, we use $...
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Unexpected Zero Variance for an Unbiased Estimator: Is the Estimator Consistent?

$\newcommand{\szdb}[1]{\!\left[#1\right]}\newcommand{\szdp}[1]{\!\left(#1\right)}$ Problem Statement: Let $Y_1, Y_2,\dots,Y_n$ denote a random sample from the probability density function $$f(y)= \...
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63 views

Hypothesis testing for detecting signal in Gaussian noise

I have the following two hypotheses: $\hspace{5cm}\mathcal{H}_0: y=w\\\hspace{5cm}\mathcal{H}_1: y=\sum_{i=1}^{N}h_ix_i+w$ Here $w\sim \mathcal{N}(0,1)$ represents Gaussian noise. $x_i \sim Bern(p), \...
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Find Bayesian estimator for $e^{\theta}$

Given $\{Y_i\}_n\sim U(\theta-1,\theta+1)$ and prior distribution $\theta\sim U(a,b),1\leq a<b$ is the posterior distribution conjugate? Find the absolute error estimator for $e^{\theta}$ and ...
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Delta Method around zero is a N(0, 0)

I have this problem: $\sqrt N \hat{\theta} \sim N(0, V)$ where $E(\hat{\theta}) = \theta_{0} = 0$. I must find the asymthotic distribution of $\frac{N}{V}\hat{\theta}^{2}$ but if I use the Delta ...
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64 views

Is a constant ever inadmissible?

For now, assume square loss. Let's estimate some parameter $\theta$, such as $\theta = \mu$ in $N(\mu, 1)$. Is there ever a case where there is no such $c$ to make $\hat{\theta} = c$ an admissible ...
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112 views

Deriving the risk of the Hodges-Le Cam estimator under squared-error loss

In order to better understand the behaviour of the Hodges-Le Cam estimator, $\tilde{\theta}_n$, I am trying to derive an expression for the risk $R_n(\tilde{\theta}_n, \theta)$ under squared error ...
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1answer
140 views

SEM : WLSMV or WLS?

I am conducting a SEM in R using lavaan and I am a bit lost regarding which estimator I should use. I have a model like the following : The variables a,b,c,h are binary categorical and the rest are ...
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Why do I get different values for MCSE and SD while I have good estimates in a simulation?

I am a newbie to this concept so I am not sure what exactly is causing this: I am doing a simulation study: 300 samples each which the size of 13000 to compare two estimators for the mean of the ...
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Deriving the limiting distribution of the Hodges-Le Cam estimator in Bickel and Doksum (2015)

I am trying to better understand the Hodges-Le Cam estimator, and am having difficulty rendering explicit some of the asymptotic arguments in the derivation of the estimator's limiting distribution. I ...
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Simple linear regression without intercept - Expected Value and Variance Estimator (Slope) [duplicate]

I'm solving an exercise while studying for exam, I have been asked to find the estimator of simple linear model without intercept estimator, its expected value and variance. I got for the Estimator B1 ...
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How can I find the best common divisor from an error-bound dataset?

I am currently trying to evaluate the best value for the charge of an electron I can get from a dataset of 38 trials of Millikan's experiment. In order to realize this, I need to find a common divisor ...

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