Questions tagged [estimators]

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

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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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Is it logical to combine cross-validation estimator like RidgeCV with cross_val_score in sklearn?

I was going through solutions for a regression problem competition on Kaggle here. Many solutions for the problem are combining cross-validation estimators like RidgeCV, LassoCV with cross_val_score ...
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Calculating "relative efficiency" of a single estimator in two samples from different processes

I want to assess how well the multilevel AR(1) model performs on not-so-long time series wherein the AR(1) model assumptions are violated. I have three data generating processes (say, $P_1$, $P_2$, ...
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Method of moments estimator for a probability of an event

I need to find the method of moments estimator for $P(pois(\lambda)=0)$. I already worked out the MME $\hat{\lambda}=\bar{X}$ but I'm not sure how to proceed here because I can only see how this ...
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Finding the consistency of an estimator?

Suppose Y1, Y2,...,Yn is a random sample from the exponential pdf, fY(y; λ) = λe^(-λy), y> 0. a. Show that λn = Y1 is not consistent for λ. b. Show that λn = sum of Yi, from i=1 to n, is not ...
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Monte Carlo in R simulation for Efficiency [closed]

An exercise displayed in the image below shows example of finding the efficiency of an estimator. I am trying to replicate this example in R using monte carlo. Y1,Y2,Y3 are random samples of normal ...
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Showing the sample mean estimator is consistent

I need to show that the sample mean estimator $(\sum x_i)/n$ calculated over the first n samples of $x_1,x_2,...$ iid of infinity size is consistent. Now it's a little bit difficult for me to ...
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Regression specification: what if one regressor is function of another

Consider the following regression model $$ Y_i=D_{i}(\alpha_1+\beta_1 X_i)+(1-D_i)(\alpha_2+\beta_2 X_i)+\epsilon_i $$ where $D_i$ is a binary variable. Suppose the researcher has an i.i.d. sample $\{...
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Fisher Information for a Cauchy-distributed Variable (MLE, Variance) [duplicate]

I think, I'm close but I'm still having issues with the following problem: I'm looking at a Cauchy distributed random variable, with: $$ f(x,\gamma,\theta)=\frac{1}{\gamma\pi}\frac{1}{1+(\frac{x-\...
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Will removing a regressor from a model reduce the variance of the remaining regressor

Let's say our full model is a mean centered: $$ y= B_0 + B_1(x_1-\bar x_1) + B_2(x_2-\bar x_2) + e$$ I know $B_0$ works out to be equal to $\bar{y}$, and so $SS_{Reg}(B_0) = 0$ My question is if we ...
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Closed form equations for simple linear regression estimators

I'm learning specifically about different forms of simple linear regression including ordinary least squares, median absolute deviation, and Theil-Sen. I have no background whatsoever in linear ...
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Why when the number of data increase the consistency can’t guarantee that the bias induced by the estimator diminishes

Consistency ensures that the bias induced by the estimator decreases as the number of data examples increases. However, the converse is not true asymptotically, an unbiased estimator does not imply ...
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What is an estimator and how to construct it?

The definition of an estimator "rule that tells how to calculate an estimate " as given here is not clear to me. If I make measurements of some quantity, say age in a group of N people, my ...
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Find posterior distribution and Bayes estimator [duplicate]

Suppose that an observation $x \in (-1,1)$ comes from a sample model with a parameter $\theta$, with density function: $$ f(x\mid\theta) = \begin{cases} \theta\ &\text{if }\ -1 < x < 0\\ ...
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How prove that this estimator is consistent

Is my first post, I hope do well. I have this estimator, from Rao-Blackwell: $t(t-1)/n(n-1)$ where $t= \sum_{i}^n X_i$. And $X_i \sim \text{Bernoulli}$. And don't know how prove the consistency. ...
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Relative efficiency of two estimators for the cardinality of a set from which we sample without replacement

I would like to check my understanding of the relative efficiency for the following estimators for the problem of estimating the cardinality of a set, from which you sample without replacement; i.e. ...
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Maximum likelihood estimator for power law with negative exponent

Background I have data that roughly follows a power law with a negative exponent (up to a point; also, the parameters of the "fit" were just guesstimated by eye as a demonstration): Now I ...
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M-estimator: There is no "of something" in the definition

I see that when talking about estimator, we have "of something", where "something" refers to a fixed parameter. For example, we say that the sample mean is an estimator of the ...
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consistency of maximum likelihood estimator

For population with n size and following density function $$f(y, a)= (1/6a^4)y^3e^{-y/a}$$ For that, I have found the maximum likelihood estimator of a which is $\hat{a}= \bar{y}/4$ I have also shon ...
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Do robust estimators like M-estimator still have higher variance than OLS in presence of non-normal errors and/or outliers?

In my studies I've learned that even with non-normality of the errors, the OLS estimator is still considered BLUE (Best Linear Unbiased Estimator). The texts also suggested using M and L estimators ...
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Why for Least square estimators for Multiple Linear Regression will not be affected after shifting the variable with its mean

Suppose we have $Y = \beta_{0} + \beta_{1}X1 + \beta_{2}X2 + \epsilon$ we have a estimator $\beta$ for this model. Now we substitute $\tilde{Y} = Y - \bar{Y}$( Y - mean of (Y)) and $\tilde{X1} = X1 - \...
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Can you explain LINEAR in BLUE?

I have hard time understanding the LINEAR part. Found something like this: Linear property of OLS estimator means that OLS belongs to that class of estimators, which are linear in Y, the dependent ...
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Estimating $\theta$ based on censored data when $X_i\sim \text{Uniform}(0,\theta)$ with $\theta\ge 1$

Suppose $(X_i)_{1\le i\le n}$ are i.i.d $\text{Uniform}(0,\theta)$ random variables where $\theta \ge 1$. We observe $Y_i=\min(X_i,1)$ instead of $X_i$. I wish to estimate $\theta$ based on the data $(...
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Correlating two matrices $A,B$ with stochastic dependency structure imposed by cross-validation

Consider a labelled data set $$D = \{(x_1, y_1),...,(x_n, y_n)\} $$ on which we want to evaluate a machine learning algorithm using $k$-fold cross validation with $m$ different random seeds. This ...
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Consistency for the estimator in a mixture of objective function

Current we have two discrepancy functions $f_1(x_1,x_2,y_1,y_2)$ and $f_2(x_1,y_1)$. $f_1$ reaches minimum when $x_1=y_1$, $x_2=y_2$; $f_2$ reaches minimum when $x_1=y_1$. We consider an objective ...
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How can I derive OLS predicted error term ^ei as a function of ei?

First of all, I'd like to say that any kind of help would be really helpful, whether it's a hint or a good grad/undergrad book. Right now I'm working with Econometric Analysis of Cross Section and ...
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Why is 50% the best breakdown point for an estimator?

As stated in Wikipedia: Intuitively, we can understand that a breakdown point cannot exceed 50% because if more than half of the observations are contaminated, it is not possible to distinguish ...
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Estimating a statistic by combining two different data sources

Say you want to estimate a statistic $\theta$ and have two data sources. A sample from data source A can be treated as a low-variance, somewhat biased estimate of $\theta$. A sample from data source B ...
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estimate a binomial parameters (n and p) from a distribution sample

I have found this function def find_np(data): that try to estimate p,n out of a binomial distribution sample: ...
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Confidence interval for exponentially distributed estimator

We have an estimator $\hat{\theta}\geq 0$ for $\theta$, with distribution function $P\{\hat{\theta}\leq t \}=1-e^{-t/\theta}$, which we can recognize as the cdf of the exponential distribution. Our ...
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Instrumental variables - OLS - estimation

I have a question regarding the OLS estimation, in the case of an estimation with instrumental variables: We assume the linear model $𝒚= 𝑿\beta+𝒖$ with $Z$ = instrumental variables. Multiplying the ...
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Replace parameter with estimate for confidence interval. Case Beta distribution

I'm trying to get a confidence interval for the mean of a beta distribution $B(\theta,1)$, using $[\hat\theta - z_{1-\alpha/2}\hat\sigma_{\hat\theta};\hat\theta + z_{1-\alpha/2}\hat\sigma_{\hat\theta}]...
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Probability that sample mean of poisson is closer to lambda than sample variance

Let's say you have a random variable $X$ which follows Poisson distribution with parameter $\lambda$. Now, if you have a sample of size $n$, you could estimate $\lambda$ with either the sample mean $\...
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