# 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.
1 vote
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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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1 vote
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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{...
1 vote
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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, 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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### 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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### 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 ...
1 vote
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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: ...
1 vote
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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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