"A rule, method, or criterion for arriving at an estimate of the value of a parameter."

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3answers
24 views

How to Prove Unbiased Estimator

I'm unsure of how to convince myself that $$\hat{\beta} = \frac{\sum X_i Y_i}{\sum X_i^2}$$ is an unbiased estimator when the regression model $$Y_i = \beta X_i + \epsilon_i$$ follows basic OLS ...
-1
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1answer
20 views

Difference between sample mean and sample median in estimator [on hold]

I need the main difference between sample mean and sample median in estimator
4
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1answer
54 views

“Consistent estimator” or “consistent estimate”?

Question: Are both expressions "consistent estimator" and "consistent estimate" meaningful? The quote below is intended to be illustrative; however, I am interested in the question above in a ...
1
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0answers
21 views

robust estimator when $\mu/\sigma$ is constant

I have a large set of measurements consisting each of about 100 points, (normally distributed), with up to 20% outliers. The outliers are all shifted towards positive values. From the physics, I know ...
0
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0answers
21 views

Minimum mean squared error linear combination of random variables

Consider the following objective function: $$ \mathbb{E}((Y-X\beta)^2)\rightarrow \min_\beta $$ where $Y$ and $X$ are (generally not independent) random variables and $\beta$ is a constant. That is, ...
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2answers
45 views

Estimating unconditional variance in time series

Consider a time series process with a well-defined, finite unconditional variance. Given a realization of the process (a time series) and a model for it, there are at least two ways of estimating the ...
1
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0answers
32 views

When to assume that $\Pr(X=x, Y=y) = |\mathcal{Y}|^{-1} \Pr(X=x|Y=y)$?

Background Let $\mathcal{X} = \{x_1, x_2, \ldots, x_n\}$ be a set of samples, each that corresponds to a label in the set of labels $\mathcal{Y}$. Ideally, our objective is to find the joint ...
4
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1answer
36 views

Cramér-Rao Lower Bound for Exponential Families

I am having a problem with applying the Cramér-Rao inequality to identify the lower bound for the variance of an unbiased estimator and hoped that you guys could help me. The problem is the following: ...
1
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1answer
42 views

How to get the maximum likelihood estimator of $U(\theta,\theta +1)$?

I know how to find the MLE for $U(0,\theta)$ but I am in trouble with this one. let $X_1,\dots,X_n$ be a random sample from $U(\theta,\theta +1)$. Consider the following three estimators for ...
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0answers
27 views

Show that weighted least squares estimator for a specific model is not consistent

Here is the background for this problem: $\qquad\qquad\qquad$ $Y_{1},...,Y_{n}$ iid $N(\mu,c^2\mu^2)$, $\,\,$ $c^2$ known. $\,$ The problem is as following: Consider the above model. Define ...
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0answers
8 views

Negative of log-likelihood on test data decreases but the parameters Mean Square Error(MSE) increases. How to justify the situation?

We develop an EM algorithm to model a problem. We generate some synthetic data of the model with parameters $\Theta$. We call the data $\text{D}$ which is decomposed it into two separate sets, ...
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1answer
28 views

Normal approximation of parameter $p$ of $Bin(n,p)$? [closed]

I've seen normal approximation applied for approximating a binomial distribution $B(n,p)$. However, if one estimates the parameter $p$, then can the parameter $p$ be "normally approximated" just as ...
8
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0answers
93 views

Anscombe-like datasets with the same box and whiskers plot (mean/std/median/MAD/min/max)

EDIT: As this question has been inflated, a summary: finding different meaningful and interpretable datasets with the same mixed statistics (mean, median, midrange and their associated dispersions, ...
3
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0answers
28 views

Relationship beween SEs of two point estimates

Good evening everybody! There's a very odd question from an old exam in Introductory Statistics that has been preoccupying me for a couple ofhours: What is the relationship between the Standard ...
0
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0answers
18 views

Sample Properties for Estimation

Newbie here. I want to estimate the nationality of a person from the nationalities of his family. I am looking for ideas/pointers for the following: How can I know if his family is a good sample ...
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0answers
12 views

Trouble understanding equations with a mix of variances, expected values and means

Let's assume we don't know the real value $\mu$ of the average of a of random variable $x$. We can find an estimator for the variance using the Bessel correction: $$\widehat{V(x)} = ...
1
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1answer
39 views

Help to understand this. Expected value of $S^\alpha$ in Gaussian distribution

Lets $X_1,\cdots,X_n$ be simple random sample from $\mathcal{N}(\mu,\sigma)$. $\overline{x}$ is sample mean. Let $$S^2=\begin{cases}\sum_{i=1}^n (x_i-\mu)^2, \mathrm{ where\ } \mu \mathrm{\ is\ ...
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0answers
22 views

Estimating multimodal (1-3 modes) signals

I am trying to estimate measured signal, which has multimodal behaviour, usually 1-3 modes (see trimodal sample frequencies below for example), but in one experimental setup it's 1, 2 or 3 all the ...
1
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1answer
180 views

Is this an unbiased estimator?

I'm working in the following problem: Let X be a sample of size = 1 from a Poisson distribution with parameter $\lambda$, and let $h(\lambda) = e^{-3\lambda}$. a.) Check if $T = (-2)^X$ is an ...
5
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1answer
44 views

why is $1/n \sum_{i} (X_i -10)^2$ unbiased

Let $\{ X_1,X_2,...,X_n \}$ be n observations randomly drawn from normal distribution with mean $10$ and unknown variance. Prove that the estimator $1/n \sum_{i} (X_i -10)^2$ is unbiased. Why is this ...
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0answers
28 views

How to prove the unbiased and biased estimator of autocovariance function?

These two estimators are commonly referenced as sample autocovariance functions. I'm curious how you're to show the first is an unbiased estimator, while the second is a biased one. And how would ...
4
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4answers
172 views

What is the difference between bias and inconsistency?

I am trying to learn about bias in simple linear regression. Specifically, I want to see what happens when the $cov(e,x) = 0$ assumption of the simple regression is violated. If this assumption is ...
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0answers
25 views

The variance of a biased estimator

This builds on an an earlier question from Math SE. I am just starting to learn about the simple regression model. In particular, I am trying to understand what happens to $\hat{\beta_1}$ when the ...
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2answers
94 views

a question on negative mean square error

for simple random sampling, i have calculated some mean square errors of many types of ratio estimator. Well, i obtained negative mean square error. Is there a mistake? negative MSE is a normal ...
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3answers
1k views

Is p-value a point estimate?

Since one can calculate confidence intervals for p-values and since the opposite of interval estimation is point estimation: Is p-value a point estimate?
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0answers
17 views

Is there any paper about the distribution of difference of log-normal variable?

I am working on the problem relating to the difference of log-normal distribution. I have found several papers about this topic, however, none of them gives me the answer I want. More specifically, ...
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0answers
17 views

Pandey and Dubey estimator.

I am studying sampling theory Pandey and Dubey(1988) proposed the following product estimator. $\bar y_{PD} = \bar y \left( \frac{\bar x + C_x}{\bar X +C_x}\right)$ And its Mean square error is ...
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0answers
21 views

Likelihood over a arbitrary set of hypotheses

I have the following issue. I have finite, large, enumerated set $H$ of hypotheses that maps a time series to an integer, let's say days to integers i.e. $H \subseteq (\mathrm{Day} \to \mathbb{N})$. ...
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1answer
47 views

How should I solve the following simultaneous equations?

I have the set of simultaneous equations below from the paper Parameter estimation for 3-parameter generalized pareto distribution by the principle of maximum entropy (POME) by VP Singh and H Guo: $$ ...
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0answers
29 views

Is a Monte Carlo approximation to a consistent estimate itself a consistent estimate?

Let $A(x)$ be a consistent estimate of some population quantity $A_0$, where $x$ is the data and there are $N$ observations. However, $A(x)$ is difficult to calculate directly, but can be ...
0
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1answer
44 views

Mean Squared Error (MSE) equivalence to trace + bias on multivariate case

I'm trying to show that: $$ MSE(\theta)\equiv E[(\hat{\theta}_n-\theta)'(\hat{\theta}_n-\theta)]=tr(V(\hat{\theta}_n))+Bias(\hat{\theta}_n)'Bias(\hat{\theta}_n) $$ where ...
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1answer
296 views

Why is asymptotic normality important for an estimator?

Is it because it allows for easy construction for confidence intervals? Isn't it still possible to construct confidence intervals without this property i.e. if it converged to another distribution? ...
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0answers
190 views

Prove that OLS estimator of the intercept has minimum variance

Let $$y_i=B_0+B_1X_i+\epsilon_i$$ where $\epsilon_i\sim N(0,\sigma^2)$. Find the least squares estimator of $B_0$ and show that it is unbiased and has minimum variance. I will not write in ...
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0answers
73 views

Tail VaR/Conditional Tail expectation/Expected Shortfall Well Known estimators

I remember looking at an estimator in a stats course a while ago for TVaR/CTE that went like this: there would be say 500 random data then the estimator for the 1% TVar would be the average of the top ...
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0answers
64 views

Find an unbiased estimator from a random sample from the logistic distribution

So I'm not sure where to start on this one, so any hints will help me because I'm a little lost on finding unbiased estimators. So suppose $X_1, X_2,... X_n$ is a random sample from the logistic ...
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0answers
44 views

Derivation of OLS estimator

Dear mathematicians/statisticians, I have a question regarding the derivation of the OLS estimators. In one explanation I have found the following step: However, I don't understand why it's -b_1 ...
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1answer
99 views

Estimation of the covariance matrix

Assume we have $n$ iid random vectors $y_1, y_2, …, y_n$, normally distributed with zero mean and unknown covariance matrix $M$. Each vector is of size $p$. I know a lot of methods that provide a ...
8
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1answer
140 views

What is the variance of this estimator

I want to estimate the mean of a function f, i.e. $$E_{X,Y}[f(X,Y)]$$ where $X$ and $Y$ are independent random variables. I have samples of f but not iid: There are iid samples for $Y_1,Y_2,\dots Y_n$ ...
12
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2answers
486 views

Correlation between OLS estimators for intercept and slope

In a simple regression model, $$ y = \beta_0 + \beta_1 x + \varepsilon, $$ the OLS estimators $\hat{\beta}_0^{OLS}$ and $\hat{\beta}_1^{OLS}$ are correlated. The formula for the correlation ...
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0answers
27 views

Conceptual doubt in prediction intervals in time series forecasts

Background: In the second chapter of Dr. Hyndmans book on Forecasting, he mentions the use of prediction intervals to define a range of possible values demand can take in a future interval. The ...
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0answers
31 views

Does removing variables decreases the variance of the random error estimator?

Lets say if have a regression of $y= f(x_1,x_2,x_3)$; if I remove $x_2$,$x_3$, I'll get regression of $y=f(x_1)$ By removing these variables, did I decrease variance of the random error estimator? Or ...
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1answer
223 views

constant $\times$ distribution

I know that if $U\sim\chi^2(k)$ then $aU\sim \Gamma(k/2,2a)$ for $a>0$. But i read about the estimator and its distribution $$\hat{\sigma}_k^2=\frac{1}{2k}\sum_{i=1}^k ...
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0answers
24 views

M-Estimation with Biases

I am currently trying to solve an m-estimation problem with a bias component. To illustrate, I will use a quick example. Let's say you have a laser range finder at position $x$ and a wall at position ...
0
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0answers
26 views

In which languages can I estimate a VMA model?

In which languages/environments are there tools to estimate a VMA model of a given order? That is, given $q\in\mathbb{N}$ and a multivariate time series $y_t\in\mathbb{R}^d$, $t=1,\dots,T$, a function ...
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0answers
44 views

What is the requirement for Kalman filters

I have few conceptual Questions about Kalman filters and their role. When classical estimation techniques like Maximum Likelihood estimation exists which assumes some information such as the states ...
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2answers
84 views

How can maximum likelihood be used to estimate parameters for a Weibull distribution? [duplicate]

Consider the following survival data (cumulative): Month 0 - 100% Month 1 - 50% Month 2 - 33% Month 3 - 25% Month 4 - 20% (meaning 20% of all initial units have survived by the end of Month 4) ...
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2answers
179 views

What is the difference between bias and residuals?

I'm aware of the bias variance trade off. Intuitively I understand how as the model becomes more complex the variance decreases and the bias increases, after a certain point. But I don't really ...
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0answers
15 views

Need help understanding this algorithm for a robust estimator for Geom. dist

I am trying to figure out a way to estimate the parameter for a Geometric distribution, using a random sample that is influenced by outliers. Searching through previous questions/answers, I saw this: ...
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0answers
33 views

Establishing consistency

I need to establish the (weak) consistency of an estimator of the mean, $T=a+b\bar{X}$. I tried to apply Chebyshev's inequality, but I couldn't do much because the parameter that subtract in the ...
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0answers
10 views

Empirical Density of Model Parameters

Given a parametric model, what are some methods to determine the distribution of (uncertainty in) its parameters? It seems like the naive way is as follows. Let's say we have not just one random ...