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Refers to an estimator of a population parameter that "hits the true value" on average. That is, a function of the observed data $\hat{\theta}$ is an unbiased estimator of a parameter $\theta$ if $E(\hat{\theta}) = \theta$. The simplest example of an unbiased estimator is the sample mean as an ...

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Poisson distribution. Statistic system. Parametric function [on hold]

X~Po(θ).There are statistic W as a system W={1 if Xn=1 and 0 if other} what is found by random sample X1,...Xn. What this denotes? Do I have to do replacement x=1 into Poisson distribution formula? ...
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1answer
26 views

Finding UMVUE of a family of continuous random variables

Let $X$ has probability density function $f_X(x;\theta) = a(\theta)b(x)I_{(0, \theta)}(x)$ (where $a(\theta)$ and $b(x)$ are nonnegative). I have to find the UMVUE of $\theta$ or show that one doesn't ...
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22 views

Unbiasedness and consistency

Assume the simple regression model satisfying all Gauss-Markov assumptions. Somebody suggests the estimator Why may someone consider such an estimator? Why will this estimator be consistent? Why ...
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1answer
44 views

Why is it important that estimators are unbiased and consistent?

I am clear on the definition of unbiasedness and consistency. But why are these the criteria we use to judge whether an estimator is a good one? There are other criteria, of course, like the variance ...
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1answer
48 views

Why is the birthday problem a biased estimator?

Can anyone tell me why the calculated birthday match probability is a slightly biased estimator when simulated? Taking a group of 30 people, theory tells us that the probability of at least 2 having ...
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8 views

Applicability of Trimmed Estimators in estimating population parameters

I have been recently focused on trimmed estimators. I read a couple of articles but they seem to be giving a somewhat conflicting output. Below I have outlined the two different conclusions I have ...
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14 views

Proof of multivariate Cramer Rao bound?

The Cramer-Rao lower bound is a presented in most statistical textbooks, but they tend to provide a proof for the univariate case, and let the multivariate case to the reader. I wonder if there is any ...
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3answers
188 views

Consistent unbiased estimator for the location parameter of Cauchy (theta, 1)

Given Cauchy distribution with pdf $p(x) = \frac{1}{\pi ((x - \theta)^2 + 1)}$ how can I find a consistent unbiased estimator for $\theta$? My reasoning so far Tried MLE, but there seems to be no ...
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1answer
59 views

Is $\mathbb{E}(\exp(-\hat{\mu})) = \exp(-\mu)$, when $\mathbb{E}\hat{\mu}=\mu$?

Say I have a biased estimator for $\xi$, say $\hat{\xi}$. But what I know is $\mathbb{E}(\hat{\mu}) = \mu$(unbiased), and $\xi = \exp(-\mu)$. So I wish to do the following bypass. Is $\exp(-\hat{...
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22 views

Perform a wald's test to check the statistical significance of biasedness of MLE

I have done a simulation to see that MLEs are asymptotically unbiased. I want to know whether I can perform a wald test here to check the statistical significance of the difference between the mean of ...
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1answer
54 views

Expected value of squared least squares estimator

I am trying to prove $E(\hat{\beta} '\hat{\beta}) = \beta'\beta+\sigma^2 *\sum_{k=1}^K\lambda_k^{-1}$ where $\lambda_k$ denotes the eigenvalues of the matrix $(X'X)$ with dimensions $K\times K$. $\...
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20 views

Estimator for repeated sampling and fitting

Say I have a Normal distribution $\mathcal{N_1}(\mu_1,\sigma_1)$. Now I will sample $N$ samples $X_1$ from this distribution, and use estimators for $\hat\mu_2$ and $\hat\sigma_2$ to fit a new ...
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86 views

Proof Verification: $\tilde{\beta_1}$ is an unbiased estimator of $\beta_1$ obtained by assuming intercept is zero

Consider the standard simple regression model $y= \beta_o + \beta_1 x +u$ under the Gauss-Markov Assumptions SLR.1 through SLR.5. Let $\tilde{\beta_1}$ be the estimator for $\beta_1$ obtained by ...
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1answer
21 views

Solving an equation to find two unknown weights given an unbiased estimate

Apologies if this is a simple question; I am reviewing out of Seber and Lee's book on regression and I am pretty rusty in my linear algebra Suppose that $X_1, ..., X_n$ have a common mean $\mu$ and ...
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26 views

Unbiased estimator for $L^2$ probability distance norm

I am trying to find an unbiased estimator for (what looks like) the $L^2$ Wasserstein distance between two probability measures. I'm pretty sure that by bickel-lehmann, there is an unbiased estimator. ...
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9 views

Unbiased Coefficient of Variation for Log-Normally Distributed Data

For a normally distributed population, the coefficient of variation is $ \frac {σ}{µ} $, and if you don't know those values for the population, you can multiply by $1 + \frac {1}{4n}$ to correct for ...
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16 views

Unbiased estimate of sign of mean

Consider the set $\mathcal{P}$ of probability distributions that have a finite first moment and define the function $\operatorname{sgn} :\mathcal{P} \to \mathbb{R}$ as $$ \operatorname{sgn}(\mu) = \...
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34 views

Is an unbiased estimator based off multiple complete sufficient statistics also UMVUE?

If $T(X)$ is a complete sufficient statistic such that $ET(X) = \sigma^2$, then $T(X)$ is the UMVUE estimator of $\sigma^2$. My question is, suppose $\tau (T(X),W(X))$ is an unbiased estimator of $\...
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10 views

Defintion of CRLB for time-dependent R.Vs/Time series

I have been reading about estimating frequency/phase of a sinusoid. The CRLB is mentioned in a number of papers but I don't understand how it applies since each data point isn't from the same random ...
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47 views

Constructing an unbiased estimator

Suppose we have a finite population $I_N$ of size $N$ on which we define a variable $\mathcal{Y}$. We also have a generic sampling design $(\mathcal{S},p)$ with first and second order inclusion ...
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1answer
119 views

Looking for an unbiased version of the empirical cumulative distribution function that I can interpolate

Most definitions of the ECDF define it as (#elements <= threshold) / #elements. Matlab and R both implement their ecdf() functions using this formula. In my testing, however, I find that there is ...
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25 views

Why don't we use the n-1 correction for standard error of sample proportion?

From my understanding, when we construct a confidence interval for a sample mean with a sample size of n, we try to estimate the standard deviation of the sampling distribution $$σ_{\overline{x}} = \...
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34 views

Unbiased estimates of parameters from pareto samples

If $X_1$, $X_2$... $X_n$ is the sample of the above distribution, so what's the unbiased estimate of $\frac{1}{θ}$
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21 views

How to calculate confidence bounds for this median unbiased estimate

I am aiming to calculate confidence bounds for the median unbiased estimates, retrieved via the procedure described in Stock and Watson (1998). The authors that I am following state the following ...
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1answer
25 views

Pre-treatment period in difference-in-differences model

I want to evaluate the consequences of a policy change using a diff-in-diff setup. I have quarterly data over ten years before the treatment ($t_{-10}$, $t_{-9}$, ..., $t_{-1}$) and ten years after ...
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16 views

Composition of groups in difference-in-difference models

Assume a difference-in-difference setup where the control and treatment groups are heterogeneous (different observable characteristics) but The parallel trend assumption is verified in the pre-...
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47 views

Sampling from characteristic/moment generating function

Suppose I am given a probability distribution only via its characteristic or moment generating function and I want to sample from that distribution to generate paths in a Monte Carlo simulation. Is ...
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2answers
100 views

UMVU estimator for non-linear transformation of a parameter

Let $X_1, ..., X_n$ be iid. and $X_1\sim N(\mu,1)$. $\gamma(\mu)=e^{t\mu}$ for $t\neq 0$ My question is how to find an UMVU estimator for $\gamma(\mu)$ My concern is not so much about the specific ...
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296 views

Improving the minimum estimator

Suppose that I have $n$ positive parameters to estimate $\mu_1,\mu_2,...,\mu_n$ and their corresponding $n$ unbiased estimates produced by the estimators $\hat{\mu_1},\hat{\mu_2},...,\hat{\mu_n}$, i.e....
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1answer
48 views

Comparing variances of two unbiased estimators

This question is from a Ph.D Qualifying Exam for Mathematical Statistics. Main reference is Casella & Berger's Statistical Inference. Let $W_1$ and $W_2$ be unbiased estimators of a parameter $\...
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38 views

Find unbiased estimators for $\lambda$ and $\lambda^2$.

For the spatial homogeneous Poisson process, find unbiased estimators for $\lambda$ and $\lambda^2$. Attempt: Since the homogeneous Poisson process is over an area, how i would i go about ...
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1answer
52 views

Unbiased Estimation of $\mu^2$ under certain conditions

Let $X_1,X_2,....,X_n$ be a random sample of size $n$ from a population with cdf $F()$. Let $E(X)=\mu$ exist. Then estimate $\mu^2$ unbiasedly for the following three cases:- (i) $Var(X)=\sigma^2$ ...
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36 views

Is there a UMVUE for arbitrary distribution with density and variance?

Let F be the family of all distributions with probability density and finite variance, and $X_1, ..., X_n$ be random samples from F. Does UMVUE for variance exists for this situation?
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31 views

Unbiased estimators of the log odds

In the book of Lehmann and Casella (2003) page 83, a random variable $X$ is distributed according to the binomial distribution $Bin(n,p)$, $n$ the number of trials and $p$ the success probability. ...
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29 views

Conceptual questions on efficient estimators for MA model

I am trying to estimate parameters of a MA(p) system where p is the order. E.g., $$y[n] = \sum_{i=1}^p {\theta}_i u[n-i] + e[n] = \mathbf{\theta}^T\mathbf{u}[n] + ...
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718 views

why does unbiasedness not imply consistency

I'm reading deep learning by Ian Goodfellow et al. It introduces bias as $$Bias(\theta)=E(\hat\theta)-\theta$$ where $\hat\theta$ and $\theta$ are the estimated parameter and the underlying real ...
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43 views

Estimating Kelley Skewness

Groeneveld et al have proposed the following measure of Skewness: $$\mathcal S(x, u) = \frac{F^{-1}(u; x) + F^{-1}(1 - u; x) - 2 F^{-1}(1/2; x)}{F^{-1}(u; x) - F^{-1}(1-u; x)}$$ where $F^{-1}(u; x)$ ...
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183 views

UMVUE for Bernoulli

Let $X_1,..,X_n$ be independent and $Bin(1,\theta)$ distributed. I would like to find the UMVUE for $\phi(\theta)=\theta^3$. I have a complete and sufficient statistic in $T=\sum_iX_i$, and a unbiased ...
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37 views

Unbiased and consistent estimate

I am just asking for a hint. I need help with this example: Let $ {X} $ is a random variable with density $ f (x, \theta) = (\frac{2}{\pi})^{\frac{1}{2}} \theta^{-1} e^{\frac{-x^{2}}{2\sigma^{2}}}, ...
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20 views

Standard errors of OLS estimate if regressor is a stochast?

Assume the model classical linear regression model (with for simplicity only one regressor) $$y=X\beta +u,$$ with $u$, $X$ independent, and $\operatorname{Var}(u|X)=\sigma^2I_n$. Assume for ...
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1answer
26 views

Empirical Risk formulation

I am trying to get an intuitive understanding of the empirical risk below. Based on my understanding, we are unable to compute a function that minimizes the expected risk because we do not have access ...
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4 views

Estimating mean of squares between mutants and parents

Context Consider a numerical organism $i$. When you develop this organism, you get a phenotype (like the height of the individual for example), $p$, following some distribution $P_i(p)$. Parent ...
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45 views

How to make this Estimator Unbiased?

My understanding is that the expected value of the estimate is biased because $\theta \neq \frac{\theta}{n}$. But, I don't understand how to make this unbiased. As n approaches infinity, then the ...
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1answer
65 views

Do tree based methods like random forest and gradient boosting produce unbiased estimates?

Could anyone point me to literature that discuss properties of tree based estimators? For example, are they unbiased, consistent, maximum likelihood, efficient, etc?
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1answer
63 views

For OLS to be unbiased, do we need $x_i$ to be uncorrelated with $\epsilon_i$ or with $\epsilon_s$ for all $s$?

In some textbooks I've read, it is said that an assumption for OLS to be unbiased in the standard cross-sectional model $y_i=\alpha + \beta \cdot x_i +\epsilon_i$, we can use the assumption $E(\...
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1answer
37 views

Is $R^2_{adjusted}$ both unbiased and consistent under the alternative in simple regression?

Consider a simple regression model $$ y_i = \beta_0 + \beta_1 x_i + \varepsilon_i. $$ and suppose it is the correct model for the data. As far as I know, $R^2_{adjusted}$ is an unbiased estimator of ...
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100 views

Minimizing bias in explanatory modeling, why?

This question references Galit Shmueli's paper To Explain or Predict. Specifically, in section 1.5, "Explaining and Prediction are Different", Professor Shmueli writes: In explanatory modeling the ...
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27 views

Find an unbiased estimator for theta

The random variable $X$ assumes values 1; 2; 3 with probabilities: $P(X = 1) = \theta^2$ $P(X = 2) = 2\theta (1-\theta ),\quad 0 < \theta < 1$ $P(X = 3) = (1 -\theta)^2$: If in a random sample ...
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1answer
105 views

Dividing by degrees of freedom [duplicate]

When estimating parameters such as (I don't care about this specific instance particularly) Variance of a random variable X, one usually adopts Bessel's correction, i.e. using the formula $\hat{Var}{(...
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1answer
83 views

Variance of estimator seemingly lower than CRLB?

While practicing for a mid-term, I came across a question where I was asked to investigate the variance of $\frac{(n+1)Y_{n}}{n}$ where $Y_{n}$ is the largest observation of a random sample of size $n$...