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Questions tagged [unbiased-estimator]

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 estimator of the population mean.

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Determining variance of UMVUE

Let $X_1,...,X_n$ be iid with pdf given by $f(x;\theta)=\frac{log\theta}{\theta^{x-1}}I(x>1)$. My task is to determine if the $\mu=E[X]=1+\frac{1}{log\theta}$ can be estimated efficiently, i.e. if ...
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Are unbiased efficient estimators stochastically dominant over other (median) unbiased estimators?

General description Does an efficient estimator (which has sample variance equal to the Cramér–Rao bound) maximize the probability for being close to the true parameter $\theta$? Say we compare the ...
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Is there any case in which unbiased but larger MSE estimator preferred to biased and smaller MSE one?

Let saying we are interested in a population mean $\mu$ and we have two estimators $\hat{\mu}_{n}^b$ and $\hat{\mu}_{n}^{u}$ defined on $n$ samples such that $\hat{\mu}_{n}^b$ : biased (i.e, $\mathbb{...
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Unbiased estimator of binomial PMF

Is there an unbiased estimator of PMF of a random variable $Y=\sum_{i=1}^{n} X_n $ where $X_i$ are independent Bernoulli trials with probability $p$, that is, the estimator of: \begin{equation}\tag{1} ...
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Unbiased Estimator for $\log\left[\int p(x\mid z)p(z) \, dz\right]$

The naive Monte Carlo estimator is an unbiased estimator for $\int p(x\mid z)p(z) \, dz$, is there a convenient unbiased estimator for $\log \left[\int p(x\mid z)p(z)\,dz \right]$
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Unbiased Estimate of a squared difference between sample of random matrices

In the description below, IER stands for "Inhomogenous Erdos Renyi" random matrix, which is basically saying that the entries in the matrix are Bernoulli distributed and iid. I didn't get the part ...
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The UMVUE of ratio of parameters for two uniform distributions,

Let $X_1,\ldots,X_m$ be i.i.d. having the uniform distribution $U(0, \theta_x)$ and $Y_1,\ldots, Y_n$ be i.i.d. having the uniform distribution $U(0, \theta_y)$. Suppose that $X_i$’s and $Y_j$’s are ...
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Bias-corrected Property for Jackknife's Pseudo Values

I come across the following formula from a note, saying that we could think of jackknife as a bunch of independent pseudo values with the following form: The notes further comment that the sample ...
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Understanding Rao-Blackwell [duplicate]

From Casella and Berger: Let $W$ be an unbiased estimator of $\tau(\theta)$ and let $T$ be a sufficient statistics for $\theta$. Define $\phi(T) = E[W|T]$. Then $E_{\theta}[ \phi(T)] = \tau(\theta)$ ...
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How can I not show the initialization of the estimation in the Extended Kalman Filter?

I'm making estimates through the Extended Kalman Filter and I have a problem related to the vertical axis of my figure, it's too big, so I can not see population dynamics. However, I wish it did not ...
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Estimating standard deviation when observing means of various block sizes

I have a sequence $X_i$ of iid random variables (you may assume gaussian distribution if you like) but I only observe the mean value of disjoint blocks of various sizes of $X_i$. E.g. $M_1 = \frac14 \...
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Estimator of ratio of central moments

In the context of Control Variates one has to estimate, for example, the following ratios of central moments: $$\frac{\mu_{1,1}}{\mu_{0,2}} \quad \text{and} \quad \frac{\mu_{1,1}^2}{\mu_{0,2}}$$ ...
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How to find an unbiased estimator of $\mathsf{Uniform}(-\theta/2,\theta/2)$

How to find an unbiased estimator of $\mathsf{Uniform}(-\theta/2,\theta/2)$. Is it a function of the order statistics?
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Unbiased estimator of p in geometric distribution

The answer to this question given by my professor was statistic T(x)= 1when X=0 and T(x) = 0 otherwise. Can I consider E(x) = (1-p)/p and then cross multiply and take 1/(1+x) as an unbiased estimator ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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}}}, ...