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

learn more… | top users | synonyms

0
votes
0answers
18 views

What is the plim estimator of $\beta$, What would we need to obtain a consistent estimator of $\beta$?

Suppose that the true model is $$Y_i = βZ_i + u_i$$ However, the researcher/analyst can only observe $X = Z + w$, where $w$ is a measurement error with zero mean and constant variance $\sigma^2_w$. ...
5
votes
2answers
158 views
+50

Drawing numbered balls from an urn

PROBLEM There is an urn with a set of balls where each ball is labeled with a different integer. The numbers on the balls are known and are not a range of integers. For example the set of balls could ...
3
votes
0answers
94 views

Better estimator of expected sum than mean

I am trying to find the optimal estimator for the maximal expected $\Sigma X_i$ where $X_i$ is sampled from an unknown distribution which is chosen to be maximal. To clarify and simplify, there are ...
0
votes
0answers
16 views

Robust Estimator for GARCH in R programming

I just want to know if R programming provides estimator estimation beside QMLE, because I want to compare these estimators' performance toward outliers. What package provides a function for this ...
0
votes
0answers
16 views

Maximum Likelihood estimators in reation to linear models

Consider two simple linear models. $y_{1j}=\alpha _1+\beta_{1}x_{1j}+\epsilon_{1j}$ and $y_{2j}=\alpha _2+\beta_{2}x_{2j}+\epsilon_{2j}$ , $ j=1,2,...,n>2$ where $ ...
2
votes
1answer
72 views

How to calculate the scale parameter of a Cauchy random variable

Let $(X_n)$ be iid random variables and suppose they have mean 0 and follow Cauchy distribution. I know I can set the location parameter to 0. My question is how to find the corresponding scale ...
1
vote
0answers
38 views

Acronyms to use for Bayesian posterior predictive distribution estimators

I am considering writing an article that discusses the Bayesian MMSE and MAP of the posterior predictive distribution. I was wondering if there are acronyms that have been used so that instead of ...
1
vote
0answers
18 views

Conditional variance model with external regressors

I need to estimate the following conditional variance model $$ y_t = \sigma_t\,\varepsilon_t $$ where $y_t$ are the observed data, $\varepsilon_t$ are iid Normal shocks (zero-mean and unit-variance) ...
1
vote
0answers
25 views

Non-Measure Theoretic Argument for Var(X) = 0 iff X is constant (X continuous RV)

I am studying out of DeGroot and Schervish trying to carefully understand the math of prob/stats. In ch 4.3 on variance, they state the theorem that given X a RV whose mean and var exist, then Var(X) ...
0
votes
0answers
59 views

Problems estimating covariance matrices with small $n$, smaller $p$?

It is well known that estimating large covariance matrices from small samples is problematic. For instance, the $p \times p$ sample covariance matrix $\Sigma_n$, estimated from $n$ samples, is not ...
0
votes
1answer
20 views

Definiton of the distribution of estimators, sampling and simulation methods

I have a question regarding the definiton of estimators. In the german wikipedia it says that the distribution is determined by g($X_1,...,X_n$) where by g is the estimator function and $X_1,..X_n$ ...
0
votes
2answers
44 views

Sufficiency of two Poisson disributions

If $X_1,X_2$ constitute a random sample of size n=2 from a Poisson Population show that the mean of the sample is a sufficient estimator of the parameter $\lambda$ . Since the sum of Poissons is ...
1
vote
1answer
57 views

Consistency of an order statistic in exponential distribution

I have two questions. 1) If $X_1,X_2,X_3,...,X_n$ constitute a random sample of size $n$ from an exponential distribution, show that $\bar X$ is a consistent estimator of the parameter $\lambda$. ...
2
votes
1answer
60 views

Showing an estimator is consistent

Show that $Y_1$, the first order statistic is a consistent estimator of the parameter $\alpha $ of a uniform distribution with $\beta = \alpha+1$ Here $f_{Y(1)}(y_1)= \begin{cases} ...
0
votes
0answers
50 views

Measure theory interpretation - textbook advice

I have seen the other questions about measure theory book advice, and I don't think any of them fit what I am looking for. The vast majority of measure theory textbooks are (naturally) math-based and ...
1
vote
1answer
35 views

Maximum likelihood estimation (MLE) for Markov Chain initial distribution?

I am working on using MLE to estimate a Markov Chain, I have successfully estimated the transition matrix $A$, using the method provided in ...
0
votes
0answers
5 views

More examples of inconsistent, but unbiased data and vice versa? [duplicate]

I'm starting to understand the distinction, but I'm having trouble envisioning an example where one's estimates have one, but not the other of these properties. Thanks for the help!
0
votes
0answers
15 views

Identifiability and unbiasedness

How do you show that if my model parameter $\theta$ (scalar) is U-estimable (i.e. if there exists an unbiased estimator of $\theta)$, then $\theta$ is identifiable? This makes sense intuitively, but ...
1
vote
1answer
111 views

Method of moments and maximum likelihood problem

I would like to ask a question on a practice problem from a textbook. The practice problem is about finding estimators of $\theta$, first by using method of moments and then by using a maximum ...
2
votes
1answer
59 views

Finding estimator for a one-parameter Weibull distribution

I'm doing some practice problems on methods of moments from a textbook. I am stuck on the following question: The pdf of a one-parameter Weibull distribution is given by: $f(x) = \begin{cases} ...
2
votes
0answers
54 views

Bayesian inference with the wrong distribution

When an observation $x$ is generated by $P(x|\theta)$ for a parameter $\theta$ the Bayesian optimal estimator for the value of $\theta$ is $\hat\theta_{BEST}=\mathbb{E}[\theta|x]=\frac{1}{P(x)}\int ...
0
votes
0answers
20 views

Compute mean from discrete normally distributed data [duplicate]

I am looking for an estimator of the mean of normally distributed data (with known variance) in the regime where the sampling grid is much coarser than the variance. I.e. in the worst case consider ...
2
votes
0answers
73 views

Comparing estimators in Cauchy distribution

Given $n$ observations with a Cauchy distribution (location=t,scale=1), I would like to compare estimators such as the mean and median by simulating sample means and mean square errors. This is an ...
2
votes
0answers
37 views

Estimator of Bessel function?

I am trying to estimate the parameters of the modified Bessel function of the first kind for integer order case. $I_n(wt) = \sum\limits_{m=0}^\infty \frac{1}{m!(m+n)!}(\frac{wt}{2})^{2m+n}$ In ...
3
votes
1answer
281 views

ML vs WLSMV: which is better for categorical data and why?

I was wondering which is a better estimator to use for categorical data: ML or WLSMV. I saw on a discussion on the Mplus website that they recommend WLSMV for categorical data but didn't explain why. ...
1
vote
0answers
20 views

Admissibility and domination for estimators

Watching a video by the "mathematicalmonk" on the web, I was wondering how to answer this kind of questions: Given $X_1,\ldots,X_n\sim \mathcal{N}\left(\mu,\sigma^2\right)$. Assume that $\mu$ is ...
4
votes
1answer
75 views

Obtaining a confidence interval from an estimator

"Consider an estimator $T(X), X = \{X_1, X_2, ..., X_n\}$, for a parameter $μ$. For $T(X)$, it is given that, $P(3T(X) + μ > 8) = 0.025$ and $P(−3T(X) − μ < −2) = 0.975$. Calculate a $95\%$ ...
0
votes
0answers
29 views

An alternative form of $L$-estimators

$L$-estimators are based on the ordered observations $X_{(1)} \leq X_{(2)} \leq \ldots \leq X_{(n)}$ of the random sample $X_1, X_2, \ldots , X_n$. The general $L$-estimtor can be written in the form: ...
0
votes
0answers
41 views

Fitting a Non-Central t-Distribution with Location and Scale Transformations

I am trying to fit a distribution function to empirical observations that have the following properties: Non-zero mean Non-unit variance Heavy tails Asymmetric about the mode I am considering ...
1
vote
0answers
15 views

Plim of an estimator

What are the steps necessary to calculate the plim of an estimator, when only the general equation form is given? I have looked at resources online, but can't understand how to approach this.
2
votes
0answers
33 views

Estimating the mean with least median of squares

I have a set of real numbers $x_{1}...x_{n}$ and would like to estimate their mean (let's call it y) so that $median(x_{i}-y)^{2}$ is minimal. Is there an algorithm and a correctness proof for ...
3
votes
1answer
133 views

Variance estimation in Random Effects model

I'm studying panel data models in my introductory econometrics class, especially random effects models. Consider the model: $$y_{it}=x_{it}'\beta +c_i+u_{it}$$ with the assumptions ...
1
vote
0answers
98 views

Parameters of $ y_i = \beta_0^2 + \beta_0 \beta_1 x_i$

I have this model, nonlinear in the parameters $ y = \beta_0^2 + \beta_0 \beta_1 x_i $ exist a known estimation of parameters ?
1
vote
1answer
112 views

Estimators, sufficiency, consistency, and bias

A random variable is said to have the Pareto distribution with parameters $\alpha$ and $\beta$, $P(\alpha, \beta)$, if its cumulative distribution function is given by $$F(x)= 1 - ...
0
votes
0answers
16 views

Find the least squares estimator of the parameter B (beta) in the following regression model: y= B + u What is the variance of the estimator? [duplicate]

Find the least squares estimator of the parameter B (beta) in the following regression model: y= B + u What is the variance of the estimator?
1
vote
0answers
145 views

How does one show that there is no unbiased estimator of $\lambda^{-1}$ for a Poisson distribution with mean $\lambda$?

Suppose that $ X_{0},X_{1},\ldots,X_{n} $ are i.i.d. random variables that follow the Poisson distribution with mean $ \lambda $. How can I prove that there is no unbiased estimator of the quantity $ ...
0
votes
1answer
96 views

Huber sandwich estimator in quantile regression

I need the description of Huber sandwich estimate method for quantile regression. I found this "a Huber sandwich estimate using a local estimate of the sparsity function". Sparsity function looks ...
0
votes
1answer
86 views

Why do we examine the rate of convergence when we find a confidence interval?

I would like to help me with this. I don't understand why do we examine the rate of convergence. Also what do we mean by saying "the error is usually dominated by the variance, not the bias" ...
1
vote
0answers
30 views

Fisher information $J_y(\theta)$ for transformation $y=F(x)$

Consider a multivariate random variable $x$ with density function $P_x(\theta)$ for a scalar parameter $\theta$. Assume the Fisher information $J_x(\theta)$ is known. Now, for a transformation ...
3
votes
1answer
143 views

Correlated Bernoulli random variables

I have about $50$ Bernoulli random variables $X_i$ whose joint distribution is unknown, but I can generate a sample of size on the order of $10^4$. They are not independent, but I think the dependence ...
1
vote
1answer
52 views

Is it possible that grid search would fail in two dimensional feature space?

Grid search suffers from the curse of dimensionality. But is there any case(any hypothetical distribution of data) in a two dimensional feature space where the data’s binary classification using Grid ...
2
votes
0answers
145 views

Why don't asymptotically consistent estimators have zero variance at infinity?

I know that the statement in question is wrong because estimators cannot have asymptotic variances that are lower than the Cramer-Rao bound. However, if asymptotic consistence means that an estimator ...
0
votes
1answer
143 views

Asymptotic normal distribution via the central limit theorem

I have a sample $n = 100$ with two "successes" (Two kids having a disease among 100). So we obviously have a binomial distribution. First I had to compute the maximum likelihood (ML) estimator ...
1
vote
1answer
93 views

Bayesian MMSE estimators from a transformation of the observations

Consider a random variable X whose value we want to estimate using a Bayesian MMSE estimator. Let $O_1(X)$ be a set of observations which depend on $X$ in some complex way (captured by $P(O_1|X)$) ...
0
votes
0answers
72 views

MADE and MSE pros and cons

When assessing the performance of an estimator, in which scenarios should one prefer the use of the Mean Absolute Deviation Error (MADE) over the Mean Squared Error (MSE) and vice versa? Edit / ...
11
votes
1answer
144 views

Is there a statistical application that requires strong consistency?

I was wondering if someone knows or if there exists an application in statistics in which strong consistency of an estimator is required instead of weak consistency. That is, strong consistency is ...
5
votes
1answer
232 views

Is the residual, e, an estimator of the error, $\epsilon$?

This question has come up in another thread that I started so I thought I would get more people's opinions on it. My question is Is the residual, e, an estimator of the error, $\epsilon$? The reason ...
1
vote
0answers
71 views

Maximum likelihood: Show that ML estimator is solution of an equation of degree $2n - 1$

My problem: Let $X_1,\dots,X_n \overset{\mathrm{i.i.d}}{\sim} \mathrm{Cauchy}(\theta,1)$ and suppose we want to estimate the location parameter $\theta$. Find the log-likelihood function of the ...
0
votes
1answer
96 views

Deriving Linear MMSE Estimator

I am trying to verify a certain derivation of linear minimum mean square error estimator as it appears in [1] (also called linear a posteriori estimate, since the estimate is based on measurement). ...
3
votes
0answers
52 views

Forming an unbiased estimator of the maximum of several parameters, given independent estimators of each parameter?

Say I have $K$ independent normals, $X_i \sim \mathcal{N}(\mu_i, \sigma_i)$ for $i = 1,...,K$. How can I form an unbiased estimator of $\max_i \mu_i$ using $X_i$'s?