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

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27 views

Pointwise probability limit

Suppose we have a joint distribution of a random sample $(y_n,x_n)\in R^2$: $$ y_n = \beta_0 x_n + \epsilon_n $$ and the estimator $$ \hat \beta = argmin_\beta \frac{1}{N} \sum_{n=1}^N(y_n - \beta ...
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24 views

Finite variance of harmonic mean estimator when samples are bounded

Harmonic mean estimator is notorious for the possibility of having infinite variance. Now I want to show that it has finite variance when samples are bounded. I am wondering whether my following ...
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9 views

Generalized minimum chi-square estimators?

I need to implement the generalized minimum chi-square estimators (alternative to L-moments method and maximum likelihood estimate (MLE)) for estimate the parameters of the gamma distribution. My ...
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1answer
25 views

Deriving the formula for the mean of a stratified sample

The formula for the mean of a stratified sample $\bar Y_s$ is: $$\bar Y_s = \frac 1 N \sum_i N_i \bar Y_i$$ where $N$ is the sample size for all strata, and $N_i$ and $Y_i$ are the sample size and ...
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29 views

Help in calculating OLS estimators under certain constraints/modifications

I am new to these forums and to econometrics as a whole. I was hoping someone would be able to give me a nudge in the right direction with this problem. I've done extensive research both online and in ...
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1answer
86 views

How can you prove that the naive estimator is less efficient than the OLS estimator

The "naive estimator" is an estimate of the slope obtained by joining the first and last observations and dividing the increase in the height by the horizontal distance between them. Given that the ...
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2answers
71 views

Obtaining an estimator via Rao-Blackwell theorem

Let $X_1,...,X_n$ be iid with pdf $$f(x|\theta) = exp(\theta -x) I(x)_{(\theta, \infty)}$$ It is asked to find an unbiased estimator for$ \theta $ that is function of a sufficient statistical for ...
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12 views

What are the canonical data sets used for testing robust linear fitting?

The UCI database (link) is one of the repositories for canonical data. It has ~295 data sets for use. There are many others. (link) While data can be useful not all data is relevant for all ...
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1answer
118 views

Two stage GMM estimator in Matlab

I am trying to create a simple GMM estimator for the mean of a normally distributed random variable using the first three odd central moments of a normal distribution (all of which should be zero ...
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19 views

Poisson counting process parameter

Two quick questions: What's the maximum likelihood estimator of the parameter of an homogeneous Poisson counting process? To estimate $\lambda$ I'm currently using number of events/total time, ...
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2answers
68 views

Limiting joint distribution of estimators; Functional Statistics; Influence curves;

Let $X_1,...,X_n$ iid r.v. with distribution F, with mean $\mu$ and median $\theta$.Assume that $Var(X_i)=\sigma^2$ and $F'(\theta)>0$. If $\hat{\mu}_n$ is the sample mean, and $\hat{\theta}_n$ the ...
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1answer
41 views

Conceptual questions on Entropy and estimation

Learning Informative Statistics: A Nonparametric Approach paper presents an approach to parameter estimation by entropy minimization. There are other related works "Minimum-entropy estimation in ...
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28 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 $\sigma^2_w$. ...
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2answers
194 views

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 ...
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116 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 ...
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17 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 $ ...
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1answer
132 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 ...
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42 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 ...
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22 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) ...
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32 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) ...
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66 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 ...
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1answer
22 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 $X_1,..X_n$ ...
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2answers
59 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 ...
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1answer
93 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 $\lambda$. ...
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82 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} ...
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52 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 ...
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1answer
48 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 ...
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6 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!
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1answer
33 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 ...
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1answer
126 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
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1answer
74 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} ...
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59 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 ...
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22 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 ...
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1answer
89 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 ...
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0answers
48 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 ...
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1answer
595 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. ...
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22 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 ...
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1answer
79 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 $95\%$ ...
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30 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: ...
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49 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 ...
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16 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.
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35 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
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1answer
172 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 ...
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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 ?
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1answer
134 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 - ...
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17 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?
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167 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 $ ...
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1answer
106 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 ...
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1answer
103 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" ...
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31 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 ...