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

learn more… | top users | synonyms

0
votes
2answers
34 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
0answers
18 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
41 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
40 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
0answers
18 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
14 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
89 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
42 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
49 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
19 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
54 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
27 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 ...
2
votes
1answer
101 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
16 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
72 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
26 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
23 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
97 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
96 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
97 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
123 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
78 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
75 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
29 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
110 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
43 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
124 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
136 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
84 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
63 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
136 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
186 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
57 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
76 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
50 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?
2
votes
0answers
52 views

Transforming an estimator to overestimate

I have an estimator $\mu^*$ of a the mean $\mu$ of a certain distribution that I obtained using a variational technique (basically just establishing a bound on $\mu$ and finding a trial function that ...
0
votes
1answer
88 views

Parameters, Estimates

I lack some knowledge in the concepts of parameters, estimates and moment (math and stats). I can't find an online easy-to-understand source of information about these concepts. Would you help me with ...
1
vote
0answers
30 views

Trying to understand an example where unbiased estimators don't exist

I am new to statistics especially in the topic of estimators and sufficient statistic. I am reading a note which says "unbiasedness is a desirable (but not necessary) property of a good estimator". ...
-1
votes
1answer
92 views

Fitting GARCH Model

I'am getting more and more familiar with this kind of model (and others models too). I'm now used to fit this model with my data (rmgarch package in R). How is it done ? What is the theory behind ...
2
votes
0answers
78 views

What do people call the groups into which quantiles divide the population?

Is there a correct technical name for the group of observations between two quantiles? For example, if you have the values of the four cut-points that divide a population into five groups of equal ...
0
votes
0answers
32 views

What is the preliminary scale in M-estimates for regression

I'm trying to get aquainted with robust regression methods and there's something about M-estimators that I don't understand. In "Robust statistics" (Maronna, Martin, Yohai) it is said that if both ...
2
votes
2answers
604 views

Asymptotic distribution of MLE (log-normal)

Say we have a sample $X_{1},...,X_{n}$ from a log-normal distribution with parameters $\mu$ and $\sigma^{2}$. That is, $\ln(X)$~$N(\mu,\sigma^{2})$. Let $T_{n},Z_{n}$ denote the MLE's for ...
2
votes
1answer
141 views

Methods of moments for t distribution

The parameters of a t distribution can be estimated via 1) ML or 2) method of moments If we use the method of moments we have: $\mu=E(R)$ $\sigma^2=V(R)=\frac{\beta \nu}{\nu -2}$ $\kappa = ...
1
vote
1answer
133 views

A constant as an admissible estimator

This is a homework question so I would appreciate hints. I believe I have the first part correct, but I fail to see how the second part is different. Assume square error loss, $L(\theta ,a)=(\theta ...
11
votes
7answers
2k views

What is the difference between an estimator and a statistic?

I learned that a statistic is an attribute you can obtain from samples.Taking many samples of same size, calculating this attribute for all of them and plotting the pdf, we get the distribution of the ...
1
vote
0answers
111 views

Is high multicollinearity always an issue in OLS?

$$Y_t = a + bX_{1,t} + cX_{2,t} + dX_{3,t} + e_t$$ A high $R^2$ in $X_{1,t} = \alpha + \beta X_{2,t} + \gamma X_{3,t} + \varepsilon_t$ will always result in a higher standard error of the $b$ ...