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

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Finding a consistent estimator mathematically

This is my first post on this website so hopefully everything will go smoothly. Let me first ask the question, then go over my problem. Q: Suppose that we are given $({X_{1i},X_{2i}})$ which is a ...
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24 views

Estimator $\gamma = \sum a_i\times x_i$ , where $X_i \sim \exp(t_i \theta )$ Show $\gamma$ is unbiased if $\sum a_i/t_i = 1$

I'm getting really confused with the estimators in this question! $X_i \sim \exp(t_i \theta x)$ where $t_i$ are positive constants. The MLE for $\theta = \frac n{\sum t_i x_i}$ And $\phi = ...
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34 views

Does MCD estimator suffers from swamping effect?

If there are multiple outliers in the data set, the Mahalanobis distance suffers from masking and swamping effects. In order to rectify this problem, robust estimation of location and scale, such as ...
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12 views

Fixed parameter estimates of parent factors in a nested design

Summary: What is happening with parameter estimates of factors that are the 'parents' of nested factors? Data: My analysis involves testing the effect of different parameter settings for automatic ...
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2answers
82 views

Self-study: Finding the maximum likelihood estimates of the parameters of a density function

Consider a random sample $x_1,x_2,...,x_n$ from a newly-generated distribution, whose probability density function is given below \begin{equation} f(x|\alpha,\beta,\sigma)=\frac{1}{\Gamma \left( ...
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How to prove that the permutation of the points are the minimal sufficient statistics for Cauchy distribution?

I see this everywhere that the permutation of the samples $X_{(1)}, ..., X_{(n)}$ is the minimal sufficient statistic for the Cauchy distribution [1]. It is clear that it is a sufficient statistic,but ...
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1answer
24 views

Horvitz-Thompson estimator for two-stage cluster sampling

So I want to apply the Horvitz-Thompson (H-T) estimator to two-stage cluster sampling. The H-T estimator is defined as: $$\sum\frac{Y_i}{\pi_i}$$ where $\pi_i$ is the probability of including the ...
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6 views

Multivariate Generalization of (Fisher) Efficiency Ratio

Fisher defined efficiency as the ratio of variance of the most efficient estimator to the variance of the estimator in question. The Cramer-Rao bound allows easy computation of the variance of the ...
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16 views

Derivation of description length

In class, our professor posted the following: We will discretize $\theta$ (some model) into $1/\sqrt{n}$ distinct values. Intuitive argument: with $N$ data points, our estimation error for $\hat ...
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15 views

linearization of an estiamtor

Suppose we have two variables $x$ and $y$ defined in some population, with all values of $x$ known. A Poisson sample is drawn, with corresponding inclusion probabilities $\pi_k$ that are proportional ...
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34 views

Unbiased estimator of the variance [duplicate]

I am going through the book The Elements of Statistical Learning and I'm finding it extremely terse. I have a background in probability but not statistics so perhaps that is why. Anyway, in Chapter 3, ...
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29 views

Posterior mode estimator unchanged under coordinate transformation?

I'm looking at a data set where the posterior mode has noticeably less "bias" than the posterior mean and posterior median, and somewhat less error. However, the posterior mode is not invariant under ...
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16 views

Source of least square estimator of Poisson parameter

I have $X \sim \text{Poisson}(\theta)$ I need to see how least square estimator of $\theta$ is obtained. Is there anything online showing that?
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13 views

Risk in density estimation: grasping the definition

When generalizing estimators to an entire function what is the space in which we perform the integral to obtain the expected value (with respect to this function)? For example, when estimating ...
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1answer
64 views

What are the implications of estimating a covariance matrix from a correlated sample?

Given a sample of $n$ independent observations $x_1,...,x_n$ (where $x_i$ are $p$-dimensional column vectors), the $p \times p$ sample covariance matrix is defined as ...
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38 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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40 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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17 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
31 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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31 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
149 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
84 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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14 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
712 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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22 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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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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43 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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30 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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217 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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122 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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18 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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226 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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44 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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28 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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50 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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73 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
24 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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73 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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120 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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95 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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54 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
71 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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7 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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36 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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147 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 ...
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81 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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60 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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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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101 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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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 ...