**2.182 estimation questions.**

In a series of papers by Altonji, Elder and Taber -- for example 1 -- they check the robustness of bivariate binary choice models by seeing what values of $\rho$ (the correlation coefficient between ...

Given the following likelihood function
$$f(y|x,\tau) = \prod_{i=0}^Nf_T(u_i-x_i-\tau) \tag{1}$$
where, $f_T(t)$ is the probability density function of an Inverse Gaussian distribution given by
...

I'm solving a reinforcement learning-like problem, where I have an agent trying to survive in a 2D room. These room contains a finite and constant number of moving objects that interact with an agent. ...

consider the simplest regular statistical inference problem: $( y_1, \dots, y_n | F ) \sim$ $\text{IID}$ from a cumulative distribution function $F$ on $\mathbb{ R }$ with mean $\mu$ and finite ...

Say we have an $N \times q$ matrix $Y$ with $N>q$.
Also, we have an $N \times p$ data matrix $X$.
We are interested in a model of $Y = X \times W + \epsilon$, where $W$ is a $p \times q$ matrix ...

I am fairly new to R and is exploring simulation to estimate the parameter n:
1) Z is a vector of n draws from N(0,1)
2) Probability of max(Z)>4 equals 0.25
What is the best way in R to estimate ...

Every statement I find of the James-Stein estimator assumes that the random variables being estimated have the same (and unit) variance.
But all of these examples also mention that the JS estimator ...

I always read that every maximum likelihood estimator has to be a function of any sufficient statistic. The idea is that, if we are dealing with a random variable $X$ with mass or density function $f(...

My current project is to find out how much accumulated current have been flown into the main motor in manufacturing some particular product. In doing so, I’ll have to map [Data 1], which consists of ...

Let $f_X$ be a joint density function that comes from an $s$-parameter exponential family with sufficient statistics $(T_1, T_2, \dots, T_s)$ so that the density $f_X$ can be expressed as
$$f_{X|\...

What is the relation between estimator and estimate?

As the result of a Molecular Dynamics simulation, I have the time series of two variables, $X$ and $Y,$ and I am interested in computing the mutual information of these two random variables. I've ...

I have a distribution over the discrete set $\mathcal{A} = \{1, \ldots, d\}$ where the pmf is $p(.)$. That is, $p(i)$ is the probability of obtaining $i$ from $\mathcal{A}$. Given a dataset with $n$ i....

Problem: You have a sequence of N steps that must occur in order. Each step is "unlikely" in any given time period (say, <10%). However, you know that all N steps happened to successfully ...

After studying James-Stein estimators for a few weeks and looking at many different sources I am stuck at trying to understand how Efron and Morris
calculated the Toxoplasmosis example in their 1975 ...

suppose $X_1,X_2,\ldots,X_n$ be random sample of $N(0,\sigma^2)$. how can I calculate UMVUE of parameter $(1-\sigma^2)^{-\frac{n}{2}}$

I have a problem where I am aware that data is well-modeled by a Zeta distribution such as $P(X=x) = x^{-a}/\zeta(a)$, and would like to learn the Zeta distribution parameter $a$ from the data. More ...

everyone, this is my first time posting on this site, so if I am in violation of any standards, kindly let me know.
So I am attempting to prove that $\mu=E(X)$ is the choice of $a$ that
minimizes $E(...

For example, I have historical loss data and I am calculating extreme quantiles (Value-at-Risk or Probable Maximum Loss). The results obtained is for estimating the loss or predicting them? Where can ...

Suppose, we have an urn where each ball has one of $M$ colours and some balls have a dot. We would like to estimate the proportion $p$ of balls that have a dot. We have two experimental protocols:
We ...

Let $X = (X_1, X_2,...,X_k) \sim \text{Multinomial}(n, \theta)$ on $k$ elements (i.e. each $X_i$ is the count of the element $i$ on $n$ trials). Let $\theta \sim \text{Dirichlet}(\alpha)$, where $\...

I'm trying to understand the connection between estimators and entropy. Grasping at straws but do estimators reduce the entropy of the random variable being estimated, with better estimators (less ...

I am struggling with a pretty basic question in parameter estimation using MLE of Copula ( and its underlying distribution)
I have data X, which is independently sampled from a mixture of two ...

I have come across an Excel model that estimates the lognormal distribution parameters with maximum likelihood estimation and least squares estimation. The least squares estimation is fitted to the ...

I am working with transaction data at a adtech DMP.
I believe that the number of transactions that are attributed to my company (as a proportion of the total number of transactions a client observes ...

Usually we don't want to include correlated variables in regression model as problems with estimation and variable significance arise.
I have always thought that a major problem is that the estimates ...

I have been struggling to find a direction on how to proceed with the following problem. Given that $x$ is a zero mean (non-Gaussian) random variable with moments
E$(x^n)=\mu_n$.
I need to find the ...

I have been trying to understand the proof of the bias/variance decomposition formula, and I came across a gap that I haven't been able to fill. I will use the notation of The Elements of Statistical ...

I was hoping someone could assist me with a basic stats question. If I have $K_1$ samples of $R_1 = C_1 + n_1$, where $C_1$ is an unknown constant and $n_1\sim\mathcal{N}(0,\sigma_1^2)$. To make an ...

It's well known that $\overline{x}$ is an unbiased estimator of the exact average.
Now, we let's imagine that we want to estimate some function of the average $f(\langle x \rangle)\equiv f(X)$.
My ...

this might be a stupid question but I don't really understand why the statistic $T = \sum_{i=1}^{n} X_{i}$ is a sufficient statistic for p , for $X_{1}, ... X_{n}\sim^{iid} Binom(1,p)$.
Shouldn't it ...

A favorite example in theoretical statistics is this:
A sample of individuals are drawn independently from a distribution with density $f(x)$, where $f(x)$ is unknown, but is known to be symmetric ...

I was doing this exercise and then I checked the solution, but I got the solution wrong.
$X_{1},...X_{n} \sim^{iid} N(0,\theta)$, i.e.
$f_{X_{i}} (x)= \frac{1}{\sqrt{2 \pi \theta}} -e^\frac{x^2}{...

I am currently an intern working on an inventory/stock policy and there is this particular modeling task on which I am blocked (and I have no statistical supervisor). I have a background mostly in ...

I mean exact likelihood based estimation instead of these LS methods.
There are more general nonlinear optimization methods, but in terms of performance, are there any specific methods for this type ...

Could I have some review of the method I used to fit following SDE: dX = f(t) dt + s X dWFitting method:
Calculated sample for sdW from our data as: $sdW_t = (...

$Y_i=a+bX_i+e_i$. $Y_i$ and $X_i$ are scalar r.v. We have,
$$
V(\hat b|X)=\frac{\sigma^2}{n\left(\bar{X^2}-\left[\bar{X}\right]^2\right)}
$$
and,
$$
V(\hat a|X)=\frac{\sigma^2 \bar{X^2}}{n\left(\bar{X^...

(This question is from my pattern recognition course.) There is this exercise:
Imagine we have $N$ samples with $n$ dimensions.
First it asks to find a point $m$ where the summation of Euclidean ...

I'm stuck on deciding what would be the best approach for my current problem:
I have an array of approximately 1500 data points and I'm trying to find a best fit line through these points based on an ...

By definition, point estimator $\hat\theta_n(\mathbf{X})$ is asymptotically normal if
$$\sqrt{n}(\hat\theta_n - \theta) \, \overset{d}{\longrightarrow} \, \mathcal{N}(0, \sigma^2(\theta)), \qquad \...

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