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2.253 estimation questions.

I have measurements of an object. Let's say I have its length $L$, mass $M$, and age $t$: $$\mathbf y = (10~\text{m},\ 0.01~\text{g},\ 5~\text{s}).$$ I also have the uncertainties on my measurements ...

I am struggling with the following problem. We are given an i.i.d sample of size $n,$ with the form $X_{i}=\mu+n_{i}$, where $\mu$ is a deterministic unknown constant, and $n_{i}$ is a noise with a ...

I have a little project for which I have to estimate parameters on a PSF (Point Spread Function = response of the system to a dirac, i.e a star in my case). I have the 6 parameters to estimate : $p=(\...

I have a strange coin whose bias $p$ (probability of heads) depends on 2 environmental factors $a, b\in\mathbb{R}$ which I can control at will, and an innate hidden parameter $\theta$: $p(a,b,\theta) ...

I'm working on a project that will run $n=10000$ experiments. In this experiment, $j$ events will occur (an unknown number). Each of the events has a value $E_j$ attached to it. We expect these values ...

It is always argued that the posterior median is the Bayes estimate associated to the absolute loss function. The proofs I have come across rely on differentiating the conditional Bayesian risk and ...

Consider an $AR(1)$ process \begin{align*} y_{t} + a y_{t-1} &= e_{t} \end{align*} for $t=1,\ldots,N$, where $e_{t}$ is a white Gaussian noise with variance $\sigma^{2}$. How do I express the ...

I am very new to this subject, but i've tried searching other questions and haven't found what i need. I'm working on an application that will be promoted in physical meetings. Lets say, for example, ...

Suppose there are two vector signals $x$, $z$. The observer 1 receives a linear version of $x$ plus Gaussian noise. Observer 2 receives a linear sum of both $x$ and $z$ plus Gaussian noise as shown ...

I have been reading about the method of moments, and now I understand how to obtain the method of moments estimator for a random sample $x_1,...,x_n$ from a distribution $f(x;\theta)$, in the ...

What are the advantages and disadvantages of MMSE estimates and MAP estimates?

I have one conceptual (but sort of vague) question regarding distribution fitting. How many observations one would need to best fit any statistical distribution to given data. Like I am dealing with ...

Consider i.i.d random variables $X_1$, $X_2$, . . . , $X_n$ having pdf $$f_X(x\mid\theta) = \begin{cases} \theta x^{−2} & x\geq\theta \\ 0 & x\lt\theta \end{cases}$$ where $\theta \...

One of the purported uses of L-estimators is the ability to 'robustly' estimate the parameters of a random variable drawn from a given class. One of the downsides of using Levy $\alpha$-stable ...

I am trying to estimate the alpha parameter of a supposed $\alpha$-stable distributed set of data. I have tried from the Hill estimator to more advanced fitting method, but they are or too ...

Let's assume I have a set of samples of a random variable $$ X = Y + Z \>, $$ where $Y$ is Gaussian (with a mean of zero and variance $\sigma^2$) and $Z$ has a symmetric $\alpha$-stable ...

I see these terms being used and I keep getting them mixed up. Is there a simple explanation of the differences between them?

Let's say the initial value is A, and A is negative (A<0). The final value is B, which is positive (B>0). I know I can not use log returns, since A is negative So, if I want to calculate the ...

Let $x$ be a $\alpha-$stable distributed random variable of parameters $\alpha,\beta,c,\mu$. When $\alpha \gt 1$ I can estimate the location parameter $\mu$ of the distribution as $\mu=E[x]$ But how ...

Suppose I have a design in which persons are randomly assigned to a intervention and a control group. The persons in the intervention group can receive several treatments, most likely not randomly ...

I'm analysing data that was collected in an optics experiment. The measurements are roughly in the form of a ring. In an ideal world the center of this ring would coincide with the origin, but due to ...

I want to use a quarterly labor force survey data for my estimation purpose. The sampling method follows a rotating panel scheme, that is, the survey begins at time $t=1$. At time $t=2$ some of the ...

Suppose you throw 1000 darts where each dart has 0.5 probability of scoring. For the first 500 darts each is worth 1 point, for the second 500 darts each is worth 3 points. If you score 1500 points, ...

For $Y \sim \mathcal{N}(\mu,\sigma^2)$, $$X = \frac{\exp(Y)}{1+\exp(Y)},$$ so $X$ has a logit-normal distribution. Then $$\mathbb{E}(X^2) = \int_{-\infty}^\infty \frac{\exp(2y)}{\left(1+\exp(y)\...

Can I estimate a diff in diff model to compare the effects of two different treatments that apply in different time periods in different countries? I have 30 countries for an average time span of 34 ...

Let $(X_1,X_2,\ldots,X_n)$ be a random sample from the density $$f_{\theta}(x)=\theta x^{\theta-1}\mathbf1_{0<x<1}\quad,\,\theta>0$$ I am trying to find the UMVUE of $\frac{\theta}{1+\...

Suppose that we observe i.i.d random variables $X_1, X_2, \ldots , X_n$ having pmf $$f_{X}(x\mid\theta) =\theta(1−\theta)^{x−1}I_{\{1,2,3,\ldots\}}(x)$$ where $\theta\in(0,1)$. Consider the ...

I am estimating parameter $\beta$ as: \begin{align} \hat \beta &= \mathop{\mathrm{arg\,max}}_\beta \;\; l(\beta;X,y) - \frac{\lambda}{2}\left(\tilde y-g(\beta,\tilde X)\right)^\prime C^\prime C\...

The answer to minimax estimator explains why minimax does not imply admissibility. The relevant statement is from https://www.stat.berkeley.edu/~yuekai/201b/lec6.pdf which says, minimaxity does not ...

Suppose we have $m$ sources, each of which noisily observe the same set of $n$ independent events from the outcome set $\{A,B,C\}$. Each source has a confusion matrix, for example for source $i$: $$...

I will state the question then my methodology. Q: We have 3 random variables, $X1,X2,X3$ that are independent and identically distributed (iid). We would like to estimate $\theta = E[X1]$. Suppose ...

Let $\{ X_i | i = 1, 2, . . . ,n \}$ be a sequence of independent and identically distributed (IID) random variables from a population and define $\mu \equiv \mathbb{E}(X)$ and $\sigma^2 \equiv \...

I want to find standard errors of estimates obtained by optimizing a custom objective function, which I will explain further below: The objective function is: $L(a,b|y,X_1) - \lambda \times RSS(f(y),...

I am working on gamma distribution for my project. I want to check the total deviation of estimate from the parent distribution for both maximum likelihood and moment estimation methods. I want to ask ...

I need to use Lohr's (2d Ed) method of calculating a standard error for an estimate generated from a ratio estimator based on a stratified sample. In working out her example on page 145-146, I find ...

I have a question that might be trivial but I have not much knowledge on that method: I want to estimate a structural model with GMM and my model works in the sense that it estimated the right ...

I have a following kind of problem. I have several countries. For every country I know the cumulative value of the variable of interest after four periods. I also know the cross-country sum for every ...

Given a dataset, I want to estimate the parameters of a jump diffusion model using MLE. The model is as follows $$X_t = \mu t+\sigma B_t +\sum_{i=1}^{N_t}Y_i$$ Here $B_t$ is a one-dimensional Brownian ...

Consider the model, yijk = µ + Ai + Bij + eijk ; i, j, k= 1,...,5 I tried testing for estimability by trying to reducing the coefficient matrix. But the matrix is really big that it's getting ...

Consider the variables $x_i \text{~} \mathcal{N}(\mu, \sigma^2,a,b)$ iid with truncation points $a$ and $b$, i.e. $a < x_i < b$. Suppose all 4 parameters, namely $\mu, \sigma, a, b$ are all ...

Often times people specify the GARCH model as follows: $$ \sigma _{t}^{2}=\omega +\alpha _{1}\epsilon _{t-1}^{2}+\cdots +\alpha _{q}\epsilon _{t-q}^{2}+\beta _{1}\sigma _{t-1}^{2}+\cdots +\beta _{p}\...

Let $X_1, ..., X_n$ be iid with one of two PDFs. If $\theta = 0$, then $f(x; \theta) = 1, \ 0 < x < 1$. if $\theta = 1$, then $f(x; \theta) = \frac{1}{2\sqrt{x}}, \ 0 < x < 1$. What ...

I'm trying to calculate time-weighted Pearson correlation as described in https://www.aaai.org/ocs/index.php/FLAIRS/FLAIRS14/paper/viewFile/7817/7840 The coefficient is given by $$ \rho_t(X,Y) = \...

There are two kinds of estimates of variance from an iid sample $X1, \dots, X_n$ $1/n * \sum_i (X_i - \bar{X})^2$, which is MLE $1/(n-1) * \sum_i (X_i - \bar{X})^2$, which is unbiased. The unbiased ...

I've been reading some papers to better understand the tools we use to learn Gaussian mixture models. However, frequently a distinction is made between estimating their density, and estimating their ...

I hope you can guide me to the right place. I am estimating two equations, where each equation produces an output which is needed in the other equation as input. Something like this: $Y_t = AX_t + f(...

consider the design model $y=\theta+e$ I know we can obtain the normal equations from observation to estimate the parameters. my question is- is the estimation BLUE? Given normal equations are: $...

I have data tuples $(x_i,\varepsilon_i,t_i)$ generated from some observations and I suspect that $\varepsilon \sim \mathcal{N}\left(0,\sigma(t)^2\right)$, where $\sigma(t)$ is an increasing function ...

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