Tagged Questions

0
votes
0answers
26 views

MVUE for Binomial $p^s$

Let $Y_1, Y_2, ..., Y_n$ be a random sample from a $\text{Bin} (1,p)$ where $0 < p < 1$. Let $0 < s < n$ be an integer. Find the MVUE for $\displaystyle\phi (p)=p^s$ and ...
0
votes
1answer
25 views

Random Poisson Sample, Probability in terms of $\vartheta$

If $X_1, X_2, \ldots, X_n$ are a random sample from a Poisson Distribution with mean $\vartheta>0$, how do you find $P(X\le 1)$ in terms of $\vartheta$? I've proven that summing $X_i$ for ...
1
vote
1answer
30 views

Improving related estimates

There are three underlying quantities $x$, $y$, and $a$, where $x$ and $y$ are vectors, and $a$ is a scalar. They are related by $x = ay$. We get noisy observations, $x_0,y_0$. We want to find $a$, ...
1
vote
1answer
57 views

Maximum likelihood estimators, hypergeometric and binomial

I'm trying to solve a two part problem. The set up is as follows: consider a bag with $\theta$ red marbles and $7-\theta$ blue marbles, with $\theta$ being unknown. Let $x$ denote the number of red ...
1
vote
1answer
69 views

Rao-Blackwell unbiased estimator geometric distribution

I'm looking at review questions and having trouble with this one! Let $X_1,\ldots,X_n$ be i.i.d. geometric R.V.s with the pmf: $(1-p)^{x-1}p$, for $x=1,2,\ldots$ and $0<p<1$. I need to use ...
1
vote
0answers
29 views

Hypothesis testing problem of Normal distributions.

Consider the following Hypothesis Testing problem: Hypothesis $H_0$ : $X \sim N(\mu_0, \sigma_0)$. Mean $\mu_0$ is known but only upper and lower bounds on $\sigma_0$ are known. Hypothesis $H_1$ : ...
0
votes
1answer
25 views

Calculating the variance of an estimator (unclear on one step)

How can you go from $4V(\bar X)$ to $\displaystyle \frac{4}{n}V(X_1)$? I understand the rest of the steps...
1
vote
0answers
33 views

Cramer-Rao bound for $\chi^2$ distribution parameter estimates.

I've stuck in unpleasant problem with noncentral $\chi^2$ distribution. I work with random variables, distributed as $\chi^2_{\nu}(\lambda)$, where $\nu$ is the degree of freedom and $\lambda$ is ...
0
votes
1answer
33 views

Can we compute confidence intervals for the variance of an unknown distributions from sample variances?

Assume $X_1,\ldots,X_n$ are i.i.d. with unknown distribution $\mathcal D$ - we only know it is not normal and has finite variance. Is there a way to give confidence intervals for the variance of ...
0
votes
0answers
35 views

Numerical calculation of fisher information

I am trying to obtain numerically the fisher information. Given a likelihood function $$ f(X,\theta),$$ with $X \in [0,1]$. The fisher information is given by $$ ...
0
votes
1answer
26 views

Scale Median for MRE Estimators with Absolute Difference Error Function for Scale Families

Lehmann, in Theory of Point Estimation p.212, defines scale median as the solution to: $${E(X)I(X\le c)} = {E(X)I(X\ge c)}$$ given $X$ is a positive random variable, and ${E(X)}< \infty$. Now ...
0
votes
1answer
67 views

Clasification of parameter estimation method

Consider that $P$ is the water pressure coming out from a valve A, therefore, the population is all the valve A pressure values. Let $P_{dif}$ be defined as the difference between the maximum and the ...
0
votes
0answers
95 views

Finding the efficiency of an unbiased estimator

I have a random sample drawn from a $N(\theta,\sigma^2)$ distribution with $\sigma^2$ known. I am trying to estimate $\theta$. I need to calculate the efficiency of the unbiased estimator, ...
0
votes
1answer
51 views

How to estimate parameters of a normal distribution?

Suppose one knew that 105 workers were evaluated by their boss. Such evaluation is distributed according to a normal distribution with mean $\mu$ and std. deviation $\sigma$. We also know that 20 ...
2
votes
1answer
278 views

How do I find the MLE of $\theta$ when x is dependent on $\theta$?

Let $X_{1},X_{2},...,X_{n}$ represent a random sample from a distribution with pdf: $f(x; \theta)=e^{-(x-\theta)}, \theta \le x<\infty, -\infty<\theta<\infty$ | zero elsewhere I need to ...
1
vote
1answer
80 views

The distribution when combining two samples together?

Suppose $X\sim N(0,{\sigma}^2)$ and $Y\sim N(0,{2\sigma}^2)$ . $X_1, ..., X_m$ are the samples from $X$ and $Y_1, ..., Y_n$ are the samples from $Y$. And then combine two samples as a new sample ...
6
votes
2answers
180 views

Intuitive explanation of a definition of the Fisher information

I'm studying statistics. When I read the textbook about Fisher Information, I couldn't understand why the Fisher Information is defined like this: $$I(\theta)=E_\theta\left[-\frac{\partial^2 ...
2
votes
2answers
77 views

Expected value of a max

We have a roulette with the circumference $a$. We spin the roulette 10 times and we measure 10 distances, $x_1,\ldots,x_{10}$, from a predefined zero-point. We can assume that those distances are ...
1
vote
2answers
87 views

Show that estimates are unbiased

The following is a problem in my book that I don't really understand: We take a random sample: $x_1,x_2,\ldots,x_n$ from a population that is $N(μ,σ)$ where $\mu$ and $\sigma$ are unknown. We build ...
3
votes
0answers
91 views

Estimate number of distinct items

I have a large array of $n$ integers, some of which may be repeated, and I want to estimate how many distinct integers are in the array. Say the number of distinct integers is $N$. I can sample with ...
2
votes
0answers
135 views

Fisher Information and minimum variance estimators

I am trying to understand what can be proved about minimum variance estimators. I have changed the question to make it more specific. Let us assume we have some finite set $S$ of elements and we just ...
1
vote
1answer
53 views

biasedness/unbiasedness of an MLE.

To show whether an MLE I just found is biased/unbiased, would I need to find the expectation of the answer? Plus would I do this by integrating $\text{MLE} \cdot \text{pdf}$. My MLE is $ ...
2
votes
1answer
43 views

Parameter optimization in probabilistic models

Task: Suppose we model a variable $y = Wx + \mu$ as a linear transformation of $x$ plus some Gaussian noise $\mu\sim\mathcal N(0,\sigma I)$. Our aim is to minimize the estimation error of $x$ given ...
1
vote
0answers
48 views

Estimating the number of observations from a set of samples

I repeatedly measure a value $S_n$ which is the sum of a set of $n$ hidden inputs. The goal is to identify the number of hidden inputs. All of the hidden inputs are driven by an experimenter ...
0
votes
0answers
38 views

Identification of parameters problem

I always struggle to get the true essence of identification in econometrics. I know that we state that a parameter (say $\hat{\theta}$) can be identified if by simply looking at its (joint) ...
1
vote
0answers
57 views

Showing that statistic is unbiased

Let $X $ be observed data. Let $\hat{\theta}(X)$ be an unbiased estimate of $\theta$ and let T be a sucient statistic for $\theta$. Define the new estimator $\hat\theta^{*}$ of $\theta$, $$ ...
2
votes
1answer
156 views

Determine whether a statistic is sufficient, given the probability density

Let $X_1, X_2, \dots, X_n$ be a sample of i.i.d. random variables, with density $$f_\theta=\frac{2}{3\theta}\left(1-\frac{x}{3\theta}\right) $$ for $0 < x < 3\theta$. And $f_\theta=0$ if $ x ...
1
vote
1answer
134 views

Using the MSE criterion, which is a better estimator for $\Theta^2$?

Question: Let $T_1$ and $T_2$ be independent unbiased estimators of a parameter $\Theta$. Assume that $\operatorname{Var}(T_2) = \operatorname{Var}(T_1)$. Using the MSE critertion, define which is a ...
1
vote
1answer
48 views

Proving that the sum of Good-Turing estimators is $1$

I want to know how to go about proving that the Good-Turing estimator has a total probability of $1$. I have seen this proof (page 2) but I found unclear the first step: $$\sum_j \theta[j] = \sum_r ...
1
vote
2answers
122 views

Fast variance calculation

Suppose to have a sequence $X$ of $m$ samples and for each $i^{th}$ sample you want to calculate a local mean $\mu_{X}(i)$ and a local variance $\sigma^2_{X}(i)$ estimation over $n \ll m$ samples of ...
0
votes
1answer
70 views

How to match a discrete distribution to a continuous distribution in information theoretic sense?

Let $$ S \sim N(\mu, \sigma^2) $$ be a normally distributed random variable with known $\mu$ and $\sigma^2$. Suppose, we observe $$ X = \begin{cases} T & \text{if $S \ge 0$}, \\ -T & ...
1
vote
1answer
171 views

Gauss-Markov estimator properties

Consider a linear model $$ y = Ab+n, $$ where $b \in \mathbb{R^m}$ is a parameter to be estimated, $n \in \mathbb{R^{n}}$ is a noise with mean $\mathbb{E}n = m_{n}$ and with covariation matrix ...
2
votes
1answer
750 views

Prove the sample variance is an unbiased estimator

I'm trying to prove that the sample variance is an unbiased estimator. I know that I need to find the expected value of the sample variance estimator $$\sum_i\frac{(M_i - \bar{M})^2}{n-1}$$ but I get ...
2
votes
1answer
133 views

Maximum Likelihood Estimation

For iid random variables from a distribution with p.d.f. $$f(x;\theta_1,\theta_2)=\frac{1}{\theta_2}\exp\bigg(-\frac{(x-\theta_1)}{\theta_2}\bigg), \quad x>\theta_1, ...
0
votes
1answer
105 views

Estimation of discrete random variable

Suppose you have a discrete random variable $X_1$ with known probability mass function. I guess that choosing a variable drawn from the same pmf would be the best way to guess $X_1$ assuming all ...
1
vote
2answers
371 views

Question about unbiased estimator

$X_1,X_2,\ldots,X_n$ are iid random variables $B(1,\theta)$ where $0< \theta<1$. Let $w = 1$ if $\sum_i X_i = n$ and $0$ otherwise. What is the best unbiased estimator of $\theta^2$. Attempt: ...
3
votes
2answers
380 views

biased Maximum Likelihood estimation

Given $N$ points ($x_k$, k from $1$ to $N$) generated from a normal distribution (1-dimensional case) with known mean $\mu$, the Maximum Likelihood estimation of the variance is ...
11
votes
2answers
542 views

You see a route 14 bus on the moon. What is the most likely number of bus routes on the moon?

This question was asked on a forum and while many argued that the answer is 14 (since the probability of you seeing bus 14 is maximum in this case), I argued against it that they were working ...
5
votes
1answer
139 views

Finding the joint distribution of $X_{1:n}$ and $\overline{X}$

I need to show that, given a random sample of independent variables $X_1, ... , X_n$, each following a distribution EXP($\theta$,$\eta$), that is, ...
4
votes
1answer
127 views

Showing that $X_{1:n}$ is sufficient for $\eta$, by factorization

I'm asked to show that $X_{1:n}$ (the minimum order statistic) is sufficient for $\eta$, in the case of a random sample $(X_1, ... , X_n)$ where $X_i\sim EXP(1,\eta$) (this is the two-parameter ...
0
votes
1answer
220 views

Intuitive explanation of Fisher Information and Cramer-Rao bound [closed]

I am not comfortable with Fisher information, what it measures and how and how is it helpful. Also it's relationship with the Cramer-Rao bound is not apparent to me. Can someone please give an ...
1
vote
3answers
524 views

How to estimate failure probability from count until first failure?

What would be the formula to estimate the rate of failure of some test as a percentage chance of failure from the number of runs of the test until the first failure was seen? For example, considering ...
1
vote
1answer
128 views

Likelihood function estimating two different means

I completely understand how to find the likelihood functions of simple pdfs. However, how would you attempt to find the likelihood function of a pdf with a negative exponential function with two ...