Questions tagged [point-estimation]
Point estimation is the application of an estimator to the data in order to learn about a certain population parameter.
147
questions
4
votes
1answer
39 views
Question about Casella and Berger's proof of MLE invariance
In Casella and Berger, p. 320, they have a proof of the invariance of the MLE. Let $g: \theta \mapsto \eta$ be a function. They define the induced likelihood as
$$
L^*(\eta \mid X) = \sup_{\{\theta: ...
0
votes
0answers
16 views
Two-sided one-parameter likelihood ratio test
Consider a single-parameter distribution and a two-sided test for hypothesis $\theta_0$. Does the likelihood ratio test just reduce to:
$$
\frac{\max_{\theta \in \Theta_0} \mathcal{L}(\theta)}{\max_{\...
2
votes
1answer
28 views
Fitting a gamma distribution to truncated data
I am faced with the following truncation problem:
$$
X_i \sim \Gamma(\alpha, \beta) \\
\delta_i = \chi(X_i \le \tau_i)
$$
I can observe only $(X_i, \tau_i)$ where $\delta_i = 1$ and I have no a-...
0
votes
0answers
35 views
MVUE for Poisson Distribution
Let $X_1,...X_n$ be $\text{Poi}(\lambda)$ distributed random variables. I want to construct a minimal variance unbiased estimator (MVUE) for $\lambda$.
By the Neyman Lemma, I know that $T:=\sum_{i=1}...
0
votes
1answer
28 views
nonexistence of a sufficient statistic
Let $X_1,X_2,\dots,X_n$ be a random sample from a $\Gamma(\theta,\theta)$ distribution. Then
$$
\prod_{i=1}^n f(x_i;\theta) = \frac{1}{\Gamma(\theta)^n\theta^n}(\prod_{i=1}^n x_i)^{\theta-1}e^{-\frac{...
8
votes
1answer
172 views
When can't Cramer-Rao lower bound be reached?
The Cramer-Rao lower bound (CRLB) gives the minimum variance of an unbiased estimator. One sentence in the wiki page says "However, in some cases, no unbiased technique exists which achieves the bound....
2
votes
0answers
14 views
Signal-to-noise-ratio, Fisher information and and “estimability”
Given a parametric statistical model, is it common to study the quantity
$$ Q_{\theta} = \theta^2 I_{\theta} \, ,$$
where $I_{\theta}$ is the Fisher information? (I focus on a single parameter for ...
0
votes
0answers
31 views
How to estimate the mean knowing standard deviation?
I want to estimate the mean and I have the next parameters:
Confidence Interval: 95%
Population Standard Deviation: 16.44
Maximum Error of the Estimate: 5
0
votes
0answers
42 views
Comparing hypotheses that depend on a different number of parameters
Let us consider the following problem. You want to establish whether (1) the outcome of the tossing of a coin is independent from previous tossings AND the probability of getting Heads is $p$; or ...
0
votes
1answer
23 views
Point estimate options with highly skewed data
(Added regression tag since I think that my question overlaps with that area but not sure. Added transformation tag since I discuss log transformations. Tag recommendations welcome)
I would like to ...
1
vote
0answers
14 views
Extension of Potts model with non-constant interactions?
Is there any work that extends (allows) Potts model to have non-constant interactions between the lattice points? Specifically, the interaction matrix is a symmetric matrix that can have both positive ...
6
votes
0answers
58 views
Posterior variance vs variance of the posterior mean
This question is about the frequentist properties of Bayesian methods.
Suppose we have data ${\bf y}$ generated from a distribution with a single parameter $\theta$, equipped with a prior $\pi(\...
0
votes
0answers
32 views
Maximum Likelihood to estimate function parameters from experimental data
Problem
I have a set of $n$ measurements $V_{i}^* = V^*(t_{i})$ of the observable $V$ in function of time $t$.
I also have the associated sensibility errors (determined by the measuring device's ...
0
votes
0answers
16 views
Bias of Pearson correlation estimator of two Bernoulli variables
Crossposting link: https://math.stackexchange.com/questions/3312349/bias-of-pearson-correlation-estimator-of-two-bernoulli-variables
Suppose we have two correlated Bernoulli random variables, $X_j$ ...
0
votes
0answers
29 views
Estimating Standard Deviation with only Linear Calculations
I have an input list of numbers which are assumed to be drawn from an underlying Gaussian distribution. I need to find the mean and standard deviation of the dataset. The code that I am working with (...
1
vote
0answers
29 views
What is the name of this kind of statistics problem and where can I go to learn how to solve it?
I have a heuristic model of a physical process which is dependent on outcomes from random variables. The physical process approximated by the random number generator suggests what the distribution ...
0
votes
0answers
4 views
Correct for known frequency of random sampling in prediction
When finding the expectation of a parameter $p$, depending on another parameter $h$, one likely wants to apply the formula $\hat{p}=E[p]=\sum_k E[p|h=k] P(h=k)$. Now, often the problem allows one to ...
0
votes
1answer
29 views
$T_0$ is MVUE for $\gamma(\theta)$ and $T_1$ is an u.e. for $\gamma(\theta)$ with efficiency 0.0169. Then corr($T_0,T_1$ )?
The estimator $T_0$ is MVUE for $\gamma(\theta)$ and $T_1$ is any other unbiased estimator for $\gamma(\theta)$ with efficiency 0.0169. Then what is the correlation coefficient between $T_0$ and $T_1$...
0
votes
1answer
28 views
estimate equal distribution of few points on a line
I am trying to find the best solution to estimate equal distribution of points over a line. I know I can use relative SD or similar, but I was wondering if there are more "specific" methods that can ...
0
votes
0answers
10 views
Distribution of $n^{1/2}\{\hat T_n−T_n(F)\}$ in bootstrap problems
I've read this in a paper, and I don't know how to proof the last statement:
Let $X_1,...,X_n$
be independent identically distributed random variables with unkown distribution function F. Suppose the ...
0
votes
0answers
24 views
How to choose estimates after Bayesian regression?
In a Bayesian logistic regression with two predictor variables $x_{1}$ and $x_{2}$, I did MCMC (2000 samples) to estimate posterior distribution. Now it's done, how can I choose the final estimates ...
0
votes
0answers
8 views
predict important markov chains(paths to an event) with confidence
I need help in validating my current approach as well as find any alternative approaches I could use in order to tackle issues like below.
I have a dataset like below and I need to find important ...
2
votes
1answer
46 views
Why is the maximum risk of an estimator independent of a prior distribution over the parameter?
One way of choosing an estimator $\delta(x)$ for data $X$ distributed as $P_{\theta}(X)$, where $\theta \in \Theta$ is:
$$minimize \sup_{\theta \in \Theta} Risk(\delta(x), \theta)$$
In this case why ...
2
votes
2answers
201 views
Two tailed t test for two companies' monthly profits
I have a question revolving around two tailed tests, confidence intervals, etc, and I'm really struggling to figure out how to find the correct answers. For some reason I can't seem to properly wrap ...
0
votes
1answer
33 views
check understanding on unbiased and consistent estimator
I'm trying to understand the expected value of an estimate. Here's my understanding.
The expected value of the estimate $\bar x$ of the parameter $\mu$ is what the mean of xbar tends to as we ...
1
vote
2answers
152 views
What is the maximum likelihood estimator for $e^{-\theta} = P(X_i = 0)$?
Suppose $X_1, X_2,...,X_n$ is a random sample from a $\text{Poisson} (\theta)$ distribution with probability mass function:
$$P(X=x)=\frac{\theta^ {x}e^{-\theta}}{x!}, x=1,2,...; 0<\theta$$
What ...
3
votes
1answer
141 views
Linear regression with error dispersion dependent on the independent variable
Suppose $y=ax+z$ where $x, y, z$ are random variables with range in $\mathbf R$, $\mathbf E[x]=0$, the probability distribution $p(z|x)$ is
1) normal distribution $N(0,\sigma(x)^2)$ with mean $0$ ...
2
votes
0answers
108 views
Recommended point estimate for non-normal distribution?
I have a rather non-normal marginalized posterior for some parameters, resulting from a Bayesian MCMC. Example:
I know that the actual distribution is what truly represents the parameter, but I need ...
2
votes
3answers
1k views
Deriving likelihood function of binomial distribution, confusion over exponents
This question focuses on a specific aspect of this one:
How to derive the likelihood function for binomial distribution for parameter estimation?
In my own derivation, I start with:
$$f(x\mid p) = ...
2
votes
0answers
16 views
Estimating the number of tokens in a pile
We have a pile of tokens numbered from $1$ to $n$, where $n$ is unknown. $k$ tokens are drawn from the pile at random without replacement(i.e. the numbers that we get from tokens are unique). Say the ...
0
votes
0answers
94 views
Simultaneous estimation of a group of linear model (regression) parameters
Suppose $y=ax+z$ where $x, y, z$ are random variables with range in $\mathbf R$, $\mathbf E[x]=\mathbf E[z|x]=0$ and $a$ is a constant. Note the distribution of $z$ conditioned on $x$ depends on $x$. ...
0
votes
1answer
30 views
Estimating standard deviation when observing means of various block sizes
I have a sequence $X_i$ of iid random variables (you may assume gaussian distribution if you like) but I only observe the mean value of disjoint blocks of various sizes of $X_i$. E.g. $M_1 = \frac14 \...
2
votes
1answer
995 views
Proof that posterior median is the Bayes estimate of absolute loss?
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 ...
3
votes
1answer
163 views
MVUE is unique - wrong proof?
Here is the proof of "MVUE is unique" that my lecturer gave:
Now I understand the following:
The first expansion is done using the formula for the sum of correlated random variables (https://en....
2
votes
0answers
25 views
Unbiased estimate of sign of mean
Consider the set $\mathcal{P}$ of probability distributions that have a finite first moment and define the function $\operatorname{sgn} :\mathcal{P} \to \mathbb{R}$ as
$$
\operatorname{sgn}(\mu) = \...
1
vote
0answers
18 views
Two point estimates fall into same confidence interval
Say I have two populations, and I am computing a sample mean and a corresponding (1-$\alpha)100 \%$ confidence interval for each. Now, I understand that if the confidence intervals are disjoint, then ...
2
votes
1answer
42 views
It what situation is a distribution known to be symmetric, but about an unknown location?
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 ...
0
votes
1answer
68 views
Minimal sufficient statistic whose dimension is less than dimension of parameter
Consider following example:
Suppose $ X\sim N(0, \sigma^2) $,
consider a random sample of size one from this population.
Clearly $X$ is sufficient statistic but $ |X| $ is minimal sufficient ...
3
votes
1answer
1k views
Efficiency, Precision, Accuracy, and Consistency
Can anyone please explain me the statistical terms efficiency, precision, accuracy, and consistency in plain language with easy example (hopefully by daily life example)?
So far, my understanding of ...
2
votes
1answer
39 views
Multidimensional Bayes point estimates
Consider the posterior distribution $p(\theta|x)$. We aim to find a "good" estimate of the random variable $\theta$. The Bayes risk associated with the loss function $L(\hat{\theta}, \theta)$ is ...
1
vote
0answers
118 views
comparison of two estimators
Assume we have a data set $\mathbf{x}_{n} = (x_{1}, \dots, x_{n})$. Let $\delta_{1}(\mathbf{x}_{n})$ and $\delta_{2}(\mathbf{x}_{n})$ be two consistent estimators of some parameter $\theta \in R^{k}$. ...
9
votes
3answers
309 views
Does a Bayes estimator require that the true parameter is a possible variate of the prior?
This might be a bit of a philosophical question, but here we go: In decision theory, the risk of a Bayes estimator $\hat\theta(x)$ for $\theta\in\Theta$ is defined with respect to a prior distribution ...
5
votes
1answer
71 views
About the variance of a weighted sum
What can be said about the variance of the following quantity
$$
\frac{1}{n}\sum_{i=1}^n \left(\frac{f(x_i)}{\sum_{j=1}^nf(x_j)}-b\right)g(x_i) ?
$$
Here, $b \in [0, 1]$, and the $x_i$s are i.i.d ...
2
votes
2answers
64 views
Computing variance from a set of samples
My dataset contains a set of samples from a set of normal RVs. Each RV is normally distributed with equal variances and varying means. However, I have only two samples from each RV.
How to estimate ...
0
votes
1answer
119 views
MME for exponential family
Let $X_1, X_2,...,X_n$ be iid random variables having pdf
$$f(x|\theta) = \frac{1}{x \sqrt{2\pi\theta}}e^{(-[\log x]^2/[2\theta])} I_{(0,\inf)}(x)$$ where $\theta > 0$. Determine the MME of $\theta$...
0
votes
0answers
59 views
Determining an MLE
I am trying to find the MLE of $\theta$ for $X_1, X_2,...,X_n$ with the following pdf:
$$f(x|\theta) = \frac{\log(\theta)}{\theta - 1} \theta^{x}$$
I determined the joint density and got:
$$\prod_{...
0
votes
1answer
2k views
MLE for variance of a lognormal distribution
Let $X_1, X_2,...,X_n$ be iid random variables having pdf
$$f(x|\theta) = \frac{1}{x \sqrt{2\pi\theta}}e^{(-[\log x]^2/[2\theta])} I_{(0,\inf)}(x)$$ where $\theta > 0$.
Determine the MLE of $\...
-1
votes
1answer
42 views
Find the Method of moments estimate
So here i am getting E(X)=1/2,E(X^2)=0 and E(X^3)=Theta squared/4
How do i proceed now?How do i use the given x values ?
1
vote
1answer
61 views
What is the maximum likelihood estimator of the given distribution?
Let $X_{1},...,X_{n}$ be independent random variables with $X_{k}$ having the normal distribution with mean $k\theta$ and variance $\sigma^2$. Find the maximum likelihood estimator of $\theta$.
My ...
0
votes
1answer
107 views
Maximum likelihood estimator of $\theta$ of the following probability distribution
Let $x_{1}=-2,x_{2}=1,x_{3}=3,x_{4}=-4$ be observed values from the following density function:
$f(x|\theta)=\frac{e^{-x}}{e^{\theta}+e^{-\theta}}$ where the support is $-\theta \leq x\leq\theta$.
...
