Questions tagged [point-estimation]
Point estimation is the application of an estimator to the data in order to learn about a certain population parameter.
211
questions
0
votes
1
answer
23
views
Derivation of the formula for the asymptotic relative efficiency of two estimators with different estimands
Background
In their book, Huber & Ronchetti (pp. 2-3) compare the efficiency of the mean absolute deviation $d_n$ with the standard deviation $s_n$ with the following formula:
$$
\operatorname{ARE}...
- 26.4k
0
votes
0
answers
13
views
Comparison of samples of optimal objective function values of a deterministic optimisation problem solved by a stochastic optimisation solver
Motivating Problem
Assume that I must pack tins of soup into a box, where the box has a fixed volume. I can choose the shape of the soup tin, and the volume of the soup tin, and the packing ...
4
votes
1
answer
136
views
Unbiased estimator for $\mu_1/\mu_2$
Let $X_1,X_2,\ldots,X_n$ and $Y_1,Y_2,\ldots,Y_n$ be independent random samples from $N(\mu_1,1)$ and $N(\mu_2,1)$ populations respectively with $\mu_2\neq0$.
I need to find an unbiased estimator for $...
- 43
0
votes
0
answers
25
views
How should one re-formulate hypothesis after hypothesis testing or whether or not to discard data after significance testing or archive it?
@Nalzook summarized what he thinks Ronald Fisher did.
Ask a question.
Propose a null hypothesis based on the question.
Do some experiments, and collect some data.
Assuming the null is true, calculate ...
- 782
8
votes
4
answers
393
views
How would a bayesian estimate a mean from a large sample?
What would a bayesian do if she wanted to do inference for the mean with a large sample but has no idea of the underlying distributions?
A frequentist statitician would use the sample mean as a point ...
- 1,629
1
vote
1
answer
66
views
comparison of proportion to a population CI
I am comparing the % of minorities from my organization to a population % of minorities, to see if it is high or low. I have data for my whole organization (not a sample) so I do not show CIs. The “...
- 197
0
votes
0
answers
14
views
Friendly reference for point estimation that covers the following
I took a mathematical statistics class in college which covered the following:
Cramér–Rao bound
Lehmann–Scheffé theorem
Rao–Blackwell theorem
I found these theorems very beautiful and want to ...
0
votes
0
answers
24
views
Is it appropriate to use linear regression when the response has a confidence interval?
I am looking to predict a response, $\hat{y}$, and have some information about how the response is generated. I know, for example, that it's a score that is computed from an array of weighted factors. ...
0
votes
0
answers
21
views
What is known, in principle, about the possibility of approximating the random discrepancy between a statistical estimate and its parameter?
The difference between the value of a statistical estimate and its parameter's value is almost never exactly $0$. For example, $r - \rho$, for a unique sample $r$, is likely to be some non-zero ...
- 122
2
votes
0
answers
32
views
An estimation method/algorithm for estimating the value of a specific parameter in a convex function
I am looking for an estimation/iteration process to estimate the value of a specific unobserved parameter of a convex function that fits the observed data of the other variables closely. Specifically, ...
- 21
-1
votes
1
answer
350
views
MLE - CDF vs PDF as the likelihood-function?
Would maximum-likelihood estimation: with the cumulative-distribution function as the likelihood-function and the probability-density function as the likelihood-function, yield the same/equal ...
- 206
2
votes
0
answers
90
views
Large sample properties of classical estimator for single scale parameter
This question was first posted on Math Stackexchange and I was told in the comment it would be a good question on Stats Stackexchange, since it comes from the well-established theory of point ...
- 91
0
votes
1
answer
41
views
Is this point estimate for mean biased?
I was wondering if this point estimate for mean: $\frac{1}{n+1}\sum_{i = 1}^{n}x_i$ is biased?
My first thought was that $\frac{1}{n+1}\sum_{i = 1}^{n}x_i \neq \frac{1}{n}\sum_{i = 1}^{n}x_i$, so then ...
0
votes
0
answers
49
views
Efficiency of two estimators for a sample from a Bernoulli population
Given a Bernoulli population, I have two estimators for a random sample of size $n$:
$T_1=\frac{\sum\limits_{i=1}^n X_i + 2X_n}{n+2}$
$T_2=\frac{\sum\limits_{i=1}^{n-2} X_i + 2X_n}{n+2}$
I want to ...
- 210
0
votes
0
answers
31
views
Why do we divide by n when solving for the Cramer-Rao Lower Bound here?
"Let $X_1,...,X_n$ be iid Bernoulli(1,$p$), with $p$ unknown. Find the CRLB for the variances of unbiased estimators of $p$."
With pdf $p^x(1-p)^{1-x}$, the derivative of the log function is ...
- 1
1
vote
1
answer
23
views
Forming a consistent estimator for the area under the regression line
I am trying to solve the following problem:
Take the following simple linear regression model, where $x_i \in \mathbb R$:
$y_i=\beta_0 + x_i \beta_1 + \epsilon_i$
Given that:
$\mathbb E[\epsilon_i]=...
- 135
0
votes
2
answers
83
views
Estimating $1/a$ for following pdf using method of moments estimation
A random sample of size $n$ is being drawn from a population with pdf as:
$$f(x) = \begin{cases} (a + 1)x^a & \text{for }0<x<1, \\
0 & \text{otherwise.} \end{cases}$$
Can we express the ...
- 107
0
votes
0
answers
17
views
estimating a population-average model with known mean and standard deviation
I have a model with some differential equations describing the effect of a drug. There are 100 rat samples, we only know the mean value and its standard deviation for measured drug response. Now I ...
- 1
0
votes
1
answer
112
views
Generalized Bayesian estimator (rule) of θ
Question:
Let $X_1, · · · , X_n$ be a random sample from $Poisson(θ)$. The prior for θ is $G(α, β)$
Find the Bayesian estimator (rule) of θ under the SEL(squared error loss).
Find the generalized ...
- 165
0
votes
1
answer
60
views
Conjugate Prior for Alpha Power Inverse Weibull Distribution
Let $X$ has Alpha Power Inverse Weibull (APIW) distribution with pdf
$f(x) = \frac{\log \alpha}{\alpha - 1} \lambda \beta x^{-(\beta+1)} e^{-\lambda x^{-\beta}} \alpha^{e^{-\lambda x^{-\beta}}}, \; x&...
- 41
0
votes
0
answers
38
views
A hypothesis test that conditional expectation (i.e. regression line) is above some number in a region of the factor space
In my work we want to know whether some variable of interest satisfies some threshold. Maybe imagine that we ask questions like whether a widget has at least a 60% probability of functioning. ...
- 1,173
2
votes
2
answers
95
views
How to combine population proportions and confidence intervals?
Say I've looked at three random samples, one each from three populations of students. I found that a few students in each sample didn't turn in a fieldtrip permission slip, leaving me with the ...
- 21
6
votes
3
answers
732
views
What happens if I change the range of a flat prior for Bayesian inference?
I am working through an example on doing Bayesian inference on binomial distribution using a flat prior, and trying to understand the impact of choosing a prior. I know that if we work with a flat ...
- 61
4
votes
2
answers
277
views
Bayesian point estimate of a random sample
I am new to statistics and some concepts are not clear to me. I have a random sample that is distributed as a Binomial with parameters $k=2$ and $\theta$ unknow.
Using a Bayesian approach I must give ...
- 151
1
vote
0
answers
25
views
What are the implications of a low coverage in multiple imputation?
When testing multiple imputation algorithms in simulations, the bias of the examined estimates and the 95% coverage rate are often used as a quality metric. I understand that it is generally ...
- 11
0
votes
0
answers
32
views
Is it possible to estimate the Hessian as the covariance of primal and cotangent?
Let's say we have a function
$$f: \mathbb R^n \to \mathbb R.$$
Can we numerically approximate the Hessian $f''(x)$ as
$$\textrm{Var}(a)^{-1} \textrm{Cov}(a, f'(a))$$
where
$$E(a) = x?$$
- 14k
3
votes
1
answer
262
views
Computing the Bayes estimator under weighted squared error loss - interchanging derivatives and integrals
I am revisiting some self-study assignment questions in elementary theoretical statistics that I previously had difficulty with. I would appreciate some clarity on a few points in the following ...
- 2,236
1
vote
0
answers
33
views
Standard error of estimate of $\lambda^2$
In a problem, given $n$ observations from $Poisson(\lambda)$ , I have to get an unbiased estimator of $\lambda ^2 $ and the corresponding standard error.
I used the efficiency test to get the unbiased ...
- 23
0
votes
1
answer
89
views
What Cramer-Rao bound should I use?
I have been researching about the Cramer-Rao bound and I have found two inequalities:
$$\text{Var}\left(\hat{\theta}\right)\geq\frac{1}{\text{E}\left[\left[\frac{\partial}{\partial\theta}\ln f(X;\...
- 3
0
votes
1
answer
59
views
Estimation with MCMC [closed]
I would like to ask some high-view questions about MCMC. I do not have a specific example, I just want to get a general intuitive idea.
Suppose I have a data set $X$ and a rather complex model with ...
- 65
1
vote
1
answer
45
views
Why Do Distributional Forecasts Need to Produce Normally-Distributed Forecasts to be Ensembled/Combined?
I am forecasting a collection of different types of items, using many different forecasting techniques. Some of the techniques I use take the input data as is to produce a distributional forecast. ...
- 31
2
votes
0
answers
71
views
Winsorized mean - trimming furthest points instead of both endpoints
I'm wondering if the Winsorized mean can be improved by trimming the 5% farthest points from the mean instead of trimming 5% on each endpoint. Concretely:
Consider the Winsorized mean, where we ...
- 6,204
2
votes
1
answer
70
views
Estimating Means of a Bivariate Normal Distribution where some parameters are known
I am trying to figure out how to estimate means of a bivariate normal distribution from a sample when some of the parameters are already known.
let
$$
\boldsymbol{x} = \begin{bmatrix}
x\\
y\\
\end{...
- 35
0
votes
0
answers
51
views
Unbiased estimator and getting estimate from estimator
I got a unbiased estimator but I don't know how to interpret it and adjust it to get estimate.
The original problem is to find out the unbiased estimator for $\lambda$ in Zero-truncated Poisson ...
- 11
4
votes
1
answer
436
views
Trimmed, weighted mean
The trimmed mean (or truncated mean) is a robust version of the mean, designed to be robust to outliers. I am wondering what is the right trimmed version of a weighted average.
If I have a sample ...
- 6,204
1
vote
0
answers
23
views
Mean-square convergence of maximum likelihood estimators: Examples?
From what I've gleaned from the literature, Cràmer, in his 1947 monograph Methods of Mathematical Statistics, proved convergence in probability of an MLE under certain regularity conditions. ...
- 265
1
vote
1
answer
78
views
Point estimate and confidence interval for the difference in $x_1$ between two groups for which a particular $y$ is achieved
I have two variables (continuous $x_1$, control/treatment $x_2$) that I want to use to predict a probability. Domain knowledge suggests that the relationship is roughly linear in the log-odds, so I am ...
- 35.7k
0
votes
0
answers
31
views
Expression for the Likelihood Function in Point Estimation
I came across this question in my statistics textbook, but I'm struggling to come up with an expression for the likelihood function. Here is the question:
Assume that there are three possible traits ...
2
votes
1
answer
39
views
Determining the minimum number of tosses, for heads to be twice more likely than tails in the next toss
I would like some help with the following statistical problem.
We have a coin with probability $\theta$ for heads, with prior for $\theta$ being a Beta(a,a) distribution (a is a known parameter).
...
- 21
0
votes
0
answers
13
views
Is there a term for an estimator's probability of estimating an impossible estimand value?
This is similar to Mean Squared Error and Mean Absolute Error but in this case the loss function assigns estimates to $0$ when they are a possible estimand and $1$ when they are impossible.
As a ...
- 1
0
votes
1
answer
3k
views
What's the advantage of a point estimate over an interval estimate?
A point estimate is :
A single numerical value that is used to estimate the corresponding population parameter.
Whereas an interval estimate is :
An estimate that consists of two numerical values ...
- 37
2
votes
0
answers
86
views
Consistent estimator of $p^2$
$(X_1, X_2,...,X_n)$ is a random sample of size $n$ from $Bernoulli(p)$ distribution. $S_n=\sum_{i=1}^nX_i$. I have to check whether $\frac{S_n(S_n-1)}{n(n-1)}$ is a consistent estimator for $p^2$.
$...
- 225
1
vote
0
answers
15
views
Is the population parameter more likely to occur closer to the sample statistic? [duplicate]
When we use the sample statistic to find a confidence interval, is there any reason to using the sample statistic after that, when we can instead refer to the confidence interval?
For example, let's ...
- 111
1
vote
1
answer
643
views
Most Efficient Estimator and Uniformly minimum variance unbiased estimator
I am studying Estimation theory from "Introduction to theory of statistics" by "Mood and Graybill".
After completing I thought I understood UMVUE (uniformly minimum variance ...
0
votes
0
answers
80
views
Taking Expectation Over Inverse Sum of Indicator Functions?
I'm working with a zero inflated Poisson distribution that has the following pmf:
$$f(y|w,\lambda)=wI[y=0]+(1-w)\frac{e^{-\lambda}\lambda^{y}}{y!}$$
I would like to find the expectation of the ...
- 134
2
votes
1
answer
21
views
Variance Estimator Change if we know Population Mean? (Normal dist. example)
For a normal distribution $N(\mu, \sigma^2)$ a commonly used unbiased and consistent estimator of variance is
$$\hat \sigma^2=\frac{\sum_ix_i^2 + n(\bar x)^2}{n-1}=\frac{\sum_i(x_i-\bar x)^2}{n-1}$$
...
- 134
3
votes
1
answer
48
views
Outlier detection in point estimates
I have to perform outlier detection on population estimates for certain variables at the city level. For example, I might be estimating median income for a city and I want to know if there are any ...
- 1,408
1
vote
0
answers
34
views
How to estimate a proportion as a step function of time
There is a black-box mechanical process that, at any time, may be either succeeding or failing. It is known that after an initial disturbance, the process will fail $X\%$ of the time for $N$ seconds, ...
- 111
4
votes
1
answer
66
views
Strong consistency in quantum estimation problem
I'm reading the paper: Strong consistency and asymptotic efficiency for adaptive quantum estimation problems by Akio Fujiwara.
In this paper, describes the next adaptive scheme of estimation:
"...
- 53
2
votes
1
answer
1k
views
It is possible to find point estimate of population mean and population variance when confidence interval of population mean is given?
Let's say that somehow $100(1-\alpha)\%$ confidence interval of population mean $\mu$ is known as $(a,b)$ and the number of samples is $n$. Is it possible to infer point estimates of population mean ...
- 23