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Questions tagged [point-estimation]

Point estimation is the application of an estimator to the data in order to learn about a certain population parameter.

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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 ...
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$T_0$ 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 $T_1$...
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21 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 ...
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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 ...
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23 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 ...
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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 ...
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41 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 ...
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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 ...
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1answer
26 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 ...
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52 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 ...
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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 $0$ ...
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76 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 ...
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3answers
802 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) = ...
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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 ...
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78 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 $x$. ...
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29 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 \...
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419 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 ...
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Restricting Properties of Point Estimators in Discrete, Unordered Parameter Spaces

A common principle used to deal with the fact that there are typically no uniformly best estimators (in the sense that they uniformly have least frequentist risk) seems to be to restrict the space of ...
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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....
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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) = \...
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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 ...
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1answer
39 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 ...
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53 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 ...
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533 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 ...
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1answer
33 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 ...
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58 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 $\theta \in R^{k}$. ...
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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 ...
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1answer
66 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 ...
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2answers
59 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 ...
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1answer
84 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 $\theta$...
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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_{...
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1answer
1k 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 $\...
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39 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 ?
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54 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 ...
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86 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 $-\theta \leq x\leq\theta$. ...
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Are exponential families uniformly continuous?

It is common to consider log-likelihood $$l(x_1,\cdots,x_n,\theta) :=\frac{1}{n}\sum_{i=1}^n \log f\left(x_i,\theta\right)$$ where $f$ is density function of $X$. Under what conditions is the ...
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What is the intuitive meaning of autocorrelation - when is $r_xx[0] = 1$

Consider a linear model $y_n = \beta x_n + v_n$ where $n$ represents the time index, $x$ is the input data source $\hat{\beta}_{OLS} = (X^TX)^{-1}X^Ty$ is the ordinary least square estimate of the ...
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1answer
74 views

Estimating a GCD

I have two related questions. Question 1: Let $k_1, \ldots k_n$ be positive integers, and $\alpha_1, \ldots \alpha_n \in (0,1)$ be such that $\sum_{j \leq n} \alpha_j = 1$. Suppose $\langle X_m \...
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1answer
53 views

Best plausible estimate of parameter of hypergeometric distribution

Consider the Urn problem where a hypothetical urn contains a finite number of $m$ balls, $r$ of which are black and $m-r$ are white. We take a random sample without replacement of $n$ balls and ...
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1answer
57 views

How would you find the MLE of this pairwise exponential distribution?

$$X_1 \dots X_n \sim f_\theta(x) = \begin{cases} \exp(\theta-x) & x\geq\theta\\ 0& otherwise \end{cases}$$ We have $$L_x(\theta) = \begin{cases} \exp(n\theta-\sum_{i=1}^n x_i) & x_i\geq\...
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1answer
263 views

Deriving confidence intervals from a LOOCV of a GAM

Say that I have some observations, $y_1, y_2, ...y_n$ that are described using a generalized additive model (GAM). A Leave-One-Out Cross-Validation (LOOCV) is then performed where each observation $...
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sufficient statistic for n-point distribution that depends of one parameter

I'm studiying the concept of sufficient statistic, but I have a question that I can't resolve... I hope someone can help me. My question is the next: I suppose that I do n independent trials with m ...
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1answer
88 views

Does nonparametric bootstrapping work on correlated data?

I have $m$ weakly stationary observations $X_1,X_2,\cdots,X_m$ from a Markov chain. I want to estimate the variance of the mean. My idea was to use nonparametric bootstrapping to make $n$ bootstrap ...
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149 views

uniqueness of MVUE

Let $(X_1,...,X_n)$ be a random sample from an exponential distribution with pdf $f(x,\theta) = \frac{1}{\theta}\exp{\frac{-x}{\theta}}I_{(0,\infty)(x)}$ , which of the following estimators is the ...
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1answer
390 views

Numerically computing the MLEs using Newton's method and the invariance proprty

I'm currently addressing maximum likelihood estimation for a three-parameter probability distribution with all parameters being positive real-valued. I'm using Newton's method to calculate the MLEs ...
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What should the estimated proportion be for the population when the sample proportion is 1?

I am auditing insurance claims for fraud. In my random sample, all the claims are fraudulent. Could I say for the population that the estimated proportion of fraud is 1? I know how to compute a ...
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46 views

Is it possible to have a point estimate outside of confidence interval? [duplicate]

Is it possible to have a point estimate outside of confidence interval? Could anyone give me an example?
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462 views

Find the UMVUE of $6\theta^2$ given $f(x\mid\theta) = \frac{1}{2\theta^2} e^{\frac{-\sqrt{x}}{\theta}} I_{(0,\infty)}(x)$

Given $X_1, X_2,\ldots, X_n$ are i.i.d rvs with pdf $f(x\mid\theta) = \frac{1}{2\theta^2} e^{\frac{-\sqrt{x}}{\theta}} I_{(0,\infty)}(x)$ for $\ \theta > 0$. Find the UMVUE of $\ 6\theta^2$, and ...
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2answers
275 views

MLE estimator of $P(X \leq c)$ for $X~ normal(\theta,1)$

I need to find MLE estimator of $P(X \leq c)$ where X is $Normal(\theta,1)$, c is fixed. Note: $X_1,...,X_n$ are a random sample drawn from $Normal(\theta,1)$ Any hint on how to approach this ...
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1answer
709 views

What is the difference between complete statistics and complete family of distributions?

I fail to understand when we call a family of distribution is complete and when a statistic is complete. What is the difference between both?, Is there a relation between them? Please provide examples ...