Questions tagged [statistical-inference]

The area of statistics that focuses on taking information from samples of a population, in order to derive information on the entire population.

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Cake slicing hypothesis problem

A cake weighing one kilogram is cut into two pieces, and each piece is weighed separately. Denote the measured weights of the two pieces by $X$ and $Y$ . Assume that the errors in obtaining $X$ and $Y$...
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44 views

Find an estimator of $N$ (hypergeometric)

A forest has $N$ (unknown) monkeys. A random sample of $n$ monkeys is selected from the forest, tagged and released back into the forest. After a few days, a random sample of $m$ monkeys is selected ...
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1D Bayesian Inference clarification

I'd like some help making sure I understand a 1D Bayesian inference problem. Stats.stackexchange wasn't helpful. Say I have a data vector which is an array of the number of flu cases reported weekly ...
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53 views

Are there generalised rules for generating heuristics from data?

Heuristics seems to be more of an art than a science, like a gut-feel supported by data; I might be wrong. Are there algorithms for mathematically generating heuristics from data, like pruning a ...
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Karlin Rubin Theorem UMP (Uniformly most powerful test ) Is it wrong?

Suppose $X_1, X_2, X_3,\ldots, X_n$ are i.i.d. random variables with a common Poisson$(\lambda)$ distribution. $$X=(X_1, X_2, X_3,\ldots, X_n)$$ and $g(λ)=\lambda(1 - e^{-λ})$ , with $(λ>0)$ ...
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25 views

What is the value of the likelihood ratio test statistic for a given poisson distribution?

In past weeks, the average number of life insurance policies sold per week was $θ = 3$. However, this week he has sold 6 life insurance policies. Based on this single observation of $x = 6$, what is ...
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20 views

How does one represent an econometric model? (an application of the ergodic theorem to stationary time series).

THE FRAMEWORK: Let $X_1 , X_2 , \dots, X_n$ be observed random variables. Then one could decide to model them as IID Gaussians with variance one and mean $\mu$. That is we choose as our model the ...
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Why is the bottom limit of the conditional probability $x$ in Bayesian Statistics?

I am learning bayesian statistics and was stuck when trying to understand the following example: Romeo and Juliet start dating, but Juliet will be late on any date by a random amount X, uniformly ...
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42 views

UMVUE of two normal distributions

Let $X_1, . . . , X_n$ be a random sample from $N(µ_X, σ^2_X)$ and let $Y_1, \ldots, Y_n$ be a random sample from $N(\mu_Y , \sigma^2_Y$), where $\mu_X \in \mathbb R$, $\mu_Y \in \mathbb R$, $\sigma^...
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45 views

Basu's theorem to show independence

Using Basu's theorem, prove that $\sum\limits_{i = 1 }^n {(X_i - X_{(1)}) }$ and $X_{(1)}$ are independent for any $(\theta, \lambda)$. You may assume that $X_{(1)}$ is complete and sufficient for $θ$ ...
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73 views

UMVUE of $P(X_1 ≥ t)$ for a two-parameter exponential distribution

I'm attempting to find $(a)$ The UMVUE of $λ$ when $θ$ is known. $(b)$ The UMVUE of $θ$ when $λ$ is known. $(c)$ The UMVUE of $P(X_1 ≥ t)$ for a fixed $t > θ$ when $λ$ is known. I'm new to the ...
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48 views

Use MGF to show $\hat\beta$ is a consistent estimator of $\beta$

Suppose that $X_1,....,X_n$ is a random sample from a gamma distribution with parameters $\alpha= 2, \beta$. \begin{equation} f(x)= \frac{x e^{(-x/ \beta)}}{\beta^2}, x>0 \end{equation} (a) Find ...
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54 views

The Cramer-Rao lower bound of $e^{-(x-\theta)}\exp(-e^{-(x-\theta)})$

Let $f(x;\theta ) = e^{-(x_i-\theta)}exp(-e^{-x_i-\theta)})$ How do I find the Cramer-Rao lower bound? the log likelihood is $l(\theta;x)=\Sigma_{i=1}^n{[-(x_i-\theta )-e^{-(x_i-\theta )}]}$ and ...
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26 views

Find supremum of Type II error in Neyman-Pearson framework

Let $X_1,\dots,X_n$ be an iid sample from an $N(\theta,1)$ distribution. We want to test $H_0:\:\theta=0$ against the alternative $H_1\:\theta \neq 0$ using the test statistic $$T_n(X_1,\dots,X_n) = \...
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73 views

Training a Boltzmann Machine (Non restricted)

I'm reading through Neural Network and Deep Learning by Charu C. Aggarwal, more specifically chapter 6 which treats Restricted Boltzmann Machines, section 6.3. treats Boltzmann Machines. I'll try to ...
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24 views

How to find the Most likelihood estimator for

$f\left( x\right) =\begin{cases}e^{-\left( x-\theta \right) }, x \ge \theta\\ 0\end{cases}$ We want to find the estimator for $\theta$. I already have the solution but I don't understand it. They ...
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33 views

Proving t distribution in small sample size and population variance known case.

I am asking a follow up question to this question. Why prefer the t-score when the sample size is low? I have seen mathematical proofs that in variance unknown case, the t-statistic follows a t-...
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33 views

are the vectors obtained from PCA of a non-negative matrix always non-negative? [closed]

I'm curious as to whether this is the case, and whether knowing this fact can help us develop better "PCA analogues". By vectors, I mean the weight vectors obtained from PCA.
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Calculating the Kullback Lieber information Criterion (Quasi Maximum Likelihood Method)

Let the observed random variables $X_1,X_2$ be independent and normally distributed with zero mean and variance respectively $\sigma_1^2$ and $\theta \sigma_1^2$, where $\theta \in \mathbb{R}$. The ...
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87 views

Find the UMVUE for $\mu ^{2}$ by assuming $\sigma ^{2}$ is unknown.

Suppose that $X_1,X_2,...,X_n$ is a random sample from normal$(\mu, σ^2)$. Find the UMVUE for $\mu ^{2}$ by assuming $\sigma ^{2}$ is unknown. My approach: The distribution of the sample mean, ...
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Estimating population size of endangered parrots

I am trying to find the best way to analyse the results of a study I have just coordinated. In short, I want to be able to estimate the total population size of a species of parrot based on ...
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How to show that the multivariate normal depends on the data only through $\Sigma x_n$ and $\Sigma x_n x^T$

I've shown something similar for the 1-dimensional case. That $\Sigma x_n$ is the sufficient statistic of the gaussian mean and $\Sigma x_n$,$\Sigma x_n^2$ are the sufficient statistics of the ...
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43 views

Correct interpretation of confidence interval

I think this question may have been asked in one form or another, but I did not find an answer that I understood or thought was satisfactory...so I apologize if it has been explained. My ...
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How reasonable is the linearity assumption in regression analysis?

As someone who moved from physics to economics, there are some things about regression analysis that have always bothered me. One of the most basic assumptions in econometrics is that for studying the ...
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1answer
47 views

Find minimal sufficient statistic for truncated exponential distribution

Let $X_1, ..., X_n$ be iid $f(x; \theta, \lambda) = \dfrac{\lambda e^{-\lambda x}}{1-e^{-\lambda \theta}}$ for x $\in [0, \theta]$. I want to find A minimally sufficient statistic for $\theta$ ...
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1answer
103 views

Inference regarding the mean lifetime of a bulb using a new technique

The lifetime in hours of each bulb manufactured by a particular company follows an independent exponential distribution with mean $\lambda$. We need to test the null hypothesis $H_0: \lambda=1000$ ...
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39 views

Testing order statistics and finding if it is MP test using Neyman Pearson lemma.

Problem: Suppose X1,X2,...Xn follows exponential (mean=1). Only the largest values of Xi are recorded and denoted by T. To test H0:n=5 vs H1:n=10, show that the most powerful test of size 0.05 rejects ...
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How to perform hypothesis testing on samples formed by matrices of measurements?

I need to test if a treatment has taken effect in a certain group of patients for which I make a measurement of the relevant variables before doing the treatment and afterwards. Usually, I would do a ...
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Sampling distribution of a functional T

While studying the bootstrap method, I came across with the following definition of the sampling distribution of a functional T: Let's say $X_1,...,X_n$ are i.i.d with distribution function $F$, then ...
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1answer
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Prove or disprove that $Y_n/n$ converges in probability to 0.

Let $X_1, X_2, ... , X_n$ denote a random sample from a distribution with pdf f(x), and let $Y_n$ denote the maximum of the sample. Prove or disprove that $Y_n/n$ converges in probability to 0 for ...
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Hypothesis testing for the lifetime of bulbs using subsystems

The lifetime in hours of each bulb manufactured by a particular company follows an independent exponential distribution with mean λ. To test the null hypothesis H0 : λ = 1000 against the alternative ...
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Does a Best Unbiased Estimator for parameter of exponential distribution exist?

For an exponential distribution $$X \sim \exp(\lambda) = \lambda \ \exp(-\lambda\ x),\ x>0$$ Does there exist an Best Unbiased Estimator (BUE) for $\lambda$ i.e. can it achieve the lower bound ...
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How to find a confidence interval for difference in response rates?

I found this question on a past exam: I know how to find a confidence interval with one mean: $\bar x \pm z_{\alpha/2} \frac{\sigma}{\sqrt n}$ and I also know how to do it with two means however ...
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Find the best unbiased estimator for $\mu^T \mu + 1^T \mu$.

Let $X_1, X_2, \dots, X_n$ independent n-dimensional vectors with the same distribution $N(\mu, I)$. Find the best unbiased estimator for $$ \mu^T \mu + 1^T\mu $$ where $1^T = (1, 1, \dots, 1)$. ...
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Real Examples of Misleading Statistics

I need to give a presentation to a group of students on Tuesday about why one needs to be careful when examining statistics or mathematical results in the media or online. In his book How Not To Be ...
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Is the minimum-variance unbiased estimator (MVUE) for P^2 of Bernoulli(p) asymptotically normally distributed

let$ ~X_1, ...X_n ~$be iid Bernoulli$(p)$ let $W_n$ is the MVUE of $p^2$ Find $W_n$ and determine it is asymptotically normally distributed in the sense that $ \sqrt{ n}(W_n − \mu)$ converges in ...
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Software based on EM algorithm (or downloadable free code) for mixture model of nonnormal (preferably customizable PDF) component PDFs

Don't know whether this is the right place to ask the question. I am looking for an EM algorithm based software (or computer program, like EMMIX.f by Mclachalan et al. (1999)) to do mixture modelling ...
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1answer
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On a particular day let $X_1,X_2,X_3$ be the number of boys born

On a particular day let $X_1,X_2,X_3$ be the number of boys born before the first girl is born in hospitals $1,2,3$ respectively.If the observations are $X_1=0$ ,$X_2=3$ and $X_3=2$, find the most ...
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How to find correlation when two random variable W1 =4X + 3Y and W2 = 4X + 9Y are given as a linear equation of x and y. [closed]

How to find correlation when two random variable $W_1 =4X + 3Y$ and $W_2 = 4X + 9Y$ are given as a linear equation of $X$ and $Y$.
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Tobit Binary Variables

I am considering the censoring model $$ Y^* = a + Bx + u \\ Y = \max\,\{Y^*; 0\} \\ u \sim N(0,1). $$ So it is a Tobit model. I derived the expression for $E[Y\mid x, Y^*>0]$, which is the Tobit ...
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Sufficient statistics not depending on the parameter

A statistics $T(X)$ is sufficient statistics for $\theta$ if the conditional distribution of the sample $X$ given the value of $T(X)$ does not depend on $ \theta$. ( this is the definition of ...
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70 views

Finding an unbiased estimator of $(1+\lambda)e^{-\lambda}$ for Poisson distribution

If $X_1,X_2,\ldots,X_n\sim \mathrm{Pois}(\lambda)$, find an unbiased estimator of $(1+\lambda)e^{-\lambda}$. I am actually supposed to find the UMVUE of $(1+\lambda)e^{-\lambda}$. but I first have ...
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Clarification regarding Parameter Estimation (Andriy Burkov's book)

So I recently decided to read Andriy Burkov's "The 100-Page Machine Learning Book" and got confused in Chapter Two (Page 11) where he discusses Parameter Estimation techniques. A picture of the ...
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62 views

UMVUE of a Bernoulli distribution

Suppose that $ (X_1,X_2, \ldots, X_n) $ be a random sample from Bernoulli (p). Find the UMVUE of $$\tau(p)= (1-p)+ e^{2}p $$ My approach: I know $T(X) = \sum_{i=1}^n X_i$ is a complete sufficient ...
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Cdf of order statistic [duplicate]

In the general case, if we have a random sample of iid continuous random variables $X_1,...,X_n$ and we define $Y_i=F(X_{(i)})$ (the cdf of the order statistic) for i=1,..,n, then what type of ...
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Let $Y_1,…,Y_n$~Ber(p). Determine if $\hat{p_2}^2=\bar{Y}^2$ is a consistent estimator of $p^2$.

Let $Y_1,...,Y_n$~Ber(p). Determine if $\hat{p_2}^2=\bar{Y}^2$ is a consistent estimator of $p^2$.So, first I tried to see if $\hat{p_2}^2$ was unbiased. It follows $E(\hat{p_2}^2)=E(\bar{Y}^2)=Var(\...
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64 views

Sufficient statistic for Double Exponential

Let $X_1,X_2,...X_n$ be a random sample from $f(x,\theta)=\frac{1}{2 \theta}e^{\frac{-|x|}{\theta}}$.We know by Factorisation theorem that $\frac{\sum |X_i|}{n}$ is sufficient for $\theta$. But can ...
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104 views

Find the maximum likelihood estimator for Pareto distribution and a unbiased estimator

Let $X_1,...X_n$ be a random sample from the Pareto distribution with parameters $\alpha$ and $\theta$, where $\alpha$ is known. Find the maximum likelihood estimator for $\theta$ and say if it is ...
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Hypothesis Testing: Null and Alternative are two separate means?

I have a bit of experience in hypothesis testing where the null hypothesis is a certain mean and the alternative is an inequality, either greater than or less than the null hypothesis, but I have ...
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38 views

Independence of the linear combinations of normal random variables

Are linear combinations of independent normally distributed random variables also independent? Suppose we have two independent standard normal variables $X\sim N(0,1)$, $Y\sim N(0,1)$. Let $U=X+Y$ and ...