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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Find the required Chi-square score for an arbitrarily low p-value (2 degrees of freedom)

I'm trying to use the Chi-Square test to find the significance of data that suffers from the multiple testing problem. Because I have this multiple testing problem, the required p-value to view a test ...
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2answers
14 views

statistical significance in probabilities

Imagine I am conducting an experiment, and I record whether $n$ individuals of different nationalities, say $A$, $B$, and $C$, either like or dislike a product. In the end I have the respective ...
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0answers
15 views

Struggling to understand multi-class logistic regression

It is well defined that given a data set of $N$ $i.i.d$ observations $\mathbf{X} = \{\vec{\mathbf{x}}_1, \dots, \vec{\mathbf{x}}_n\}$, along with corresponding target values $\vec{\mathbf{t}} = {t_1, ...
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1answer
25 views

Estimating the number of classes in a finite population [on hold]

Suppose I have N smarties, each of which is one of C distinct colours. Suppose further that N is known and largish (10,000) but C is not, and that for each colour C there are $c_i$ smarties of that ...
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2answers
29 views

Variance of least square estimator

I have two random variables X and Y with $X\sim Exp(a)$ and $Y \sim Exp(\frac a2)$. I have a least square estimator $a=\frac {2x +y}{2.5}$. I want to calculate the variance of the estimator and to do ...
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1answer
44 views

Prove that risk function is analytic?

I'm considering the statistical minimax estimation problem of the bounded normal mean: Specifically, the problem is to find the minimax estimator of $X \sim N(\theta,1)$ where $\theta \in ...
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1answer
33 views

Recursive Variance

What will be the distribution or features about the following $x$? $x=\mu+\epsilon$ where $\epsilon\sim N(0,x^{-1})$. It seems interesting in econometrics if we allow $x$ being a time series and ...
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2answers
34 views

Difference between two proportions in a Confidence Interval

Ten engineering schools in the United States were surveyed. The sample contained $250$ electrical engineers, $80$ being women; $175$ chemical engineers, $40$ being women. Compute a $90\%$ confidence ...
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27 views

How does a median have a value that is a decimal which isn't exactly half of an integer if the data should consist of only integer values?

I real an article which said the average man accumulated 6.1 sexual partners while the average woman accumulates 3.6. If the statistic talked about the average, surely the numbers would be equal-so it ...
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1answer
48 views

Transforming the probability distribution that have unknown form [closed]

I have the following expression, which is difficult to compute as the explicit form of the probability distribution is unknown. $\int_{0 \leq y < t} y^2 Q(y)dy$. The density $Q(y)$ is for $y ...
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26 views

How is the Variance of this estimator equal to $\theta$?

Currently going through solutions of a worksheet and I don't understand the jump between two lines of working. "$\hat{\theta}_1$ and $\hat{\theta}_2$ are independent unbiased estimators for an ...
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42 views

How do I get a English version of an article in French? [closed]

Do anybody know how to get the English version of this article? Contribution a l'etude du bonus pour non sinistre en assurance automobile by P. Thyrion DOI: ...
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0answers
7 views

Global hypothesis testing with multiple (non-IID) observations

I have a set of $\{x_i\}_{i=1}^m$ observations from $m$ independent binomial trials with $(a)\space X_i \sim Bin(n_i, p)$ and $(b)\space X_i \sim Bin(n_i, p_i)$. I want to test a global hypothesis ...
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1answer
304 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 $$ ...
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2answers
410 views

Improper Uniform Prior Distribution

In Bayesian method, choosing the prior distribution is an important step when using the Bayesian method. When choosing prior, we consider the prior knowledge to choose which prior distribution is the ...
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21 views

Estimation, errors and hypotheses [closed]

Distinguish between the following terms: (i) Point and interval estimation. (ii)Type I and type II error. (iii)One-sided and two-sided hypothesis.
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1answer
34 views

In terms of $a, b,$ and $\theta$, what is the biased $b(\hat \theta)$?

The Statement of the Problem: Let $\{P_{\theta}: \theta \in \Theta \}$ be a statistical model. Suppose that $\hat \theta$ is an estimator for a parameter $\theta$ and $E_{\theta}(\hat \theta) = ...
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1answer
32 views

For a Poisson model, show that the sample mean $\overline X$ is an unbiased estimator of $\lambda$.

The Statement of the Problem: For a Poisson model $\{\text{Pois}(\lambda): \lambda \in (0, \infty) \}$ show that the sample mean $\overline X$ is an unbiased estimator of $\lambda$. What I Did: I ...
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1answer
27 views

Hypothesis testing with order statistics

I know there have been a lot of questions asked on this forum relating to order statistics, so, hopefully, this is not going to be a duplicate. I am trying to understand how I should go about ...
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0answers
17 views

Does it make sense to interpret autocorrelation and box test on 5 data points?

I am trying to see if after I trade a stock the price movements at 2, 5, 7, 10, 30 and 60 seconds after exhibit any autocorrelation. Below I have the returns from my trade price to the trade 2,5,7,10 ...
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1answer
17 views

studying set I of population

let x be the mean and € the standard deviation of the statistics ×1,,,,,,, xn. let I=(x-3€, x+3×€) and the number of items not in I is k. prove that n greater or equal to 9k. prove that the ...
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1answer
19 views

F-Testing ; constant returns to scale

$lnQ=1.37+0.632lnK+0.452lnL$ (0.257). (0.219) $cov(bk,bl)=0.055, R^2=0.98$ $H_0: bk+bl=1$ How can I proceed f-test even though I can't find df and RSS?
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1answer
25 views

Two types of errors, type-$1$ error and type-$2$ error, can not be minimized simultaneously when the sample size $n$ is already fixed. How?

I read in some of the books that the two types of errors, type-$1$ error and type-$2$ error, can not be minimized simultaneously in Neyman Pearson Theory of testing of hypothesis when the sample size ...
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1answer
23 views

How to find which treatment is most effective in gene data given one standard method and 3 variations

Sorry I am a biologist and it appears am not quite confident enough for statistical analysis. I have datasets that represent different treatments on a biological system. It records how many genes have ...
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1answer
288 views

One-tailed two-sample T-test OK?

I'm trying to conduct a one-sided hypothesis test between two random variables which are both asymptotically normally distributed with different variances. The variances are not known but have been ...
2
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1answer
23 views

Almost sure convergence of $\hat{\sigma^2}$

Let $Y \sim N(X\beta,\sigma^2I)$ where $Rank(X_{n\times p})=p \leq n$. The least square estimate of $\sigma^2$ is $\hat{\sigma^2}=\frac{Y'(I-P)Y}{n-p}$ where $P=X(X'X)^{-1}X'$ is the projection matrix ...
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1answer
27 views

Expected value of a sample

I am unsure of how to solve this question. I know from examples questions that expected value of a sample is usually very close to the population mean. However, it says to compute the expected value ...
2
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3answers
51 views

Confidence interval for sample

I have a sample of size $n=19593$ of count data ...
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1answer
22 views

calculating UMVUE of parameter $(1-\sigma^2)^-\frac{n}{2}$.

suppose $X_1,X_2,\ldots,X_n$ be random sample of $N(0,\sigma^2)$. how can I calculate UMVUE of parameter $(1-\sigma^2)^-\frac{n}{2}$. I know $T=\sum_{i=1}^n X_i^2$ is Sufficient and complete ...
2
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1answer
341 views

Finding MLE of $f(x;\theta) =1$ if $\theta-1/2<x< \theta+1/2$

Let $X_1,...,X_n$ have density: $f(x;\theta) = \begin{cases} 1 &\mbox{if } \theta-1/2<x< \theta+1/2 \\ 0 & otherwise \end{cases}$ Let $Y_1=min \lbrace X_1,...,X_n \rbrace$ and ...
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18 views

How to measure the stability of datas

The background: I have a server handling $n$ kinds of requests, denoted by $k_1, ..., k_n$, at a certain time, many requests has been processed, the average time it takes to process $k_i$ is $t_i$, ...
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2answers
43 views

Confidence interval for Poisson distribution coefficient

This is an exam question, testing if water is bad - that is if a sample has more than 2000 E.coli in 100ml. We have taken $n$ samples denoted $X_i$, and model the samples as a Poisson distribution ...
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16 views

Approximation of Mahalanobis distance

If $A$ is a symmetric positive definite $n\times n$ matrix then the square Mahalanobis norm of a vector $v\in \mathbb{R}^n$ is given by $$\lVert v \rVert_A^2=v^t A^{-1} v.$$ Now I have a situation ...
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24 views

variance of a sum of independent random variables

I don't get why here https://en.wikipedia.org/wiki/Standard_error, T/n = 1/n²*(n*sig²) Is there a side knowledge to have here ?
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1answer
37 views

Same Expected Value but different variances. Is $E[U(X)] \ge E[U(Y)]$?

Let $U: \mathbb R -> \mathbb R$ be a concave function, and let $X$ be a random variable with a normal distribution, expected value $\mu$, and standard deviation $\sigma$. Let $\lambda \gt 1$, and ...
2
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1answer
43 views

Prove that $E[U(X)] \ge E[U(Z)]$

Let U: $\mathbb R$ -> $\mathbb R$ be a concave function, let X be a random variable with a finite expected value, and let Y be a random variable that is independent of X and has an expected value 0. ...
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1answer
24 views

construct confidence interval from proportions

Suppose you have a population of count data, i.e., $1,2,3, \dots, k$, you have a sample of the population of size $n$, and you have a confidence interval for the proportion of $1$'s , $2$'s,\dots$n$'s ...
2
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1answer
30 views

The asymptotic equivalence of LR, Wald and score tests

Suppose that $Y_1, \ldots, Y_{n}$ are iid from a Bernoulli distribution with parameter $p$ and consider $H_0 : p = p_0\,.$ The test statistics are $$ T_W = \frac{n ({\widehat p} - p_0)^2}{{\widehat ...
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0answers
12 views

What is the asymptotic value of the smoothed probability in a HMM model?

If I have a HMM model with a hidden markov chain $(S_t)_t$ with 3 states and if I assume that the distribution of the observation knowing in which state it is, is a normal. Do I know what is the value ...
2
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0answers
47 views

Sample median of Cauchy distribution is consistent. How?

When we use chebyshev's inequality to show whether an estimator is consistent or not, we require the mean square error of the estimator and I do not know sample median's probability distribution. So ...
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1answer
56 views

Some true/false statements about MLE and UMVUE for a normal distribution

Let $X_1,X_2,...,X_n$ (assume $n\geq 2$) be a random sample from an $N(\mu,\sigma^2)$ population where $-\infty<\mu <\infty$ and $\sigma^2>0$ are unknown. Which of the following statements ...
2
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2answers
27 views

Estimating grader bias/variance and MLE test scores given multiple graders assigned to grade each test

Suppose we have $m$ graders and $n$ students, and we want to grade a test so that $k$ graders are assigned to grade to each test, and all graders grade the same number of tests. (I realize $m,n,k$ ...
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2answers
29 views

A problem in method of moment - in my Quiz

Let $X_1,...,X_n$ be an i.i.d. sample from the uniform distribution on ($-\theta$, $\theta$). (a) Find a method of moments estimator of $\theta$. By integration of second moment, ...
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1answer
22 views

Sum of Squares From Regression Formula in Matrix Form

I am trying to show that the regression sum of squares, $$SS_{reg}=\sum(\hat{Yi} - \bar{Y})^2 = Y'(H - \frac 1 nJ)Y$$ where $H$ is the hat matrix and $J$ is a matrix of ones. I can do this using the ...
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0answers
14 views

How does one estimate the order of a Markov chain empirically (given the data)?

I have a string of symbols $x_1, x_2, ...., x_n$, ($n$ very large), belonging to a finite alphabet. I know that they are a result of a Markov process, but I want to find out the order of the process. ...
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1answer
21 views

How do I show that the sum of residuals of OLS are always zero using matrices

I am trying to show that $$\sum_{i=1}^ne_i = 0$$ using matrices (or vectors). I have two hints, so to speak: $$ HX = X$$ where $H$ is the hat matrix, and that $$\sum_{i=1}^ne_i = e'1$$ My previous ...
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0answers
15 views

Sufficient Statistics Multi Conditional values

I am trying to find $\mathbb{E}\{X_1| X_1+X_2, X_1+X_3\}$ where all are non negative independent r.v.'s (e.g. Poisson). I am not clear about the concept of sufficient statistics, is't it enough in ...
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1answer
18 views

In OLS is the vector of residuals always 0? [duplicate]

I am trying to show that $$\sum_{i=1}^ne_i = 0$$ I have two hints, so to speak: $$ HX = X$$ where $H$ is the hat matrix, and that $$\sum_{i=1}^ne_i = e'1$$ My solution is as follows: $$e'1 = ...
0
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1answer
34 views

calculate The maximum likelihood estimator of parameter $\mu$ according to $T$

suppose $X_1,X_2,\ldots,X_n$ be a random sample of $N(\mu,1)$. if $T=\sum_{i=1}^n I_{(X_i<0)}$ how can I calculate The maximum likelihood estimator of parameter $\mu$ according to $T$. ($\Phi$ is ...
0
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4answers
44 views

Line Of Regression given x? [closed]

You have found the regression line for a set of data points to be: ŷ = 30.23x + 173.52. Use the line to predict the value of y when x = 48.