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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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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20 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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2answers
34 views

Confidence interval for sample

I have a sample of size $n=19593$ of count data ...
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34 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 ...
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40 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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23 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 ...
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2answers
38 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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1answer
29 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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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 ...
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28 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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2answers
24 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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1answer
21 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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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
18 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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2answers
26 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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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 = ...
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4answers
40 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.
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Estimate Sample [duplicate]

You wish to estimate, with 99% confidence, the proportion of drivers who want the speed limit raised to 130 kph. Your estimate must be accurate to within 5%. How many drivers must you survey, if your ...
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1answer
14 views

Point Estimates Using C.I.

0.680 < p < 0.800 What is the point estimate for p, and the margin of error from which the C.I. was formed? I am confused as to what "p̂" and "E" are equal to. Normally, I would use the ...
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$p$-value in hypothesis testing

Find $p$-value, make appropriate conclusion about $H_0$. Left tailed test ($H_a$ is $<$), $z= -1.28$, $\alpha= 0.05$ Two-tailed test ($H_a$ is $\neq$), $z= 1.28$, $\alpha=.01$ ...
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1answer
20 views

Statistical Estimation

There is an example problem in my book that doesn't explain how they got to this answer: sample: $217$ sample mean: $132.5$ standard deviation: $10$ "The $95$ part of the $68-95-99.7$ rule for ...
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1answer
38 views

Calculating MSE of the estimate $T=\max\{X_1,X_2,\ldots,X_n\}$ of $\theta$.

The variables $X_1,X_2,\ldots X_n$ are i.i.d uniform distributed on $[0,\theta]$. $$T=\max\{X_1,\ldots,X_n\}$$ is the estimate of $\theta$. I need to calculate MSE. I know that ...
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21 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
9 views

Prediction intervals with OLS and indicator variables

Suppose I have a model like so, call it the first model: $$E[y] = \beta_0+\beta_1x+\beta_2x_m+\beta_3(x\cdot x_m) $$ where $x_m$ is an indicator variable. I fit it using ordinary least squares. ...
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Estimating Prevalence of Disease, Defects, or Spam With Screening Tests

Background. A screening test is a relatively quick and easy or inexpensive preliminary test that gives preliminary warning of undesirable condition $D$, such as disease in a patient, defect in a ...
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66 views

Reasoning for confidence interval

Suppose $$X_1,\dots,X_{20} \sim f_X(x;\beta)$$ where $$f_X(x;\beta) = \frac{1}{\beta} e^{-\frac{x}{\beta}},\quad x>0;\beta>0$$ It can shown that ("details omitted") $$P(0.52 \bar{X} \leq \beta ...
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66 views

Homogeneous polynomials on sphere - need an example that is used in machine learning.

My question is about an example of use of homogeneous polynomials on sphere as a hypothesis space in learning problem. In order to ask a question I need to make a quick introduction: I'm reading an ...
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26 views

Pareto distribution confidence interval

$X$ is distributed by Pareto with $$f_X (x) = \frac{\alpha k^{\alpha}}{x^ {\alpha +1}},\alpha,k>0,x>k.$$ Derive a 95% confidence interval for $k $. My friend said I gotta do this $$Pr ...
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General method to find exact confidence interval?

Say we have a beta random variable with pdf, $f_X (x) = e^{-x/\theta}/\theta$ for positive $x $. Find the exact confidence interval of $\theta$ with 95% confidence. A solution to this says ...
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54 views

How to use Laplace method to get the asymptotic expansion of multiple integral

I meet difficulty when I try to get the asymptotic behaviour of multiple integral as x tends to plus infinity. And $-1<$p$<1$ $$\int_x^{+\infty}\int_x^{+\infty}e^{-{\frac{1}{2\sigma^2(1-p^2)}\ \ ...
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1answer
28 views

Confidence interval and symmetric distribution

Let $X_1,...,X_n$ random sample of $X$~$U[\theta-\frac{1}{2};\theta+\frac{1}{2}]$.Consider $[X_{(1)};X_{(n)}]$ a confidence interval for $\theta$. Find their confidence level and show that ...
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64 views

Probability about confidence interval

Let $X_1,...X_n$ be iid $N(\theta,1)$. A 95% confidence interval for $\theta$ is $\overline{X}\pm\frac{1.96}{\sqrt{n}}$.Let p denote the probability that an additional independent ...
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7 views

Pitman Asymptotic Relative Efficiency

Let $T_1$ and $T_2$ are two test for testing the same $H_0$ versus $H_1$. Let $t_1$ and $t_2$ are test statistic correspond to $T_1$ and $T_2$, respectively. If $t_1$ and $t_2$ have noncentral chi ...
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33 views

Reference request, statistical inference

Good morning, I'm looking for a good reference for study on statistical inference, the main topics that will study are Tests of Hypotheses Interval estimation I recommend taking a look at Mood ...
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Method of moments estimation for not-idd rv:s

I'm trying to solve a problem in which I have a sample from to random variables X and Y with $X\sim Exp(a)$ and $Y \sim Exp(\frac a2)$ (or actually it says $X \in Exp(a)$ and $Y \in exp(\frac a2)$ but ...
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56 views

How to write R program to solve the confidence interval?

The problem: let $X_1,\ldots,X_n$ be random variable from $\mathrm{Poisson}(\theta)$. Under $H_0: \theta=\theta_0$, we want to find the $(1-\alpha)100\%$ confidence interval for $\theta$ by using the ...
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Log-likelihood approaches +Inf for a Gaussian process

I am trying to do a standard likelihood maximization for the hyperparameters of a Gaussian Process (details in Chapter 5, Rasmussen & Williams: ...
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1answer
37 views

Sampling distribution of sample trimmed (truncated) mean

It is elementary probability theory that the sample mean of an i.i.d. sample follows normal distribution, if the background distribution is normal. But what about the trimmed mean? Is there any result ...
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12 views

testing significant difference on species richness

I measured the species richness (number species) in three different sites. Now, I want to test whether there is significant difference between each site in terms of species richness. The species ...
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21 views

Relation between the Coefficient of Multiple Correlation and Coefficient of Simple Correlation

Consider the regression model $Y=\beta_1 X_1+\beta_2 X_2+\epsilon$, with a sample of size $n$, $Y_i=\beta_1 X_{i1}+\beta_2 X_{i2}+\epsilon_i$, $\epsilon_i \in N(0,\sigma^2)$. Suppossing ...
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Proof that the least square estimators are normally distributed

In my book I have the following proof showing that one of the least square estimators is normally distributed: $\hat\beta_i$ = $\frac {S_{xy}}{S_{xx}}$ = $\frac {1}{S_{xx}}\sum_1^n({x_i}- ...
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1answer
14 views

Integration to get expected mean and variance

I have the following question: The times in minutes needed to collect the tolls from motorists crossing a toll bridge has the probability density function $$f(x) = 2 exp(−2x), 0 ≤ x < ∞$$ A ...
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1answer
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Continuum random variable distribution: integral and trapezium-rule methods give different result

Suppose p.d.f. is $\frac{1}{63}x^{2}$. Find the $P(4<x<5)$. I've tried with integration method and trapezium-rule, but they give me different result. With integration, ...
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1answer
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Ratio laying within the confidence interval still being depicted as having an influence?

I keep seeing this in research papers. The researchers claim that there is a positive correlation between A and B then subsequently show that they odds ratio/sample mean etc. is IN the confidence ...
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Hypothesis-test; test about equal sample means

I'm asked to formulate and do a test of the hypothesis that the sample mean (average grade) of math students and economic students are equal. Data below: I'm not sure how to "attack" this problem. ...
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what is the influence of the specific statistical model selection in a practical project

I hope this is the right place to ask this question. But if it is not, please feel free to migrate. There is a famous quote, which is like "all models are wrong, but a few are useful". So, I was just ...
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1answer
63 views

What is a easy way to draw from distribution $f(x|u) \propto x^{\alpha-1}I(x<u)$?

We know $u$ and want to draw $x$ from the conditional density $f(x|u) \propto x^{\alpha-1}I(x<u), \alpha>0$. One way is that first draw $r$ from uniform(0,1), and then set $x=u ...
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673 views

How to quantify the differencen between 2/4 and 20/40?

Assume I have two methods to do prediction. The first method makes 4 predictions and 2 out of 4 are correct. The second method makes 40 predictions and 20 out of 40 are correct. The prediction ...
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1answer
53 views

Is there a method to check if two curves (non-linear) are identical

I have two data sets of pollutant concentration on simultaneous days. I have to check whether these two curves follow similar pattern or not ( there might be some time lag between both) on daily ...