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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Analysis of rank/likert scale data

If age groups/gender represents independent variable(s) and responses have been collected for 5-6 items in the form of ranking (most preferred - rank 1 to least preferred rank - 5). Which test is an ...
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19 views

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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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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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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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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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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2answers
35 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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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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30 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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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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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
28 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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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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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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20 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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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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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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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
28 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 ...
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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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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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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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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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51 views

Confidence interval for sample

I have a sample of size $n=19593$ of count data ...
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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 ...
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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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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 ...
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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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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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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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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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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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1answer
23 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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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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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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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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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
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.
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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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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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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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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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29 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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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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95 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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34 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 ...