Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Mathematical theory of statistics, concerned with formal definitions and general results.

0
votes
0answers
20 views

Convexity of CDF

Let $X$ be a random variable with distribution function $F$ and mean $m_x$. Consider another random variable $Y$ (with same support as that of $X$) with distribution function $G$ and mean $m_y$. I ...
2
votes
1answer
23 views

When are correlated Normal random variables multivariate Normal?

I know that there are many example of correlated normal random variables which are not jointly (multivariate) normal. However, are there conditions which state when correlated normal random variables ...
0
votes
0answers
23 views

reA function of sufficient statistic

I'm following notes at onlinecourses and I got confused on transformation of sufficient statistics. For example, if $X$ is a sufficient statistic for $\mu$, why $Y=X^2$ is not a sufficient statistic ...
0
votes
0answers
7 views

Rules based Model (Function) - Derive Probability & Ensembling

Basically, let's assume I have a simple rules-based function/model (if weight >= 150) -> return true. Simple binary answer (true or false) from a single feature input. If I have a range of samples/...
1
vote
0answers
22 views

How to model and estimate interference and subsequent shocks on panel data?

I have the following setting: In my factory, we have mutliple assembly lines(>10). Each line produces an amout of itmes every day, with some weekly and mothly production peeks. Thus its basically a ...
0
votes
0answers
12 views

What is the correct approach to know if something exists at any time, with given mean/SD persistence time and start date (but exact time is unknown)?

I am doing research on food resource competition between two predator species, and I am not exactly an expert on statistics so I'll thought I'll come here and ask. I have information on mean and SD ...
0
votes
0answers
9 views

Any sub-vector of an isotropic sub-gaussian random vector is an isotropic sub-gaussian random vector.

Any sub-vector of an isotropic sub-gaussian random vector is an isotropic sub-gaussian random vector. Note that a random vector $x\in\mathbb{R}^n$ is sub-gaussian if for any $a\in\mathbb{R}^n$, $a\...
1
vote
0answers
22 views

Does Hoeffding or Bernstein inequality provide a more accurate bound for this problem?

Let $X_i$, $i=1,\dots, n$ be independent random variables with $EX_i = \mu_i$, $\operatorname{Var}X_i=\sigma^2$ and $|X_i - \mu_i| \leq b$, $i=1,\dots,n$. Let $X=\sum X_i$ and $\mu = \sum \mu_i$. Let $...
5
votes
2answers
87 views

How to determine the number of possible combinations of letters that contain a degenerate substring

I've been racking my brain for a couple of days to work out a series or closed-form equation to the following problem: Specifically: given all strings of length $N$ that draws from an alphabet of $L$ ...
1
vote
1answer
41 views

How many sample needed for 95% confidence of a rule

I've got an electronical component measuring some voltage. The documentation of this component say the maximum error of measurement of this component is 3%. Assuming a chosen component always do the ...
-1
votes
0answers
19 views

How can you present String value into numerical value? [on hold]

I have the text below: NT_ARFH25_ARCHER_580205_180919 How can I present this text into numerical value that calculated using a statistical formula?
2
votes
2answers
41 views

Does the random sample from the population always have the same distribution as the population?

Let $Y_1,Y_2,...Y_n$ be a random sample of size $n$ from a population with distribution X. From this information, can I also always conclude that $Y_1,Y_2,...Y_n$ will have distribution X? Or is this ...
0
votes
0answers
21 views

Are there occasions when both the two-sample t-test and the Wilcoxon rank-sum test are appropriate?

As the title says, Are there occasions when both the two-sample t-test and the Wilcoxon rank-sum test are appropriate? What kind of problem would be correct to use both on ?
3
votes
1answer
46 views

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....
0
votes
0answers
10 views

Confidence Interval in a linear model with restrictions

I am reading the paper by Hannes Leeb and Benedikt Potscher "Model Selection: Facts and Fictions",(available as a working paper here https://www.wiwi.uni-frankfurt.de/profs/klump/D/koll/...
3
votes
0answers
32 views

Check Computation of MME and MLE

Let $X_1$, . . . , $X_n$ be i.i.d random variables having pdf $$f(x\mid\theta) = (\theta+ 1)x^{\theta}I_{(0,1)}(x)$$ where $\theta \gt−1$ (a) Give a MME of $\theta$ based on the first ...
1
vote
0answers
39 views

Baseball game duration distributions [on hold]

The following represents the game time, in minutes, for the Major League Baseball games played in the United States between March 30 and April 9, 2003, sorted by league. Compare the times between the ...
0
votes
0answers
52 views

Stuck on a term in $\operatorname{Var}\left[ \widehat{\beta}_0 \right]$ proof

So I was trying to prove that $\operatorname{Var}[\hat{\beta}_0] = \dfrac{\sigma^2n^{-1} \sum{(x_i)^2}}{\sum{(x_i-\bar{x}})^2}$ And I got stuck with the part $\dfrac{-2\bar{x}}{\sum{(x_i-\bar{x})^2}} ...
0
votes
0answers
9 views

Determining the Relationship Between Monte Carlo Breaks and Model Volatility

I'm looking for a statistical test to understand the relationship (if any) between the model volatilities of a stochastic process, and the occurrence of a'break', defined as an instance when an ...
0
votes
0answers
13 views

How to calculate image similarity between 2 images using DTV?

I try to implement Differential total variation (DTV) as described in (Wu, Y. et al 2017) or (Li, Y. et al 2015). (Wu, Y. et al 2017) describe DTV as following My DTV Matlab function is ...
0
votes
1answer
42 views

How to calculate the integral of Normal CDF and Normal PDF?

I'm trying to find $\int_{\frac{a-b}{B}}^\infty\Phi\left(tA+ABx\right)\phi(x)\,dx$ where $A = \frac{\sqrt{\gamma_{3}+\sigma_3^2}}{\gamma_{3}},\ B = \frac{\gamma_{2}}{\sqrt{\gamma_{2}+\sigma_{2}^...
2
votes
0answers
19 views

Optimization textbooks for statistics and data analytics

Any statistical analysis, machine learning or data science involves some sort of optimization at the end of the day. I'm looking for good linear and nonlinear optimization textbooks for self ...
1
vote
0answers
42 views

MLE Where Observations are Frequency Counts

Consider independent trials, such that for each trial a type 1 outcome will occur with probability $p_1= \frac{1}{2} + \frac{\theta}{4}$ a type 2 outcome will occur with probability $p_2= ...
1
vote
1answer
23 views

Question on complete sufficient statistic

In my textbook, I have this example which says $X\sim U(0,\theta)$ we show that the family of PDFs of X is complete. We need to show that $E(g(x))=\int_{0}^{\theta}\frac{1}{\theta}g(x)dx=0\ \ \ \ \ \...
1
vote
0answers
47 views

Sampling without replacement - Normal sampling distribution [duplicate]

Most of the introductory stats textbooks, treat the sampling distribution of the mean as a normal distribution when sampling is done without replacement and n/N > 0.1. They just use of the finite ...
5
votes
1answer
84 views

MLE of $f(x\mid\theta) = \theta x^{\theta−1}e^{−x^{\theta}}I_{(0,\infty)}(x)$

$X_1$, $X_2$, and $X_3$ are i.i.d random variables having pdf $$f(x\mid\theta) = \theta x^{\theta−1}e^{−x^{\theta}}I_{(0,\infty)}(x)$$ where $\theta \gt0$. Using the observed values of these ...
0
votes
1answer
22 views

How to draw one standard deviation range around the mean of a skewed distribution [closed]

I have a distribution of data with a positive skew, shown in the image. The standard deviation is 1.34 and the mean is 2.01. I want to illustrate on the graph the range of values that are within one ...
-1
votes
0answers
21 views

Probability calculation for posterior information [closed]

Assume there are 400 athletes in a training camp, who are required to attend the morning drill starting at 4 am. The attendance in morning drills is 70%, i.e. on an average, 280 athletes are present. ...
1
vote
0answers
28 views

Show that $X \sim \chi^2(1)$, the chi-squared random variable with one degree of freedom, is a sub-exponential $(4,4)$ random variable

Show that $X \sim \chi^2(1)$, the chi-squared random variable with one degree of freedom, is a sub-exponential $(4,4)$ random variable. For this purpose, I first want to show that if $|\lambda|\leq ...
2
votes
2answers
24 views

Prove minimum argument of square error function is equal to expected value

Consider a probability density function $f(x)$ defined over the interval $[a,b]$ where $-\infty<a<b<\infty$. The square error function is defined as $J(y)=\int_a^b (x-y)^2f(x)dx$. The ...
1
vote
0answers
8 views

How to interpret forestplot with pymc on standard devisions of two groups

I'm using pyMC3 to do Bayesian estimation supersedes the t test (BEST) and I was wondering how to actually interpret this result. I see both groups have significantly different stds because the bar ...
0
votes
1answer
33 views

What percent of males in the sample own their own home?

I said it was (males who own a house) divided by total number of males: 132 / (132 + 50) which equals to 72.5% However, my professor seems to think it is the # of male who own the house divided by ...
2
votes
1answer
37 views

How to evaluate $\int_0^\infty m^{x+1}e^{-2m}dm$ as $\Gamma(x+2)\frac{1}{2}^{x+2}$?

$\int_0^\infty \frac{m^{x+1}e^{-2m}}{\Gamma(x+1)\Gamma(2)}dm =\frac{\Gamma(x+2)\frac{1}{2}^{x+2}}{\Gamma(x+1)\Gamma(2)}$ How does the left side equal the right side? I understand that the gamma ...
2
votes
1answer
113 views

Maximum Likelihood Estimate for a likelihood defined by parts

Suppose $X_1$, . . . , $X_n$ are i.i.d random variables having pdf $$ f(x\mid\theta)= \begin{cases} \frac{4}{\theta}-\frac{4x}{\theta^2} & \frac{\theta}{2} \lt x \lt \theta \\ \frac{4x}{...
0
votes
0answers
17 views

Making data results equal/proportional

I have a question on how to make my stats on an equal level if you will. As we can see below days rest has 5 different categories. The ...
2
votes
2answers
375 views

Why Standard Deviation is more popular than Mean Absolute Deviation? [duplicate]

$$ Mean\;Absolute\;Deviation = \frac{1}{n}\sum_{i=1}^n |x_i-mean(X)| $$ Almost all textbooks and papers are using Standard Deviation as a measurement of dispersion. And of course, almost all the ...
0
votes
0answers
9 views

Kernel ridge regression with matrix-vector data set $S := \{ X_i, y_i \}_{i=1}^{N}$?

Please notice that this question was asked in MO, but it seems that it doesn't interest MO community. So, I have got a comment to post in this community in the hope that I may get some attention to ...
1
vote
1answer
75 views

t-test for differently-sized sample sets

First, Please understand my lack of English skills. I have a problem dealing with statistical hypothesis testing. I'm in a situation to evaluate two independent samples t-test. There are two ...
0
votes
0answers
41 views

how to estimate confidence interval when p is close to 0 for binomial distribution? [duplicate]

I was trying to get confidence interval for $\hat{p}$. However, the estimated p in the population is close to zero. So I am wondering how I can calculate the confidence interval when p is that small? (...
0
votes
0answers
45 views

Latin Hypercube around set points

Is there a method for constructing a LHC around set points? For example, I want to test a component in a system and I have enough time to run 20 conditions. I expect that in the system, the component ...
0
votes
0answers
6 views

Finding ratios such that two sets of data is mean reverting

I wanted to know how to find the optimal ratio between two variables X & Y with prices x and y, such that when XY is plotted the resultant data set is mean reverting. XY will have the following ...
0
votes
0answers
13 views

Determining If a certain value could be rejected at specific confident level for two methods?

We have performed a test to determine the calcium level in the blood. We were taken replicate measurements for the old method A and for the new method B which we tested to find the calcium level in ...
0
votes
0answers
9 views

LPP vs Convex optimisation for Neural Networks

I am studying for my statistics exam and we have Linear Programming,Simplex and graphical method, Duality Programming and Integer Programming. I have a great interest in Convex optimisation and wanted ...
0
votes
1answer
19 views

How to calculate 95%CI for F1 score and Matthews correlation coefficient?

How can I calculate the 95%CI for F1 score, Matthews correlation coefficient and fraction correct ((TN+TP)/(TP+TN+FP+FN)) when I evaluating a perfomance of binominal diagnostic test?
0
votes
0answers
17 views

How to find dependency of variables over time series and the point of change?

There are have 3 variables over time series.Temperature(T) , Cold Molecules(C) , Warm Molecules(W) . Visually via graph it can find out that there is a dependency between T with (C and W) . The ...
0
votes
2answers
53 views

Showing $max(X_i)$ is the consistent estimator for pdf $f(x;θ) = 2x / θ^2$

Suppose $X_1,...,X_n$ are $iid$ with pdf $f(x;θ) = 2x / θ^2, 0 < x ≤ θ$, I need to find the MLE for the median of the distribution and show that it is a consistent estimator. I don't understand ...
0
votes
1answer
22 views

A fair die is thrown two times independently . Let X,Y be the two outcomes of these two throws and Z= X+Y [closed]

U is the remainder obtained when Z is divided by 6. X and Z are independent ? X and U are independent ? Z and U are independent ? Y and Z are not independent ?
0
votes
0answers
61 views

Right Regression Model to use

I am trying to predict reservation count from a dataset with few features. Features are both categorical and continuous. The dependent variable reservations looks like below: My dataset size is ...
0
votes
0answers
13 views

Doubt on passage in the derivation of the equation for estimating beta coefficients in the multivariate regression

The equation that I need to derive is this: $$\hat{\beta_1}=\frac{\sum\limits_{i=1}^n\hat{r}_{i1}y_i}{\sum\limits_{i=1}^n\hat{r}_{i1}^2}$$ * $r$ are the residuals regressing $x_{i1}$ on the other ...
0
votes
0answers
15 views

Random dicrete variable in a biased coin

A coin is biased so that the probability of obtaining a tail is $0.75$.the coin is tossed four times and the random variable x is the number of tails obtained. Find i) $E[2X]$ ii) $Var(3X)$ In the ...