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Questions tagged [mathematical-statistics]

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

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1answer
30 views

An easy way to find the PDF of $XY/Z$

Suppose that we know the joint distribution $f_{X,Y,Z}$. I want to find the PDF of $W = XY/Z$ in a way that does not involve too much calculation. I have thought about first finding the Marginalized ...
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0answers
7 views

What is Importance Value?

I am reading this paper that talks about Principle Components analysis and Importance Values(IV) here Is this Importance Value a statistical concept? I could not find any information about it in ...
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1answer
36 views

Statistical test to verify if a population has a average of k times another

Consider the situation where there are 2 positive distributions with unknown expected values $\mu_1$ and $\mu_2$, We have samples $S_1 = \{x_1, x_2, ..., x_n\}$ and $S_2 = \{y_1, y_2, ..., y_n\}$. ...
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0answers
21 views

What is the intuition behind pi in the PDF of a Normal Distribution ? Is it related to some sort to a circle / sphere

The PDF of a Normal distribution is given as below I am aware of the various properties of Normal distribution and how the two parameters mu and sigma affect the shape of the distribution. What is ...
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0answers
20 views

Compute the information matries related to normal distribution

This is a problem that I have trouble with. Suppose that we have $X_{1}, \ldots, X_{m}$ are iid $N\left(\mu, \sigma^{2}\right), Y_{1}, \ldots, Y_{n}$ are iid $N\left(0, \sigma^{2}\right),$ the $X$ 's ...
7
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0answers
126 views

Random variables $X, Y$ such that $X$, $Y$ and $\sqrt{X + \sqrt{Y}}$ belongs to the same family of distributions?

Is there a family of positive distributions such that if $X$ has the distribution in question, then $\sqrt{X}$ also has a distribution from the same family. Ideally, it would be great if $X+Y$ also ...
-2
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0answers
21 views

Probability, Statistics [on hold]

1.Probability of getting two before three, 2. probability of getting 2 before 3 or 5 or 6, 3.probability of getting 2 before 3 or 4 or 5 or 6
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2answers
50 views

Outlier removal, extremes on both ends

A list of numbers and I want to remove the extremes on both ends. The standard deviation is calculated: 26.3 (rounded to 1 decimal) ...
1
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0answers
14 views

Land Method for Lognormal Distribution

The Land Method is explain here. Anyone knows how I can look up the H*1-α* It says "Tables of these values are given by Gilbert (1987, Tables A-10 and A-12) and Land (1975). " Gilbert (1987) is not ...
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0answers
9 views

Product/SKU level time series forecasting [on hold]

Predict demand of product at each outlet for next 6 months 5 – 7 years of sales data at outlet level for each and every brand is available As it is a time series problem we need to design per outlet ...
0
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1answer
14 views

Direction of a one sample Wilcoxon signed rank test

I am comparing a selection of values which come from a non-normal distribution to 0. I've done a wilcoxon in python: result = scipy.stats.wilcoxon(values) my W ...
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0answers
4 views

Demonstrate the internal laws of the circumstance when the inequalities of Sauer's Lemma are strict [on hold]

We proved Sauer’s lemma by proving that for every class H of finite VC-dimension d, and every subset A of the domain, $$ \left|\mathcal{H}_{A}\right| \leq |\{B \subseteq A: \mathcal{H} \text { ...
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1answer
24 views

Data Mining/Statistical Methods to find trends, peaks, etc

currently I am working on a project for my final exam. The data is coming from a streaming plattform. The data I am using are some logging data (data when customers have problems with the streaming ...
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0answers
17 views

Basic results on convergence in distribution

Let $\{X_i\}_{i=1}^n$ be independent zero mean random variables with finite variance and $\{r_n\},\{d_n\}$ positive monotone increasing real sequences. Assume that $$\frac{\sum_{i=1}^{r_n} X_i}{Var(\...
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1answer
25 views

Range of integration for joint and conditional densities

Did I mess up the range of integration in my solution to the following problem ? Consider an experiment for which, conditioned on $\theta,$ the density of $X$ is \begin{align*} f_{\theta}(x) = \...
1
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0answers
10 views

Is there an example that a probability distribution is having an increasing mean residual lifetime(IMRL) but not a Dreasing Failure Rate(DFR)

It is well known in risk theory that a random variable having a Decreasing Failure rate implies the Random Variable has an increasing mean residual lifetime. However, the converse is not true. Is ...
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0answers
8 views

Which model is best using the hettest for heteroscedasticity test?

I have calculated hettest using STATA and i have found two models have no Heteroscedasticity problem. but i can not identify which one is the best model for my data. can you please help me out? I ...
0
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2answers
35 views

What test should i use to prove A is better than B [closed]

I have sets of data that have number of goods produced by 2 machines, A & B for 2 months. Each machines has two numbers that is plan value and actual value. I'm not really familiar with hypothesis ...
2
votes
2answers
69 views

Computing the confidence interval for two samples but getting slightly different answers

Consider two samples $X_1,..,X_k$ and $Y_1,..,Y_m$ where $X_i \sim \mathcal{N}(\mu_x,\,\sigma^{2})\,$ and $Y_i \sim \mathcal{N}(\mu_y,\,\sigma^{2})\,.$ Say $k=m=100$ and $k+m=n$. Say that the ...
6
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2answers
87 views

Using all Metropolis-Hastings proposals to estimate an integral

Suppose we run the Metropolis-Hastings with target distribution $\mu$ to compute the integral $\int f\:{\rm d}\mu$. Usually, we use the estimator $$A_n:=\frac1n\sum_{i=0}^{n-1}f(X_i).$$ However, ...
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1answer
15 views

How to compute P values for Breusch pagan test?

I got bptest() output for my OLS model as follows; studentized Breusch-Pagan test data: modelfit BP = 3.3201, df = 3, p-value = 0.3449 I read that BP test is ...
1
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0answers
11 views

Probability of random population value being higher than sample maximum

Considering a small sample size (n < 10) from a population, I'm trying to find how likely a random population value would be greater than the maximum of the sample. Hoping ye could help me with ...
3
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1answer
49 views

linear combination and univariate normal

Show that $(X_1,X_2)$ has a bivariate normal distribution with means $\mu_1, \mu_2$, variances $\sigma _1^2 $ and $\sigma _2^2$, and correlation coefficient $\rho $ if and only if every linear ...
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0answers
26 views

What distribution can be predicted more accurately?

I am working on simple statistical prediction. However, whatever i did, i come up with a high range. I have used standard deviation and also percentile. Yes they work well in the given data but the ...
2
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0answers
28 views

Determining between using Discrete vs Continuous Survival Analysis

This is a follow up question based on what has been explained here: Survival analysis: continuous vs discrete time but I wish to further confirm it for my case. I have a dataset collected over a ...
1
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0answers
5 views

Suggesting multiple labels prediction for multi-class training data

I have a dataset of tickets the machine gets for some machine components failure. The ticket are in text form. For each failure we have around 8-10 diagnosis labels. This tells what the issue might ...
2
votes
1answer
26 views

Network analysis: Formal definition of the number of 1st order neighbours of degree 1

I am looking for a formal definition of a network metric I am using in a scientific article. Let $i$ be a vertex in a graph $G$ and $N(i)$ are the first order neighbors of vertex $i$. I am interested ...
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2answers
115 views
+50

Finding $\mathsf P(\textbf{Y}\mid\textbf{Z})$ where $Z_1,Z_2,\ldots,Z_n$ are conditionally independent

As an extension of my previous question which is described below... Assume $Y_1$, $Y_2$, $\ldots$ ,$Y_n$ are random variables over a regular lattice indexed by $i= 1,2,\ldots,n$ where $Y_i\in\{...
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0answers
11 views

Finding the expected value of Laplace distribution truncated to interval [closed]

Hello I am experiencing difficulties finding the solution for the following Problem: Find the expectation of X where X has a Laplace distribution truncated to interval [5, 9].
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0answers
15 views

Finding the expected value of Laplace distribution for specific interval

Hello I am experiencing difficulties finding the solution for the following Problem: Find the expectation of X where X has a Laplace distribution on [5, 9]. I understand how to find the expectation ...
6
votes
1answer
210 views
+50

Example of a non-measurable maximum likelihood estimator

If you have a measurable parameter space $(\Theta, \mathcal{F})$ and a parametric family of probability measures $(P_\theta)_{\theta \in \Theta}$ on a measurable space $(\mathcal{X}, \mathcal{B})$ ...
0
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0answers
8 views

Prove that in such cases, it is possible to find an ERM hypothesis for $H_n$ in the unrealizable case in time $O(mnm^{O(n)})$

Let $H_1$ , $H_2$ ,... be a sequence of hypothesis classes for binary classification. Assume that there is a learning algorithm that implements the ERM rule in the realizable case such that the ...
3
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1answer
66 views

Finding conditional probability of an individual component of a joint distribution

Assume $Y_1$, $Y_2$, $\ldots$ ,$Y_n$ are random variables over a regular lattice indexed by $i= 1,2,\ldots,n$ where $Y_i\in\{1,2,...,K\}$. Let the probability of a particular configuration $\...
1
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1answer
20 views

relation between location of a sample mean in sampling distribution and the standard error

Can someone explain the following statement with an example We'll describe the location of the sample mean by calculating how many standard errors it is away from the center of the sampling ...
2
votes
1answer
47 views

How to compute confidence and prediction intervals for (nonlinear) regression using non squared “loss function”

Does somebody know (or can point me to a reference) how can I compute a confidence and prediction intervals for (posibly nonlinear) regression using non squared "loss functions"? Let me add some ...
0
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0answers
14 views

Applying L’hôpital’s rule to an exponential function yielding different results [migrated]

I was solving problem 2.38b in Casella and Berger when I came across the below issue: $$\lim_{p \rightarrow 0} f_{P}(p) = p^{r}.[1 - (1 - p)e^{2tp}]^{-r}$$ Applying L’hôpital’s rule to: $$\lim_{p \...
0
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1answer
28 views

precision and recall for biased data

here is my assumption input data's labels are 99% true and there are only 1% false(trying to say most of the data are true). My classifier's probability is 0.5. just like unbiased coin flip. in this ...
0
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0answers
29 views

Machine Learning Features Recorded At Different Timestamps & Locations

Suppose that you are trying to model the following hypothetical situation - where each of your features is a separate time series, as is the target variable. However the feature time series occur ...
0
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0answers
34 views

Is it scientifically feasible to compare two samples of different sizes, while values in each sample are average numbers? [closed]

I want to compare result from my protein docking experiments to various protein structures of the same protein. I have two samples and in Sample 1, I have ...
0
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0answers
10 views

Principal components and how to determine how many principal components to use

Here I've been given the X vector and the means and variances for each variable, how are you suppose to figure out how many principal components would be required to explain 75% of the variance of X, ...
0
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0answers
11 views

Principal Components Analysis, Reading PC Plot

For Question 2v, can someone please explain to me why each subject is located where they are in terms of the two principal components.
0
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0answers
7 views

Significance Calculation for Difference in Percent of two samples

I have a test and control group. Each has a current year(CY) and prior year(PY) order rate, as well as sample size. From my understanding, I can calculate the significance of the Year over Year(YOY) ...
2
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0answers
25 views

Confidence Interval of a bounded variable [duplicate]

Given a random variable $X$ which can take values on $[0,1]$. I am interested in calculating the confidence values for the mean estimator. We know that the mean is normally distributed and can ...
0
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1answer
15 views

How to modify the mean?

I have data collected in consecutive time periods using two sensors (i.e. sensor1 and sensor2) as follows. ...
1
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1answer
46 views

How to compute $Pr(a \leq Y \leq b | a \leq X \leq b)$?

In general, how to compute $Pr(a \leq Y \leq b | a \leq X \leq b)$ if given $F_{Y|X}(y|x)$ and $f_x(x)$? Notice that $a$ and $b$ for the intervals of the two random variables are the same. I know how ...
0
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0answers
7 views

Statistical error in the approximation-estimation tradeoff

Show that $$E(g_\tau ^G(X)-g^* (X))^2 = E(X^T \hat{\beta}-X^T\beta^G)^2+E(X^T\beta^G-g^*(X))^2$$ where $g_\tau ^G(X) = X^T \hat{\beta}$ and $g^G(X) = X^T \beta^G$ where G is a class of linear ...
0
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0answers
21 views

Expected Optimism 0-1 Loss with 0-1 Response

Want to show that $$ E_X op = \frac{2}{n} \sum_{i=1}^n Cov_X(g(x_i), Y_i)$$ For 0-1 loss function with 0-1 response. Want I've done $$op = l_{in}(g) - l(g)=\frac{1}{n}\sum_{i=1} ^n Loss(Y_i', g(...
1
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0answers
42 views

Which statistical test to use to compare 2 tests/measurements against a gold standard

I'd like to know how best to go about comparing 2 tests/measurements against a gold standard. Say I have 100 patients with 3D printed models and CT scans of their hands, and I'll measure a couple ...
0
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0answers
11 views

Derivation of variance of Odds Ratio in a 2x2 table

Assuming that the elements of a 2x2 table (a,b,c,d) are binomial in distribution with a being $Bin(n_1,p_1)$ and c being $Bin(n_2,p_2)$. How can we show that $\hat {var(logOR)} = \frac 1 a + \frac 1 b ...
0
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0answers
14 views

Sample covariance vs population variance of means [duplicate]

Was looking at this link and wondering why for population covariance, the denominator is n, while for sample covariance, denominator is n-1. How does this 1/n(n-1) replace the 1/n^2 in the proof in ...