# Questions tagged [mathematical-statistics]

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

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### Complete statistic for joint distribution of independent variables [closed]

If $X_1, ..., X_m\sim U(0,\theta)$ and $Y_1, ..., Y_n\sim U(0,\theta ')$, then $X_{(m)}$ and $Y_{(n)}$ (the last order statistics) are complete sufficient for $X$ and $Y$. Is there a straightforward ...
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### ECM Specification of ARDL model

I have a question regarding the model reparametrization of an ARDL model. Consider the following ARDL$(p,q,q,\ldots,q)$ model: y_{it} = \alpha_i + \sum_{j=1}^{p} \lambda_{ij} y_{i,t-j}...
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### Since $f(x)$ is a pdf, we must have that $(1 / \sigma)f((x - \mu)/\sigma) \ge 0$ for all values of $x$, $\mu$, and $\sigma$? Incorrect reasoning?

Theorem 3.5.1 of Statistical Inference, second edition, by Casella and Berger, says the following: Theorem 3.5.1 Let $f(x)$ be any pdf and let $\mu$ and $\sigma > 0$ be any given constants. Then ...
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### Is there a mathematical function with a shape like an inverted sigmoid pdf but X values that go beyond 1?

I am looking for a mathematical function that can produce a shape like the one below: The above curve has probabilities ranging between 0 and 1, with the x values being able to take any positive ...
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### Skew vs one sided standard deviation [closed]

If a unimodal dataset x with mean 0 E(x) = 0 is positively skewed E(x^3) > 0, does it imply that mean squared value of positive data (y: x/ x>0) is more than negative data (z: x/ x<0) ...
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