In this paper, it was proposed to carry out a preliminary normalization of diagnostic parameters using the Johnson distribution, which with three basic distribution groups (SL, SB, SU), covers a wide class of empirical distributions. The mathematical description of the family allows us to find the approximating probability density function in an explicit form, to determine the distribution parameters for obtaining the corresponding function (curve), as well as the inverse function for finding the quantiles of the specified levels. To assess the accuracy of the obtained normalized data, they were compared with the data obtained by replacing the resulting law with a Gaussian one. Percentages of values were compared in the implementation under study, which concentrated in the limits of estimated quantiles. Implementations were obtained using the simulation method. By the same method, the correctness (relative systematic error) of determining the quantile values of the specified levels was evaluated. The error value δ was estimated between the conditionally true quantile value calculated from the generated pseudo-general complex and the value estimated using the methods considered in the paper. Obtained data show that the relative error in the calculation of quantiles using the Johnson distribution does not exceed 0.07% and decreases in two orders of magnitude than the currently accepted procedure for replacing sample laws with Gaussian.

methods

/pdf/research-of-diagnostic-parameters-of-composite-materials-45hqy990m9.pdf

The median of an exponential family and the normal law

Research of diagnostic parameters of composite materials using johnson distribution

Let $P_0$ be a probability on the real line generating a natural exponential family $(P_t)_{t\in \mathbb {R}}$. Fix $\alpha$ in $ (0,1).$ We show that the property that $P_t((-\infty,t)) \leq \alpha \leq P_t((-\infty,t])$ for all $t$ implies that there exists a number $\mu_\alpha$ such that $P_0$ is the Gaussian distribution $N(\mu_{\alpha},1).$ In other terms, if for all $t$, $t$ is a quantile of $P_t$ associated to some threshold $\alpha\in (0,1)$, then the exponential family must be Gaussian. The case $\alpha=1/2$, \textit{i.e.} $t$ is always a median of $P_t,$ has been considered in Letac \textit{et al.} (2018). Analogously let $Q$ be a measure on $[0,\infty)$ generating a natural exponential family $(Q_{-t})_{t>0}$. We show that $Q_{-t}([0,t^{-1}))\leq \alpha \leq Q_{-t}([0,t^{-1}])$ for all $t>0$ implies that there exists a number $p=p_{\alpha}>0$ such that $Q(dx)\propto x^{p-1}dx,$ and thus $Q_{-t}$ has to be a gamma distribution with parameters $p$ and $t.$

background

Location and scale behaviour of the quantiles of a natural exponential family

Let $P_0$ be a probability on the real line generating a natural exponential family $(P_t)_{t\in \mathbb {R}}$. Fix $\alpha$ in $ (0,1).$ We show that the property that $P_t((-\infty,t)) \leq \alpha \leq P_t((-\infty,t])$ for all $t$ implies that there exists a number $\mu_\alpha$ such that $P_0$ is the Gaussian distribution $N(\mu_{\alpha},1).$ In other terms, if for all $t$, the number $t$ is a quantile of $P_t$ associated to some threshold $\alpha\in (0,1)$, then the exponential family must be Gaussian. The case $\alpha=1/2$, \textit{i.e.} when $t$ is always a median of $P_t,$ has been considered in Letac et al. (2018). Analogously let $Q$ be a measure on $[0,\infty)$ generating a natural exponential family $(Q_{-t})_{t>0}$. We show that $Q_{-t}([0,t^{-1}))\leq \alpha \leq Q_{-t}([0,t^{-1}])$ for all $t>0$ implies that there exists a number $p=p_{\alpha}>0$ such that $Q(dx)\propto x^{p-1}dx,$ and thus $Q_{-t}$ has to be a gamma law with parameters $p$ and $t.$

/pdf/location-and-scale-behaviour-of-the-quantiles-of-a-natural-1h1508l0l7.pdf

Study of the collection, analysis, interpretation, and presentation of data

The median of an exponential family and the normal law

Citations

Research of diagnostic parameters of composite materials using johnson distribution

Location and scale behaviour of the quantiles of a natural exponential family

Location and scale behaviour of the quantiles of a natural exponential family

References

Séminaire de théorie du potentiel : 4e année 1959-60

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