Pearson distribution

Pearson distribution is a continuous probability distribution whose probability density is a solution to the differential equation ${\frac {df(x)}{dx}}={\frac {a_{1}x+a_{0}}{b_{0}+2b_{1}x+b_{2}x^{2}}}f(x)$ ${\ displaystyle {\ frac {df (x)} {dx}} = {\ frac {a_ {1} x + a_ {0}} {b_ {0} + 2b_ {1} x + b_ {2} x ^ {2}}} f (x)}$ $\ frac {df (x)} {dx} = \ frac {a_ {1} x + a_ {0}} {b_ {0} + 2b_ {1} x + b_ {2} x ^ {2}} f ( x)$ where are the numbers $a_{0},a_{1},b_{0},b_{1},b_{2}$ ${\ displaystyle a_ {0}, a_ {1}, b_ {0}, b_ {1}, b_ {2}}$ $a_ {0}, a_ {1}, b_ {0}, b_ {1}, b_ {2}$ are distribution parameters. ^[1] Particular cases of Pearson distribution are beta distribution (Pearson distribution of type I), gamma distribution (Pearson distribution of type III), student distribution (Pearson distribution of type VII), exponential distribution (Pearson distribution of type X), normal distribution (distribution Pearson XI type). Pearson distributions are widely used in mathematical statistics to smooth out empirical data distributions. To approximate the probability distribution of the experimental data, the first four moments are calculated by numerical methods, and then the Pearson distribution parameters are calculated on their basis. ^[2]

Properties

Pearson distributions are completely determined by the first four moments of a random variable. Let be $\mu _{k}$ ${\ displaystyle \ mu _ {k}}$ $\mu_{k}$ is an $k$ ${\ displaystyle k}$ $k$ the central moment of a random variable having a Pearson distribution. Then if $a_{1}=1$ ${\ displaystyle a_ {1} = 1}$ $a_{1}=1$ then

a_{0}={\frac {\mu _{3}(\mu _{4}+3\mu _{2}^{2})}{A}}

{\ displaystyle a_ {0} = {\ frac {\ mu _ {3} (\ mu _ {4} +3 \ mu _ {2} ^ {2})} {A}}}

a_{0}=\frac{\mu_{3}(\mu_{4}+3\mu_{2}^{2})}{A}

,

b_{0}=-{\frac {\mu _{2}(4\mu _{2}\mu _{4}-3\mu _{3}^{2})}{A}}

{\ displaystyle b_ {0} = - {\ frac {\ mu _ {2} (4 \ mu _ {2} \ mu _ {4} -3 \ mu _ {3} ^ {2})} {A} }}

b_{0}=-\frac{\mu_{2}(4\mu_{2}\mu_{4}-3\mu_{3}^{2})}{A}

,

b_{1}=-{\frac {\mu _{3}(4\mu _{2}\mu _{4}+3\mu _{2}^{2})}{A}}

{\ displaystyle b_ {1} = - {\ frac {\ mu _ {3} (4 \ mu _ {2} \ mu _ {4} +3 \ mu _ {2} ^ {2})} {A} }}

b_{1}=-\frac{\mu_{3}(4\mu_{2}\mu_{4}+3\mu_{2}^{2})}{A}

,

b_{2}=-{\frac {2\mu _{2}\mu _{4}-3\mu _{3}^{2}-6\mu _{2}^{3}}{A}}

{\ displaystyle b_ {2} = - {\ frac {2 \ mu _ {2} \ mu _ {4} -3 \ mu _ {3} ^ {2} -6 \ mu _ {2} ^ {3} } {A}}}

b_{2}=-\frac{2 \mu_{2} \mu_{4} - 3\mu_{3}^{2} - 6 \mu_{2}^{3}}{A}

,

Where $A=10\mu _{4}\mu _{2}-18\mu _{2}^{3}-12\mu _{3}^{2}$ ${\ displaystyle A = 10 \ mu _ {4} \ mu _ {2} -18 \ mu _ {2} ^ {3} -12 \ mu _ {3} ^ {2}}$ $A = 10 \mu_{4} \mu_{2} - 18 \mu_{2}^{3}- 12 \mu_{3}^{2}$ . ^[one]

Pearson Distribution Types

Depending on the distribution of the roots of the square trinomial $b_{0}+2b_{1}x+b_{2}x^{2}$ ${\ displaystyle b_ {0} + 2b_ {1} x + b_ {2} x ^ {2}}$ $b_{0}+2b_{1}x+b_{2}x^{2}$ There are 12 types of Pearson distributions. We denote $D=b_{0}b_{2}-b_{1}^{2}$ ${\ displaystyle D = b_ {0} b_ {2} -b_ {1} ^ {2}}$ $D = b_{0}b_{2}-b_{1}^{2}$ , $\lambda ={\frac {b_{1}^{2}}{b_{0}b_{2}}}$ ${\ displaystyle \ lambda = {\ frac {b_ {1} ^ {2}} {b_ {0} b_ {2}}}}$ $\lambda = \frac{b_{1}^{2}}{b_{0}b_{2}}$ . ^[one]

Type I

Pearson Type I distributions are beta distributions. Conditions: $D<0$ ${\ displaystyle D <0}$ , $\lambda <0$ ${\ displaystyle \ lambda <0}$ , $b_{0}+2b_{1}x+b_{2}x^{2}=(\alpha +x)(-\beta +x)b_{2}$ ${\ displaystyle b_ {0} + 2b_ {1} x + b_ {2} x ^ {2} = (\ alpha + x) (- \ beta + x) b_ {2}}$ , $\alpha ,\beta >0$ ${\ displaystyle \ alpha, \ beta> 0}$ Probability Density: $f(x)={\begin{cases}{\frac {\alpha ^{2m}\beta ^{2n}}{(\alpha +\beta )^{m+n+1}B(m+1,n+1)}}(\alpha +x)^{m}(\beta -x)^{n},&x\in [-\alpha ,\beta ]\\0,&x\notin [-\alpha ,\beta ]\end{cases}}$ ${\ displaystyle f (x) = {\ begin {cases} {\ frac {\ alpha ^ {2m} \ beta ^ {2n}} {(\ alpha + \ beta) ^ {m + n + 1} B (m + 1, n + 1)}} (\ alpha + x) ^ {m} (\ beta -x) ^ {n}, & x \ in [- \ alpha, \ beta] \\ 0, & x \ notin [- \ alpha, \ beta] \ end {cases}}}$ where $B(m+1,n+1)={\frac {\Gamma (m+1)\Gamma (n+1)}{\Gamma (m+n+2)}}$ ${\ displaystyle B (m + 1, n + 1) = {\ frac {\ Gamma (m + 1) \ Gamma (n + 1)} {\ Gamma (m + n + 2)}}}$ , $m>-1,n>-1$ ${\ displaystyle m> -1, n> -1}$ . ^[one]

Type II

Conditions as for type I with additional conditions $\alpha =\beta ,m=n$ ${\ displaystyle \ alpha = \ beta, m = n}$ . ^[one]

Type III

Pearson type III distributions are gamma distributions. Conditions: $D<0$ ${\ displaystyle D <0}$ , $\lambda =\infty$ ${\ displaystyle \ lambda = \ infty}$ , $b_{0}+2b_{1}x+b_{2}x^{2}=2(\alpha +x)b_{1}$ ${\ displaystyle b_ {0} + 2b_ {1} x + b_ {2} x ^ {2} = 2 (\ alpha + x) b_ {1}}$ . Probability Density: $f(x)={\begin{cases}{\frac {k^{m+1}}{\Gamma (m+1)}}(\alpha +x)^{m}e^{-k(\alpha +x)},&x>-\alpha ,k>0\\0,&x\leqslant -\alpha \end{cases}}$ ${\ displaystyle f (x) = {\ begin {cases} {\ frac {k ^ {m + 1}} {\ Gamma (m + 1)}} (\ alpha + x) ^ {m} e ^ {- k (\ alpha + x)}, & x> - \ alpha, k> 0 \\ 0, & x \ leqslant - \ alpha \ end {cases}}}$ . ^[one]

Type IV

Conditions: $D>0$ ${\ displaystyle D> 0}$ , $0<\lambda <1$ ${\ displaystyle 0 <\ lambda <1}$ , $b_{0}+2b_{1}x+b_{2}x^{2}=(\alpha ^{2}+x^{2})b_{2}$ ${\ displaystyle b_ {0} + 2b_ {1} x + b_ {2} x ^ {2} = (\ alpha ^ {2} + x ^ {2}) b_ {2}}$ . Probability Density: $f(x)=c(\alpha ^{2}+x^{2})^{-m}\exp {-\nu \arctan {\frac {x}{\alpha }}}$ ${\ displaystyle f (x) = c (\ alpha ^ {2} + x ^ {2}) ^ {- m} \ exp {- \ nu \ arctan {\ frac {x} {\ alpha}}}}$ , $x\in (-\infty ,\infty )$ ${\ displaystyle x \ in (- \ infty, \ infty)}$ , $m\geqslant {\frac {1}{2}}$ ${\ displaystyle m \ geqslant {\ frac {1} {2}}}$ where $c^{-1}=\int _{-\infty }^{\infty }(\alpha ^{2}+x^{2})^{-m}\exp {-\nu \arctan {\frac {x}{\alpha }}}dx$ ${\ displaystyle c ^ {- 1} = \ int _ {- \ infty} ^ {\ infty} (\ alpha ^ {2} + x ^ {2}) ^ {- m} \ exp {- \ nu \ arctan {\ frac {x} {\ alpha}}} dx}$ . ^[3]

V type

Conditions: $D=0$ ${\ displaystyle D = 0}$ , $\lambda =1$ ${\ displaystyle \ lambda = 1}$ , $b_{0}+2b_{1}x+b_{2}x^{2}=(\alpha +x)^{2}b_{2}$ ${\ displaystyle b_ {0} + 2b_ {1} x + b_ {2} x ^ {2} = (\ alpha + x) ^ {2} b_ {2}}$ . Probability Density: $f(x)={\begin{cases}{\frac {\gamma ^{m-1}}{\Gamma {m-1}}}x^{-m}e^{-{\frac {\gamma }{x}}},&\gamma >0,m>1,x>0\\0,&x\leqslant 0\end{cases}}$ ${\ displaystyle f (x) = {\ begin {cases} {\ frac {\ gamma ^ {m-1}} {\ Gamma {m-1}}} x ^ {- m} e ^ {- {\ frac {\ gamma} {x}}}, & \ gamma> 0, m> 1, x> 0 \\ 0, & x \ leqslant 0 \ end {cases}}}$ . ^[3]

VI type

Conditions: $D<0$ ${\ displaystyle D <0}$ , $\lambda >1$ ${\ displaystyle \ lambda> 1}$ , $b_{0}+2b_{1}x+b_{2}x^{2}=(\alpha +x)(x-\beta )b_{2}$ ${\ displaystyle b_ {0} + 2b_ {1} x + b_ {2} x ^ {2} = (\ alpha + x) (x- \ beta) b_ {2}}$ . Probability Density: $f(x)={\begin{cases}{\frac {(\alpha +\beta )^{-(m+n+1)}}{B(-m-n-1,n+1)}}(x+\alpha )^{m}(x-\beta )^{n},&x>\beta ,m-1>0,n>-1\\0,&x\leqslant \beta \end{cases}}$ ${\ displaystyle f (x) = {\ begin {cases} {\ frac {(\ alpha + \ beta) ^ {- (m + n + 1)}} {B (-mn-1, n + 1)} } (x + \ alpha) ^ {m} (x- \ beta) ^ {n}, & x> \ beta, m-1> 0, n> -1 \\ 0, & x \ leqslant \ beta \ end {cases} }}$ . ^[3]

VII type

The Pearson type VII distribution is the student distribution. Conditions: $D>0$ ${\ displaystyle D> 0}$ , $\lambda =0$ ${\ displaystyle \ lambda = 0}$ , $b_{0}+2b_{1}x+b_{2}x^{2}=(\alpha ^{2}+x^{2})b_{2}$ ${\ displaystyle b_ {0} + 2b_ {1} x + b_ {2} x ^ {2} = (\ alpha ^ {2} + x ^ {2}) b_ {2}}$ . Probability Density: $f(x)={\frac {\alpha }{B(m-{\frac {1}{2}},{\frac {1}{2}})}}(\alpha ^{2}+x^{2})^{-m}$ ${\ displaystyle f (x) = {\ frac {\ alpha} {B (m - {\ frac {1} {2}}, {\ frac {1} {2}})}} (\ alpha ^ {2 } + x ^ {2}) ^ {- m}}$ , $x\in (-\infty ,\infty )$ ${\ displaystyle x \ in (- \ infty, \ infty)}$ , $m\geqslant {\frac {1}{2}}$ ${\ displaystyle m \ geqslant {\ frac {1} {2}}}$ . ^[3]

VIII type

Conditions: $D<0$ ${\ displaystyle D <0}$ , $\lambda <0$ ${\ displaystyle \ lambda <0}$ , $b_{0}+2b_{1}x+b_{2}x^{2}=x(x+\alpha )b_{2}$ ${\ displaystyle b_ {0} + 2b_ {1} x + b_ {2} x ^ {2} = x (x + \ alpha) b_ {2}}$ . Probability Density: $f(x)={\begin{cases}{\frac {m+1}{\alpha ^{m+1}}}(x+\alpha )^{m},&x\in [-\alpha ,0],-1<m<0\\0,&x\notin [-\alpha ,0]\end{cases}}$ ${\ displaystyle f (x) = {\ begin {cases} {\ frac {m + 1} {\ alpha ^ {m + 1}}} (x + \ alpha) ^ {m}, & x \ in [- \ alpha , 0], - 1 <m <0 \\ 0, & x \ notin [- \ alpha, 0] \ end {cases}}}$ . ^[3]

IX type

Conditions: $D<0$ ${\ displaystyle D <0}$ , $\lambda <0$ ${\ displaystyle \ lambda <0}$ , $b_{0}+2b_{1}x+b_{2}x^{2}=x(x+\alpha )b_{2}$ ${\ displaystyle b_ {0} + 2b_ {1} x + b_ {2} x ^ {2} = x (x + \ alpha) b_ {2}}$ . Probability Density: $f(x)={\begin{cases}{\frac {m+1}{\alpha ^{m+1}}}(x+\alpha )^{m},&x\in [-\alpha ,0],m<-1\\0,&x\notin [-\alpha ,0]\end{cases}}$ ${\ displaystyle f (x) = {\ begin {cases} {\ frac {m + 1} {\ alpha ^ {m + 1}}} (x + \ alpha) ^ {m}, & x \ in [- \ alpha , 0], m <-1 \\ 0, & x \ notin [- \ alpha, 0] \ end {cases}}}$ . ^[3]

X type

The Pearson type X distribution is an exponential distribution. Conditions: $D=0$ ${\ displaystyle D = 0}$ , $\lambda =0$ ${\ displaystyle \ lambda = 0}$ , $b_{0}+2b_{1}x+b_{2}x^{2}=b_{0}$ ${\ displaystyle b_ {0} + 2b_ {1} x + b_ {2} x ^ {2} = b_ {0}}$ , $a_{1}=0$ ${\ displaystyle a_ {1} = 0}$ . Probability Density: $f(x)={\begin{cases}\gamma e^{-\gamma x},&x>0,\gamma >0\\0,&x\leqslant 0\end{cases}}$ ${\ displaystyle f (x) = {\ begin {cases} \ gamma e ^ {- \ gamma x}, & x> 0, \ gamma> 0 \\ 0, & x \ leqslant 0 \ end {cases}}}$ ^[2]

XI Type

The Pearson type XI distribution is the normal distribution. Conditions: $D=0$ ${\ displaystyle D = 0}$ , $\lambda$ ${\ displaystyle \ lambda}$ vaguely $b_{0}+2b_{1}x+b_{2}x^{2}=b_{0}$ ${\ displaystyle b_ {0} + 2b_ {1} x + b_ {2} x ^ {2} = b_ {0}}$ . Probability Density: $f(x)={\frac {1}{{\sqrt {2\pi }}\sigma }}\exp {-{\frac {x^{2}}{2\sigma ^{2}}}},x\in (-\infty ,\infty )$ ${\ displaystyle f (x) = {\ frac {1} {{\ sqrt {2 \ pi}} \ sigma}} \ exp {- {\ frac {x ^ {2}} {2 \ sigma ^ {2} }}}, x \ in (- \ infty, \ infty)}$ . ^[2]

XII type

Conditions as for type I with additional conditions $\alpha =\beta ,m=-n$ ${\ displaystyle \ alpha = \ beta, m = -n}$ . ^[one]

Notes

↑ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ Korolyuk, 1985 , p. 133.
↑ ¹ ² ³ Korolyuk, 1985 , p. 135.
↑ ¹ ² ³ ⁴ ⁵ ⁶ Korolyuk, 1985 , p. 134.

Literature

Korolyuk V.S. , Portenko N.I. , Skorohod A.V. , Turbin A.F. Handbook of probability theory and mathematical statistics. - M .: Nauka, 1985 .-- 640 p.

[_5872716a89a83134-1] ¹ ² ³ ⁴ ⁵ ⁶ ⁷ Korolyuk, 1985 , p. 133.

[_5872716a89a83132-2] ¹ ² ³ Korolyuk, 1985 , p. 135.

[_5872716a89a83133-3] ¹ ² ³ ⁴ ⁵ ⁶ Korolyuk, 1985 , p. 134.