THE TRANSMUTED HALF-NORMAL DISTRIBUTION WITH APPLICATION TO PRECIPITATION DATA

Menezes, André; Mazucheli, Josmar; Cardoso, Jean; Chakraborty, Subrata

doi:10.1590/0101-7438.2020.040.00216792

André Menezes ^**Corresponding author

Department of Statistics, Universidade Estadual de Maringá, DEs, PR, Brazil - E-mail: andrefelipemaringa@gmail.com

https://orcid.org/0000-0002-3320-9834

Josmar Mazucheli

Department of Statistics, Universidade Estadual de Maringá, DEs, PR, Brazil - E-mail: jmazucheli@gmail.com

https://orcid.org/0000-0001-6740-0445

Jean Cardoso

Department of Statistics, Universidade Estadual de Maringá, DEs, PR, Brazil - E-mail: jeancarlos.card@gmail.com

https://orcid.org/0000-0002-1539-1564

Subrata Chakraborty

Department of Statistics, Dibrugarh University, India - E-mail: subrata_stats@dibru.ac.in

https://orcid.org/0000-0002-6405-1486

*Corresponding author

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Figure 1
Behavior of the p.d.f. of the THN distribution.

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Figure 2
Behavior of the h.r.f. of the THN distribution.

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Figure 3
Estimated bias and mean square error of θ (θ = 1, 1 : λ = −0.9, 2 : λ = −0.7, 3 : λ = −0.5, 4 : λ = −0.3, 5 : λ = 0, 6 : λ = 0.3, 7 : λ = 0.5, 8 : λ = 0.7 and 9 : λ = 0.9).

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Figure 4
Estimated bias and mean square error of λ (θ = 1, 1 : λ = −0.9, 2 : λ = −0.7, 3 : λ = −0.5, 4 : λ = −0.3, 5 : λ = 0, 6 : λ = 0.3, 7 : λ = 0.5, 8 : λ = 0.7 and 9 : λ = 0.9).

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Figure 5
TTT Plot.

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Figure 6
The empirical and estimated means.

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Table 1
Descriptive statistics.

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Table 2
Maximum likelihood estimates (standard errors).

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Table 3
Comparison criterion.

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Table 4
Parameter estimates and 95% confidence intervals.

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Table 5
Empirical and estimated means for each ten-day period.

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Table 6
Model selection measures.

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Distribution	θ	α	λ	a	b	p
HN	242.4746	.	.	.	.	.
	(31.3031)	.	.	.	.	.
GHN	235.5162	.	.	.	.	.
	(30.4049)	.	.	.	.	.
PHN	211.7649	1.4462	.	.	.	.
	(28.9254)	(0.3307)	.	.	.	.
THN	202.6784	.	-0.6767	.	.	.
	(24.8288)	.	(0.3097)	.	.	.
GHN II	260.1236	1.2904	.	.	.	.
	(28.3470)	(0.1952)	.	.	.	.
TGHN	201.8721	0.9952	-0.6852	.	.	.
	(59.6638)	(0.3247)	(0.6450)	.	.	.
OLLGHN	262.3615	1.2245	.	1.0701	.	.
	(33.1280)	(0.4707)	.	(0.4822)	.	.
GMHN	120.00	1.2035	.	.	2.4547	.
	(324.49)	(0.9856)	.	.	(12.5272)	.
BGHN	94.6287	1.0069	.	1.1730	0.2371	.
	(189.44)	(0.4544)	.	(0.8355)	(0.6290)	.
EGHN	49.9935	0.9418	.	0.0947	1.0646	.
	(989.30)	(0.4977)	.	(3.3346)	(4.4612)	.
KGHN	640.00	8.0006	.	0.1852	3.9998	.
	(298.26)	(0.9535)	.	(0.0772)	(5.5428)	.
BGHNG	86.8732	1.1999	.	0.9728	0.0806	0.5568
	(¡0.001)	(¡0.001)	.	(0.2289)	(¡0.001)	(¡0.001)

Distribution	−2 × Log-Like	AIC	AICc	BIC	KS	AD	CvM	Total
HN	373.0016⁷	375.0016⁵	375.1444⁵	376.4028²	0.8602⁷	0.7372⁶	0.1312⁷	39⁷
GHN	371.2544⁶	373.2544¹	373.3972¹	374.6556¹	0.7476⁶	23.9874⁷	0.0954⁶	28⁴
PHN	370.5842⁵	374.5842⁴	375.0287⁴	377.3866⁵	0.3984³	0.1498²	0.0167¹	24³
THN	370.4147²	374.4147²	374.8591²	377.2170³	0.3802²	0.1500³	0.0192³	17¹
GHN II	370.4468⁴	374.4468³	374.8913³	377.2492⁴	0.4327⁵	0.1721⁵	0.0224⁵	29⁵
TGHN	370.4144¹	376.4144⁶	377.3375⁶	380.6180⁶	0.3777¹	0.1488¹	0.0189²	23²
OLLGHN	370.4232³	376.4232⁷	377.3463⁷	380.6268⁷	0.4035⁴	0.1589⁴	0.0196⁴	36⁶

		HN		GHN II		THN
	Estimate	C.I.	Estimate	C.I.	Estimate	C.I.
β ₀	5.0942	(4.7007, 5.4876)	5.0119	(4.5392, 5.4846)	5.4141	(4.9208, 5.9075)
β ₁	-0.0785	(-0.1064, -0.0507)	-0.0795	(-0.1129, -0.0462)	-0.0792	(-0.1123, -0.0462)
α	-	-	0.8348	(0.7700, 0.8996)	-	-
λ	-	-	-	-	0.7822	(0.5896, 0.9747)

Ten-Day	Empirical	HN	GHN II	THN
10	51.6089	59.3420	55.1367	54.8366
11	48.1578	54.8614	50.9212	50.6590
12	49.1844	50.7191	47.0280	46.7996
13	41.0744	46.8896	43.4325	43.2342
14	41.1256	43.3492	40.1118	39.9404
15	37.8395	40.0761	37.0451	36.8976
16	32.3732	37.0502	34.2128	34.0866
17	31.9098	34.2528	31.5970	31.4897
18	26.2780	31.6665	29.1813	29.0907

Criteria	HN	GHN II	THN
−2 × Log-Like	3631.9093	3608.9037	3597.1897
AIC	3635.9093	3614.9037	3603.1897
BIC	3643.8106	3626.7556	3615.0416
SSR	207.2568	43.8623	40.3417

h (x | θ, λ) = \frac{\frac{2}{θ} ϕ (\frac{x}{θ}) [1 + λ - 2 λ (2 Φ (\frac{x}{θ}) - 1)]}{1 - (1 + λ) (2 Φ (\frac{x}{θ}) - 1) + λ {(2 Φ (\frac{x}{θ}) - 1)}^{2}} .

\begin{array}{l} h (x | θ, λ) = \frac{[1 + λ - 2 λ (2 Φ (\frac{x}{θ}) - 1)]}{1 - λ (2 Φ (\frac{x}{θ}) - 1)} \frac{\frac{2}{θ} ϕ (\frac{x}{θ})}{2 - 2 Φ (\frac{x}{θ})} \\ \begin{matrix} = \frac{[1 + λ - 2 λ (2 Φ (\frac{x}{θ} - 1))]}{1 - λ (2 Φ (\frac{x}{θ}) - 1)} \end{matrix} h_{H N} (x | θ), \end{array}

\begin{array}{l} m (t ∣ θ, λ) = \frac{- π^{\frac{- 1}{2}}}{S (t ∣ θ, λ)} {2^{\frac{3}{2}} θ λ e^{\frac{- t^{2}}{2 θ^{2}}} γ (t) - (λ + 1) θ \sqrt{2} e^{\frac{- t^{2}}{2 θ^{2}}} - 2 t \sqrt{π} [Φ (\frac{t}{θ}) - 1] + \\ \begin{matrix}  \end{matrix} - 4 θ λ [Φ (\frac{t \sqrt{2}}{θ}) - 1] + λ t \sqrt{π} γ (t) (γ (t) - 1)} \end{array}

\begin{array}{l} m (t ∣ θ, λ) = \frac{- π^{\frac{- 1}{2}}}{S (t ∣ θ, λ)} {2^{\frac{3}{2}} θ λ e^{\frac{- t^{2}}{2 θ^{2}}} [2 Φ (\frac{t}{θ}) - 1] - 2 θ λ [2 Φ (\frac{t \sqrt{2}}{θ}) - 1] + \\ \begin{matrix}  \end{matrix} + λ t \sqrt{π} {[2 Φ (\frac{t}{θ}) - 1]}^{2} + t \sqrt{π} - t \sqrt{π} [2 Φ (\frac{t}{θ}) - 1] + \\ \begin{matrix}  \end{matrix} - λ t \sqrt{π} [2 Φ (\frac{t}{θ}) - 1] - λ θ \sqrt{2} e^{\frac{- t^{2}}{2 θ^{2}}} - θ \sqrt{2} e^{\frac{- t^{2}}{2 θ^{2}}} + 2 λ θ} . \end{array}

\begin{array}{l} m (t ∣ θ, λ) = \frac{- π^{\frac{- 1}{2}}}{S (t ∣ θ, λ)} {2^{\frac{3}{2}} θ λ e^{\frac{- t^{2}}{2 θ^{2}}} [2 Φ (\frac{t}{θ}) - 1] - (λ + 1) θ \sqrt{2} e^{\frac{- t^{2}}{2 θ^{2}}} - 2 t \sqrt{π} [Φ (\frac{t}{θ}) - 1] + \\ \begin{matrix}  \end{matrix} - 4 θ λ [Φ (\frac{t \sqrt{2}}{θ}) - 1] + λ t \sqrt{π} [2 Φ (\frac{t}{θ}) - 1] [(2 Φ (\frac{t}{θ}) - 1) - 1]} . \end{array}

E (X^{k}) = \frac{θ^{k} 2^{\frac{k}{2}}}{π^{3 / 2}} \{(λ + 1) π Γ (\frac{k + 1}{2}) - 2 λ k \sqrt{π} H g e o [\frac{1}{2}, \frac{k}{2} + 1; \frac{3}{2}; - 1] Γ (\frac{k}{2})\}

\begin{array}{l} E (X^{k}) = \int_{0}^{\infty} x^{k} \frac{2}{θ} ϕ (\frac{x}{θ}) [1 + λ - 2 λ (2 Φ (\frac{x}{θ}) - 1)] d x \\ \begin{matrix} = (1 + λ) \int_{0}^{\infty} \frac{2}{θ} x^{k} ϕ (\frac{x}{θ}) d x - 2 λ \int_{0}^{\infty} \frac{2}{θ} \end{matrix} x^{k} ϕ (\frac{x}{θ}) (2 Φ (\frac{x}{θ}) - 1) d x \end{array}

E (X^{k}) = \frac{θ^{k} 2^{\frac{k}{2}}}{π^{3 / 2}} \{(λ + 1) π Γ (\frac{k + 1}{2}) - 2 λ k \sqrt{π} H g e o [\frac{1}{2}, \frac{k}{2} + 1; \frac{3}{2}; - 1] Γ (\frac{k}{2})\} .

\begin{array}{l} S k w (X) = \frac{E {(X - E (X))}^{3}}{V a r {(X)}^{3 / 2}} \\ \begin{matrix}  \end{matrix} = [\sqrt{2} (4 - π) + λ (\sqrt{2} (18 - π) + π - 24) + λ^{2} (42 \sqrt{2} - 60) \\ \begin{matrix}  \end{matrix} + λ^{3} (28 \sqrt{2} - 40)] \times {[(π - 2) + (4 \sqrt{2} - 6) (λ - λ^{2})]}^{- 3 / 2} \end{array}

\begin{array}{l} K u r (X) = \frac{E {(X - E (X))}^{4}}{V a r {(X)}^{2}}, \\ \begin{matrix}  \end{matrix} = [(3 π^{2} - 4 π - 12) + λ (4 π (3 \sqrt{2} - 4) - 48 \sqrt{2} - 72) \\ \begin{matrix}  \end{matrix} + λ^{2} (4 π (3 \sqrt{2} - 5) + 192 \sqrt{2} - 264) + λ^{3} (288 \sqrt{2} - 408) \\ \begin{matrix}  \end{matrix} + λ^{4} (144 \sqrt{2} - 204)] \times {[(π - 2) + (4 \sqrt{2} - 6) (λ - λ^{2})]}^{- 2} . \end{array}

\begin{array}{l} M_{X} (t) = - 2 λ \sum_{n = 0}^{\infty} \frac{t^{n}}{n!} H g e o [\frac{1}{2}, \frac{n}{2} + 1; \frac{3}{2}; - 1] Γ (\frac{n}{2} + 1) \frac{θ^{n} 2^{n / 2 + 1}}{π} + \\ \begin{matrix}  \end{matrix} + (1 + λ) \sum_{n = 0}^{\infty} \frac{t^{n}}{n!} Γ (\frac{n + 1}{2}) \frac{θ^{n} 2^{n / 2}}{\sqrt{π}} . \end{array}

\begin{array}{l} M_{x} (t) = (1 + λ) \sum_{n = 0}^{\infty} \frac{t^{n}}{n!} \int_{0}^{\infty} \frac{2}{θ} x^{n} ϕ (\frac{x}{θ}) d x + \\ \begin{matrix}  \end{matrix} - 2 λ \sum_{n = 0}^{\infty} \frac{t^{n}}{n!} \int_{0}^{\infty} \frac{2}{θ} x^{n} ϕ (\frac{x}{θ}) (2 Φ (\frac{x}{θ}) - 1) d x . \end{array}

\begin{array}{l} M_{X} (t) = - 2 λ \sum_{n = 0}^{\infty} \frac{t^{n}}{n!} H g e o [\frac{1}{2}, \frac{n}{2} + 1; \frac{3}{2}; - 1] Γ (\frac{n}{2} + 1) \frac{θ^{n} 2^{n / 2 + 1}}{π} \\ \begin{matrix}  \end{matrix} + (1 + λ) \sum_{n = 0}^{\infty} \frac{t^{n}}{n!} Γ (\frac{n + 1}{2}) \frac{θ^{n} 2^{n / 2}}{\sqrt{π}} . \end{array}

\begin{array}{l} H_{S h} (X) = - \log (\frac{2}{θ}) + \frac{π (1 - \log 2 π) - 2 λ}{2 π} + \\ \begin{matrix}  \end{matrix} + 2 λ^{2} \{1 - \log [(1 + λ) (1 - λ)]\} - λ (λ^{2} + 1) \log (\frac{1 + λ}{1 - λ}) . \end{array}

\begin{array}{l} H_{S h} (X) = - \int_{0}^{\infty} \frac{2}{θ} ϕ (\frac{x}{θ}) [1 + λ - 2 λ (2 Φ (\frac{x}{θ}) - 1)] \times \\ \begin{matrix} \times \log \{\frac{2}{θ} ϕ (\frac{x}{θ}) [1 + λ - 2 λ (2 Φ (\frac{x}{θ}) - 1)]\} d x . \end{matrix} \end{array}

\begin{array}{l} H_{S h} (X) = - \int_{0}^{\infty} \frac{2}{θ} ϕ [1 + λ - 2 λ (2 Φ (\frac{x}{θ}) - 1)] \times \\ \begin{matrix} \times \{\log (\frac{2}{θ}) + \log (ϕ (\frac{x}{θ})) + \log [1 + λ - 2 λ (2 Φ (\frac{x}{θ}) - 1)]\} d x \end{matrix} \\ \begin{matrix} = - \int_{0}^{\infty} \log (\frac{2}{θ}) \{\frac{2}{θ} ϕ (\frac{x}{θ}) [1 + λ - 2 λ (2 Φ (\frac{x}{θ}) - 1)]\} + \end{matrix} \\ \begin{matrix}  \end{matrix} + \log (ϕ (\frac{x}{θ})) \{\frac{2}{θ} ϕ (\frac{x}{θ}) [1 + λ - 2 λ (2 Φ (\frac{x}{θ}) - 1)]\} + \\ \begin{matrix}  \end{matrix} + \log (1 + λ - 2 λ (2 Φ (\frac{x}{θ}) - 1)) \{\frac{2}{θ} ϕ (\frac{x}{θ}) [1 + λ - 2 λ (2 Φ (\frac{x}{θ}) - 1)]\} d x . \end{array}

\begin{array}{l} H_{S h} (X) = - \int_{0}^{\infty} \log (\frac{2}{θ}) \{\frac{2}{θ} ϕ (\frac{x}{θ}) [1 + λ - 2 λ (2 Φ (\frac{x}{θ}) - 1)]\} d x + \\ \begin{matrix}  \end{matrix} - \int_{0}^{\infty} \log (ϕ (\frac{x}{θ})) \{\frac{2}{θ} ϕ (\frac{x}{θ}) [1 + λ - 2 λ (2 Φ (\frac{x}{θ}) - 1)]\} d x + \\ \begin{matrix}  \end{matrix} - \int_{0}^{\infty} \log (1 + λ - 2 λ (2 Φ (\frac{x}{θ}) - 1)) \{\frac{2}{θ} ϕ (\frac{x}{θ}) [1 + λ - 2 λ (2 Φ (\frac{x}{θ}) - 1)]\} d x . \end{array}

\begin{array}{l} H_{S h} (X) = - \log (\frac{2}{θ}) + \frac{π (1 - \log 2 π) - 2 λ}{2 π} + \\ \begin{matrix}  \end{matrix} + 2 λ^{2} \{1 - \log [(1 + λ) (1 - λ)]\} - λ (λ^{2} + 1) \log (\frac{1 + λ}{1 - λ}) . \end{array}

\begin{array}{l} \int_{0}^{μ} x f (x | θ, λ) d x = \frac{θ}{\sqrt{π}} \{\sqrt{2} (1 + λ) + 2 λ [2 Φ (\frac{μ \sqrt{2}}{θ}) - 1]\} + \\ \begin{matrix}  \end{matrix} - θ^{2} \{\frac{2}{θ} ϕ (\frac{μ}{θ}) [1 + λ - 2 λ (2 Φ (\frac{μ}{θ}) - 1)]\}, \end{array}

\begin{array}{l} B (p) = \frac{- θ^{2}}{p μ} {\frac{2}{θ} ϕ (\frac{q}{θ}) [1 + λ - 2 λ (2 Φ (\frac{q}{θ}) - 1)] + \\ \begin{matrix}  \end{matrix} - \frac{1}{θ \sqrt{π}} [\sqrt{2} (1 + λ) + 2 λ (2 Φ (\frac{q \sqrt{2}}{θ}) - 1)]} . \end{array}

\begin{array}{l} L (p) = \frac{- θ^{2}}{μ} {\frac{2}{θ} ϕ (\frac{q}{θ}) [1 + λ - 2 λ (2 Φ (\frac{q}{θ}) - 1)] + \\ \begin{matrix}  \end{matrix} - \frac{1}{θ \sqrt{π}} [\sqrt{2} (1 + λ) + 2 λ (2 Φ (\frac{q \sqrt{2}}{θ}) - 1)]} . \end{array}

\begin{array}{l} B (p) = \frac{1}{p μ} \int_{0}^{q} x \frac{2}{θ} ϕ (\frac{x}{θ}) [1 + λ - 2 λ (2 Φ (\frac{x}{θ}) - 1)] \\ \begin{matrix}  \end{matrix} = \frac{- θ^{2}}{p μ} [\frac{2}{θ} ϕ (\frac{q}{θ}) 2 λ (2 Φ (\frac{q}{θ}) - 1) - λ \frac{2}{θ} ϕ (\frac{q}{θ}) + \\ \begin{matrix}  \end{matrix} - \frac{2}{θ} ϕ (\frac{q}{θ}) - \frac{2 λ}{θ \sqrt{π}} (2 Φ (\frac{q \sqrt{2}}{θ}) - 1) + \frac{\sqrt{2} λ + \sqrt{2}}{θ \sqrt{π}}] \\ \begin{matrix}  \end{matrix} \frac{- θ^{2}}{p μ} {\frac{2}{θ} ϕ (\frac{q}{θ}) [1 + λ - 2 λ (2 Φ (\frac{q}{θ}) - 1)] + \\ \begin{matrix}  \end{matrix} - \frac{1}{θ \sqrt{π}} [\sqrt{2} (1 + λ) + 2 λ (2 Φ (\frac{q \sqrt{2}}{θ}) - 1)] . \end{array}

\begin{array}{l} f_{j} (x ∣ θ, λ) = \frac{n!}{(j - 1)! (n - j)!} \{\frac{2}{θ} ϕ (\frac{x}{θ}) [1 + λ - 2 λ (2 Φ (\frac{x}{θ}) - 1)]\} \times \\ \begin{matrix}  \end{matrix} \times {\{(1 + λ) (2 Φ (\frac{x}{θ}) - 1) - λ {(2 Φ (\frac{x}{θ}) - 1)}^{2}\}}^{j - 1} \times \\ \begin{matrix}  \end{matrix} \times {\{1 - (1 + λ) (2 Φ (\frac{x}{θ}) - 1) + λ {(2 Φ (\frac{x}{θ}) - 1)}^{2}\}}^{n - j} \end{array}

\begin{array}{l} f_{j} (x | θ, λ) = \frac{n!}{(j - 1)! (n - j)!} [\frac{2}{θ} ϕ (\frac{x}{θ}) (1 + λ - 2 λ γ (x))] \times \\ \begin{matrix}  \end{matrix} \times {[(1 + λ) γ (x) - λ γ {(x)}^{2}]}^{j - 1} {[1 - (1 + λ) γ (x) + λ γ {(x)}^{2}]}^{n - j} . \end{array}

\begin{array}{l} f_{j} (x | θ, λ) = \frac{n!}{(j - 1)! (n - j)!} \frac{2}{θ} ϕ (\frac{x}{θ}) γ {(x)}^{j - 1} (1 + λ - 2 λ γ (x)) \times \\ \begin{matrix}  \end{matrix} \times {(1 + λ - λ γ (x))}^{j - 1} {(γ (x) λ - 1)}^{n - j} {(γ (x) - 1)}^{n - j} . \end{array}

R = P (X_{2} < X_{1}) = \int_{0}^{\infty} f_{1} (x ∣ θ, λ) F_{2} (x ∣ θ, λ) d x \begin{matrix}  \end{matrix} = (1 + λ_{1}) (1 + λ_{2}) \frac{4}{θ_{1} π} \sum_{j = 1}^{\infty} \frac{{(- 1)}^{j + 1} {(\frac{\sqrt{2}}{2 θ_{2}})}^{j + 1} θ^{2 j + 2} 2^{- \frac{1}{2} + j} Γ (j + 1)}{(2 j - 1) (j - 1)!} + \begin{matrix}  \end{matrix} - (1 + λ_{1}) λ_{2} \frac{8}{θ_{1} π^{\frac{3}{2}}} \sum_{j = 1}^{\infty} \sum_{l = 1}^{\infty} \frac{{(- 1)}^{j + l + 2} {(\frac{\sqrt{2}}{2 θ_{2}})}^{2 j + 2 l + 2} θ^{2 j + 2 l + 3} 2^{j + l} Γ (j + l + \frac{3}{2})}{(2 j - 1) (j - 1)! (2 l - 1) (l - 1)!} + \begin{matrix}  \end{matrix} - λ_{1} (1 + λ_{2}) \frac{16}{θ_{1} π^{\frac{3}{2}}} \sum_{k = 1}^{\infty} \sum_{j = 1}^{\infty} \frac{{(- 1)}^{k + j + 2} {(\frac{\sqrt{2}}{2 θ_{1}})}^{2 k + 1} {(\frac{\sqrt{2}}{2 θ_{2}})}^{2 j + 1} θ^{2 k + 2 l + 3} 2^{k + l} Γ (k + l + \frac{3}{2})}{(2 k - 1) (k - 1)! (2 j - 1) (j - 1)!} + \begin{matrix}  \end{matrix} + λ_{1} λ_{2} \frac{32}{θ_{1} π^{2}} \sum_{k = 1}^{\infty} \sum_{j = 1}^{\infty} \sum_{l = 1}^{\infty} \frac{{(- 1)}^{k + j + l + 3} {(\frac{\sqrt{2}}{2 θ_{1}})}^{2 k + 1} {(\frac{\sqrt{2}}{2 θ_{2}})}^{2 j + 2 l + 2} θ^{2 k + 2 l + 2 j + 4} 2^{\frac{1}{2} + k + l + j}}{(2 k - 1) (k - 1)! (2 j - 1) (- 1)! (2 l - 1) (l - 1)!} \times \begin{matrix}  \end{matrix} \times Γ (k + l + j + 2) .

\begin{array}{l} R = P (X_{2} < X_{1}) = \int_{0}^{\infty} \frac{2}{θ_{1}} ϕ (\frac{x}{θ_{1}}) [1 + λ_{1} - 2 λ_{1} (2 Φ (\frac{x}{θ_{1}}) - 1)] \times \\ \begin{matrix} \times [(1 + λ_{2}) (2 Φ (\frac{x}{θ_{2}}) - 1) - λ_{2} {(2 Φ (\frac{x}{θ_{2}}) - 1)}^{2}] d x . \end{matrix} \end{array}

\begin{array}{l} R = \int_{0}^{\infty} \frac{2}{θ_{1}} ϕ (\frac{x}{θ_{1}}) \{1 + λ_{1} - 2 λ_{1} [\frac{2}{\sqrt{π}} \sum_{k = 1}^{\infty} \frac{{(- 1)}^{k + 1} {(\frac{\sqrt{2}}{2 θ_{1}})}^{2 k + 1} x^{2 k + 1}}{(2 k - 1) (k - 1)!}]\} \times \\ \begin{matrix}  \end{matrix} \times \{(1 + λ_{2}) (2 Φ (\frac{x}{θ_{2}}) - 1) - λ_{2} {[\frac{2}{\sqrt{π}} \sum_{j = 1}^{\infty} \frac{{(- 1)}^{j + 1} {(\frac{\sqrt{2}}{2 θ_{2}})}^{2 j + 1} x^{2 j + 1}}{(2 j - 1) (j - 1)!}]}^{2}\} d x . \end{array}

R = (1 + λ_{1}) (1 + λ_{2}) \frac{4}{θ_{1} \sqrt{π}} \sum_{j = 1}^{\infty} \frac{{(- 1)}^{j + 1} {(\frac{\sqrt{2}}{2 θ_{2}})}^{2 j + 1} \int_{0}^{\infty} ϕ (\frac{x}{θ_{1}}) x^{2 j + 1} d x}{(2 j - 1) (j - 1)!} +

\begin{matrix}  \end{matrix} - (1 + λ_{1}) λ_{2} \frac{8}{θ_{1} π} \sum_{j = 1}^{\infty} \sum_{l = 1}^{\infty} \frac{{(- 1)}^{j + l + 2} {(\frac{\sqrt{2}}{2 θ_{2}})}^{2 j + 2 l + 2} \int_{0}^{\infty} ϕ (\frac{x}{θ_{1}}) x^{2 j + 2 l + 2} d x}{(2 j - 1) (j - 1)! (2 l - 1) (l - 1)!} +

\begin{matrix}  \end{matrix} - λ_{1} (1 + λ_{2}) \frac{16}{θ_{1} π} \sum_{k = 1}^{\infty} \sum_{j = 1}^{\infty} \frac{{(- 1)}^{k + j + 2} {(\frac{\sqrt{2}}{2 θ_{1}})}^{2 k + 1} {(\frac{\sqrt{2}}{2 θ_{2}})}^{2 j + 1} \int_{0}^{\infty} ϕ (\frac{x}{θ_{1}}) x^{2 k + 2 j + 2} d x}{(2 k - 1) (k - 1)! (2 j - 1) (j - 1)!} +

+ λ_{1} λ_{2} \frac{32}{θ_{1} π^{\frac{3}{2}}} \sum_{k = 1}^{\infty} \sum_{j = 1}^{\infty} \sum_{l = 1}^{\infty} \frac{{(- 1)}^{k + j + l + 3} {(\frac{\sqrt{2}}{2 θ_{1}})}^{2 k + 1} {(\frac{\sqrt{2}}{2 θ_{2}})}^{2 j + 2 l + 2} \int_{0}^{\infty} ϕ (\frac{x}{θ_{1}}) x^{2 k + 2 j + 2 l + 3} d x}{(2 k - 1) (k - 1)! (2 j - 1) (j - 1)! (2 l - 1) (l - 1)!}

s^{'} (x) = \frac{\frac{1}{θ} ϕ (\frac{x}{θ}) w_{1} (x)}{Φ (\frac{x}{θ}) w_{1} (x) - w_{1} (x) \{2 Φ (\frac{x}{θ}) - 1\}} = \frac{d}{d x} \{- \log (1 - Φ (\frac{x}{θ}))\}, x \geq 0,

\begin{array}{l} \frac{\partial l (θ, λ | x)}{\partial λ} = \sum_{i = 1}^{n} \frac{- 4 Φ (\frac{x_{i}}{θ}) + 3}{- 4 λ Φ (\frac{x_{i}}{θ}) + 3 λ + 1} \\ \frac{\partial l (θ, λ | x)}{\partial λ} = - \frac{n}{θ} - \frac{1}{θ^{2}} \sum_{i = 1}^{n} \frac{ϕ' (\frac{x_{i}}{θ})}{ϕ (\frac{x_{i}}{θ})} x_{i} + \frac{4 λ}{θ^{2}} \sum_{i = 1}^{n} \frac{ϕ (\frac{x_{i}}{θ})}{- 4 λ Φ (\frac{x_{i}}{θ}) + 3 λ + 1} x_{i} \end{array}

f_{5} (x) = \frac{\frac{2}{θ} ϕ (\frac{x}{θ})}{Γ (α) b^{α}} {\{- log [2 Φ (\frac{- x}{θ})]\}}^{α - 1} {[2 Φ (\frac{- x}{θ})]}^{\frac{1}{b} - 1}, θ, α, b > 0.

f_{6} (x) = \frac{a \sqrt{(\frac{2}{π})} (\frac{α}{x}) {(\frac{x}{θ})}^{α} e^{- \frac{1}{2} {(% \frac{x}{θ})}^{2 α}} {\{[2 Φ ({(\frac{x}{θ})}^{α}) - 1] [2 - 2 Φ ({(\frac{x}{θ})}^{α})]\}}^{a - 1}}{{\{{[2 Φ ({(\frac{x}{θ})}^{α}) - 1]}^{a} + {[2 - 2 Φ ({(\frac{x}{θ})}^{α})]}^{a}\}}^{2}}, θ, α, a > 0.

f_{8} (x) = \frac{2^{b} α}{a θ B (a, b)} {(\frac{x}{θ})}^{α - 1} ϕ ({(\frac{x}{θ})}^{α}) {[2 Φ ({(\frac{x}{θ})}^{α}) - 1]}^{a - 1} {[Φ (- {(\frac{x}{θ})}^{α})]}^{b - 1}, θ, α, a, b > 0

\begin{array}{l} f_{9} (x) = a b \sqrt{\frac{2}{π}} (\frac{α}{x}) {(\frac{x}{θ})}^{α} \exp (- \frac{1}{2} {(\frac{x}{% θ})}^{2 α}) {[2 - 2 Φ ({(\frac{x}{θ} %)}^{α})]}^{a - 1} \\ \begin{matrix}  \end{matrix} \times {\{1 - {[2 - 2 Φ ({(\frac{x}{θ})}^{α})]}^{a}\}}^{b - 1}, θ, α, a, b > 0. \end{array}

\begin{array}{l} f_{10} (x) = a b \sqrt{\frac{2}{π}} (\frac{α}{x}) {(\frac{x}{θ})}^{α} \exp (- \frac{1}{2} {(\frac{x}{θ})}^{2 α}) {[2 Φ ({(\frac{x}{θ})}^{α}) - 1]}^{a - 1} \\ \begin{matrix}  \end{matrix} \times {\{1 - {[2 Φ ({(\frac{x}{θ})}^{α}) - 1]}^{a}\}}^{b - 1}, θ, α, a, b > 0. \end{array}

\begin{array}{l} f_{11} (x) = (1 - p) \frac{2^{b - \frac{1}{2}} (\frac{a}{x}) {(\frac{x}{θ})}^{α} \exp (- \frac{1}{2} {(\frac{x}{θ})}^{2 α})}{\sqrt{π} B (a, b)} {[2 Φ ({(\frac{x}{θ})}^{α}) - 1]}^{a - 1} \\ \times {[1 - Φ ({(\frac{x}{θ})}^{α})]}^{b - 1} {[1 - p (1 - I_{2 Φ ({(\frac{x}{θ})}^{α}) - 1} (a, b))]}^{- 2}, θ, α, a, b > 0,0 \leq p < 1. \end{array}

[1] *Corresponding author

N. obs	30
Mean	206.93 mm
Std.	128.55 mm
Min.	8.20 mm
Q25	107.18 mm
Median	191.80 mm
Q75	207.2568
Max.	500.30 mm

Brasil

Brasil

THE TRANSMUTED HALF-NORMAL DISTRIBUTION WITH APPLICATION TO PRECIPITATION DATA

ABSTRACT