Package 'normtest'

Adjusted Jarque–Bera test for normality

Description

Performs adjusted Jarque–Bera test for the composite hypothesis of normality, see Urzua (1996).

Usage

ajb.norm.test(x, nrepl=2000)

Arguments

x	a numeric vector of data values.
nrepl	the number of replications in Monte Carlo simulation.

Details

The adjusted Jarque–Bera test for normality is based on the following statistic:

$AJB = \frac{(\sqrt{b_1})^2}{\mathrm{Var}\left(\sqrt{b_1}\right)} + \frac{(b_2 - \mathrm{E}\left(b_2\right))^2}{\mathrm{Var}\left(b_2\right)},$

where

$\sqrt{b_1} = \frac{\frac{1}{n}\sum_{i=1}^n(X_i - \overline{X})^3}{\left(\frac{1}{n}\sum_{i=1}^n(X_i - \overline{X})^2\right)^{3/2}}, \quad b_2 = \frac{\frac{1}{n}\sum_{i=1}^n(X_i - \overline{X})^4}{\left(\frac{1}{n}\sum_{i=1}^n(X_i - \overline{X})^2\right)^2},$

$\mathrm{Var}\left(\sqrt{b_1}\right) = \frac{6(n-2)}{(n+1)(n+3)}, \quad E\left(b_2\right) = \frac{3(n-1)}{n+1}, \quad \mathrm{Var}\left(b_2\right) = \frac{24n(n-2)(n-3)}{(n+1)^2(n+3)(n+5)}.$

The p-value is computed by Monte Carlo simulation.

Value

A list with class "htest" containing the following components:

statistic	the value of the adjusted Jarque–Bera statistic.
p.value	the p-value for the test.
method	the character string "Adjusted Jarque-Bera test for normality".
data.name	a character string giving the name(s) of the data.

Author(s)

Ilya Gavrilov and Ruslan Pusev

References

Urzua, C. M. (1996): On the correct use of omnibus tests for normality. — Economics Letters, vol. 53, pp. 247–251.

Examples

ajb.norm.test(rnorm(100))
ajb.norm.test(abs(runif(100,-2,5)))

---

Frosini test for normality

Description

Performs Frosini test for the composite hypothesis of normality, see e.g. Frosini (1987).

Usage

frosini.norm.test(x, nrepl=2000)

Arguments

x	a numeric vector of data values.
nrepl	the number of replications in Monte Carlo simulation.

Details

The Frosini test for normality is based on the following statistic:

$B_n = \frac{1}{\sqrt{n}}\sum_{i=1}^n{\left|\Phi(Y_i) - \frac{i-0.5}{n} \right|},$

where

$Y_i=\frac{X_{(i)}-\overline{X}}{s}, \quad s^2=\frac{1}{n}\sum_{i=1}^n(X_i-\overline{X})^2.$

The p-value is computed by Monte Carlo simulation.

Value

A list with class "htest" containing the following components:

statistic	the value of the Frosini statistic.
p.value	the p-value for the test.
method	the character string "Frosini test for normality".
data.name	a character string giving the name(s) of the data.

Author(s)

Ilya Gavrilov and Ruslan Pusev

References

Frosini, B.V. (1987): On the distribution and power of a goodness-of-fit statistic with parametric and nonparametric applications, "Goodness-of-fit". (Ed. by Revesz P., Sarkadi K., Sen P.K.) — Amsterdam-Oxford-New York: North-Holland. — Pp. 133–154.

Examples

frosini.norm.test(rnorm(100))
frosini.norm.test(runif(100,-1,1))

---

Geary test for normality

Description

Performs Geary test for the composite hypothesis of normality, see Geary (1935).

Usage

geary.norm.test(x, nrepl=2000)

Arguments

x	a numeric vector of data values.
nrepl	the number of replications in Monte Carlo simulation.

Details

The Geary test for normality is based on the following statistic:

$d = \frac{1}{ns}\sum_{i=1}^n|X_i-\overline{X}|,$

where

$s^2=\frac{1}{n}\sum_{i=1}^n(X_i-\overline{X})^2.$

The p-value is computed by Monte Carlo simulation.

Value

A list with class "htest" containing the following components:

statistic	the value of the Geary statistic.
p.value	the p-value for the test.
method	the character string "Geary test for normality".
data.name	a character string giving the name(s) of the data.

Author(s)

Ilya Gavrilov and Ruslan Pusev

References

Geary, R. C. (1935): The ratio of the mean deviation to the standard deviation as a test of normality. — Biometrika, vol. 27, pp. 310–332.

Examples

geary.norm.test(rnorm(100))
geary.norm.test(runif(100,-1,1))

---

Hegazy–Green test for normality

Description

Performs Hegazy–Green test for the composite hypothesis of normality, see e.g. Hegazy and Green (1975).

Usage

hegazy1.norm.test(x, nrepl=2000)

Arguments

x	a numeric vector of data values.
nrepl	the number of replications in Monte Carlo simulation.

Details

The Hegazy–Green test for normality is based on the following statistic:

$T_1 = \frac{1}{n}\sum_{i=1}^n{\left|Y_{i}-\Phi^{-1}{\left(\frac{i}{n+1}\right)}\right|},$

where

$Y_i=\frac{X_{(i)}-\overline{X}}{s}, \quad s^2=\frac{1}{n}\sum_{i=1}^n(X_i-\overline{X})^2.$

The p-value is computed by Monte Carlo simulation.

Value

A list with class "htest" containing the following components:

statistic	the value of the Hegazy–Green statistic.
p.value	the p-value for the test.
method	the character string "Hegazy-Green test for normality".
data.name	a character string giving the name(s) of the data.

Author(s)

Ilya Gavrilov and Ruslan Pusev

References

Hegazy, Y. A. S. and Green, J. R. (1975): Some new goodness-of-fit tests using order statistics. — Journal of the Royal Statistical Society. Series C (Applied Statistics), vol. 24, pp. 299–308.

Examples

hegazy1.norm.test(rnorm(100))
hegazy1.norm.test(runif(100,-1,1))

---

Hegazy–Green test for normality

Description

Performs Hegazy–Green test for the composite hypothesis of normality, see e.g. Hegazy and Green (1975).

Usage

hegazy2.norm.test(x, nrepl=2000)

Arguments

x	a numeric vector of data values.
nrepl	the number of replications in Monte Carlo simulation.

Details

The Hegazy–Green test for normality is based on the following statistic:

$T_2 = \frac{1}{n}\sum_{i=1}^n{\left(Y_{i}-\Phi^{-1}{\left(\frac{i}{n+1}\right)}\right)^2}.$

where

$Y_i=\frac{X_{(i)}-\overline{X}}{s}, \quad s^2=\frac{1}{n}\sum_{i=1}^n(X_i-\overline{X})^2.$

The p-value is computed by Monte Carlo simulation.

Value

A list with class "htest" containing the following components:

statistic	the value of the Hegazy–Green statistic.
p.value	the p-value for the test.
method	the character string "Hegazy-Green test for normality".
data.name	a character string giving the name(s) of the data.

Author(s)

Gavrilov Ilya and Ruslan Pusev

References

Hegazy, Y. A. S. and Green, J. R. (1975): Some new goodness-of-fit tests using order statistics. — Journal of the Royal Statistical Society. Series C (Applied Statistics), vol. 24, pp. 299–308.

Examples

hegazy2.norm.test(rnorm(100))
hegazy2.norm.test(runif(100,-1,1))

---

Jarque–Bera test for normality

Description

Performs Jarque–Bera test for the composite hypothesis of normality, see Jarque and Bera (1987).

Usage

jb.norm.test(x, nrepl=2000)

Arguments

x	a numeric vector of data values.
nrepl	the number of replications in Monte Carlo simulation.

Details

The Jarque–Bera test for normality is based on the following statistic:

$JB = \frac{n}{6}\left((\sqrt{b_1})^2 + \frac{(b_2-3)^2}{4}\right),$

where

$b_1 = \frac{\frac{1}{n}\sum_{i=1}^n(X_i - \overline{X})^3}{\frac{1}{n}(\sum_{i=1}^n(X_i - \overline{X})^2)^{3/2}},$

$b_2 = \frac{\frac{1}{n}\sum_{i=1}^n(X_i - \overline{X})^4}{\frac{1}{n}(\sum_{i=1}^n(X_i - \overline{X})^2)^2}.$

The p-value is computed by Monte Carlo simulation.

Value

A list with class "htest" containing the following components:

statistic	the value of the Jarque–Bera statistic.
p.value	the p-value for the test.
method	the character string "Jarque-Bera test for normality".
data.name	a character string giving the name(s) of the data.

Author(s)

Ilya Gavrilov and Ruslan Pusev

References

Jarque, C. M. and Bera, A. K. (1987): A test for normality of observations and regression residuals. — International Statistical Review, vol. 55, pp. 163–172.

Examples

jb.norm.test(rnorm(100))
jb.norm.test(abs(runif(100,-2,5)))

---

Kurtosis test for normality

Description

Performs kurtosis test for the composite hypothesis of normality, see, e.g., Shapiro, Wilk and Chen (1968).

Usage

kurtosis.norm.test(x, nrepl=2000)

Arguments

x	a numeric vector of data values.
nrepl	the number of replications in Monte Carlo simulation.

Details

The kurtosis test for normality is based on the following statistic:

$b_2 = \frac{\frac{1}{n}\sum_{i=1}^n(X_i - \overline{X})^4}{\left(\frac{1}{n}\sum_{i=1}^n(X_i - \overline{X})^2\right)^2},$

The p-value is computed by Monte Carlo simulation.

Value

A list with class "htest" containing the following components:

statistic	the value of the test statistic.
p.value	the p-value for the test.
method	the character string "Kurtosis test for normality".
data.name	a character string giving the name(s) of the data.

Author(s)

Ilya Gavrilov and Ruslan Pusev

References

Shapiro, S. S., Wilk, M. B. and Chen, H. J. (1968): A comparative study of various tests for normality. — Journal of the American Statistical Association, vol. 63, pp. 1343–1372.

Examples

kurtosis.norm.test(rnorm(100))
kurtosis.norm.test(runif(100,-1,1))

---

Skewness test for normality

Description

Performs skewness test for the composite hypothesis of normality, see, e.g., Shapiro, Wilk and Chen (1968).

Usage

skewness.norm.test(x, nrepl=2000)

Arguments

x	a numeric vector of data values.
nrepl	the number of replications in Monte Carlo simulation.

Details

The skewness test for normality is based on the sample skewness:

$\sqrt{b_1} = \frac{\frac{1}{n}\sum_{i=1}^n(X_i - \overline{X})^3}{\left(\frac{1}{n}\sum_{i=1}^n(X_i - \overline{X})^2\right)^{3/2}},$

The p-value is computed by Monte Carlo simulation.

Value

A list with class "htest" containing the following components:

statistic	the value of the sample skewness.
p.value	the p-value for the test.
method	the character string "Skewness test for normality".
data.name	a character string giving the name(s) of the data.

Author(s)

Ilya Gavrilov and Ruslan Pusev

References

Shapiro, S. S., Wilk, M. B. and Chen, H. J. (1968): A comparative study of various tests for normality. — Journal of the American Statistical Association, vol. 63, pp. 1343–1372.

Examples

skewness.norm.test(rnorm(100))
skewness.norm.test(abs(runif(100,-2,5)))

---

Spiegelhalter test for normality

Description

Performs Spiegelhalter test for the composite hypothesis of normality, see Spiegelhalter (1977).

Usage

spiegelhalter.norm.test(x, nrepl=2000)

Arguments

x	a numeric vector of data values.
nrepl	the number of replications in Monte Carlo simulation.

Details

The Spiegelhalter test for normality is based on the following statistic:

$T = \left( (c_nu)^{-(n-1)}+g^{-(n-1)} \right)^{1/(n-1)},$

where

$u=\frac{X_{(n)}-X_{(1)}}{s}, \quad g=\frac{\sum_{i=1}^n|X_i-\overline{X}|}{s\sqrt{n(n-1)}}, \quad c_n=\frac{(n!)^{1/(n-1)}}{2n}, \quad s^2=\frac{1}{n-1}\sum_{i=1}^n(X_i-\overline{X})^2.$

The p-value is computed by Monte Carlo simulation.

Value

A list with class "htest" containing the following components:

statistic	the value of the Geary statistic.
p.value	the p-value for the test.
method	the character string "Spiegelhalter test for normality".
data.name	a character string giving the name(s) of the data.

Author(s)

Ilya Gavrilov and Ruslan Pusev

References

Spiegelhalter, D. J. (1977): A test for normality against symmetric alternatives. — Biometrika, vol. 64, pp. 415–418.

Examples

spiegelhalter.norm.test(rnorm(100))
spiegelhalter.norm.test(rexp(100))

---

Weisberg–Bingham test for normality

Description

Performs Weisberg–Bingham test for the composite hypothesis of normality, see Weisberg and Bingham (1975).

Usage

wb.norm.test(x, nrepl=2000)

Arguments

x	a numeric vector of data values.
nrepl	the number of replications in Monte Carlo simulation.

Details

The Weisberg–Bingham test for normality is based on the following statistic:

$WB = \frac{(\sum_{i=1}^nm_iX_{(i)})^2/\sum_{i=1}^nm_i^2}{\sum_{i=1}^n(X_i-\overline{X})^2},$

where

$m_i=\Phi^{-1}\left(\frac{i-3/8}{n+1/4}\right).$

The p-value is computed by Monte Carlo simulation.

Value

A list with class "htest" containing the following components:

statistic	the value of the Weisberg–Bingham statistic.
p.value	the p-value for the test.
method	the character string "Weisberg-Bingham test for normality".
data.name	a character string giving the name(s) of the data.

Author(s)

Ilya Gavrilov and Ruslan Pusev

References

Weisberg, S. and Bingham, C. (1975): An approximate analysis of variance test for non-normality suitable for machine calculation. — Technometrics, vol. 17, pp. 133–134.

Examples

wb.norm.test(rnorm(100))
wb.norm.test(runif(100,-1,1))

---