ppeak

This function computes \(\sum_{k} P(n,k)\), i.e. the probability that a
sequence of \(n\) independent and identically distributed random variables
contains \(\ge k\) \((\le k)\) peaks, with peaks as defined in
Goldfeld65;textualskedastic. The function may be used to
compute \(p\)-values for the Goldfeld-Quandt nonparametric test for
heteroskedasticity in a linear model. Computation time is very slow for
\(n &gt; 170\) if <code>usedata</code> is set to <code>FALSE</code>.

Implements numerous methods for testing for, modelling, and
correcting for heteroskedasticity in the classical linear regression
model. The most novel contribution of the
package is found in the functions that implement the as-yet-unpublished
auxiliary linear variance models and auxiliary nonlinear variance
models that are designed to estimate error variances in a heteroskedastic
linear regression model. These models follow principles of statistical
learning described in Hastie (2009) <doi:10.1007/978-0-387-21606-5>.
The nonlinear version of the model is estimated using quasi-likelihood
methods as described in Seber and Wild (2003, ISBN: 0-471-47135-6).
Bootstrap methods for approximate confidence intervals for error variances
are implemented as described in Efron and Tibshirani
(1993, ISBN: 978-1-4899-4541-9), including also the expansion technique
described in Hesterberg (2014) <doi:10.1080/00031305.2015.1089789>. The
wild bootstrap employed here follows the description in Davidson and
Flachaire (2008) <doi:10.1016/j.jeconom.2008.08.003>. Tuning of
hyper-parameters makes use of a golden section search function that is
modelled after the MATLAB function of Zarnowiec (2022)
<https://www.mathworks.com/matlabcentral/fileexchange/25919-golden-section-method-algorithm>.
A methodological description of the algorithm can be found in Fox (2021,
ISBN: 978-1-003-00957-3).
There are 25 different functions that implement hypothesis tests for
heteroskedasticity. These include a test based on Anscombe (1961)
<https://projecteuclid.org/euclid.bsmsp/1200512155>, Ramsey's (1969)
BAMSET Test <doi:10.1111/j.2517-6161.1969.tb00796.x>, the tests of Bickel
(1978) <doi:10.1214/aos/1176344124>, Breusch and Pagan (1979)
<doi:10.2307/1911963> with and without the modification
proposed by Koenker (1981) <doi:10.1016/0304-4076(81)90062-2>, Carapeto and
Holt (2003) <doi:10.1080/0266476022000018475>, Cook and Weisberg (1983)
<doi:10.1093/biomet/70.1.1> (including their graphical methods), Diblasi
and Bowman (1997) <doi:10.1016/S0167-7152(96)00115-0>, Dufour, Khalaf,
Bernard, and Genest (2004) <doi:10.1016/j.jeconom.2003.10.024>, Evans and
King (1985) <doi:10.1016/0304-4076(85)90085-5> and Evans and King (1988)
<doi:10.1016/0304-4076(88)90006-1>, Glejser (1969)
<doi:10.1080/01621459.1969.10500976> as formulated by
Mittelhammer, Judge and Miller (2000, ISBN: 0-521-62394-4), Godfrey and
Orme (1999) <doi:10.1080/07474939908800438>, Goldfeld and Quandt
(1965) <doi:10.1080/01621459.1965.10480811>, Harrison and McCabe (1979)
<doi:10.1080/01621459.1979.10482544>, Harvey (1976) <doi:10.2307/1913974>,
Honda (1989) <doi:10.1111/j.2517-6161.1989.tb01749.x>, Horn (1981)
<doi:10.1080/03610928108828074>, Li and Yao (2019)
<doi:10.1016/j.ecosta.2018.01.001> with and without the modification of
Bai, Pan, and Yin (2016) <doi:10.1007/s11749-017-0575-x>, Rackauskas and
Zuokas (2007) <doi:10.1007/s10986-007-0018-6>, Simonoff and Tsai (1994)
<doi:10.2307/2986026> with and without the modification of Ferrari,
Cysneiros, and Cribari-Neto (2004) <doi:10.1016/S0378-3758(03)00210-6>,
Szroeter (1978) <doi:10.2307/1913831>, Verbyla (1993)
<doi:10.1111/j.2517-6161.1993.tb01918.x>, White (1980)
<doi:10.2307/1912934>, Wilcox and Keselman (2006)
<doi:10.1080/10629360500107923>, Yuce (2008)
<https://dergipark.org.tr/en/pub/iuekois/issue/8989/112070>, and Zhou,
Song, and Thompson (2015) <doi:10.1002/cjs.11252>. Besides these
heteroskedasticity tests, there are supporting functions that compute the
BLUS residuals of Theil (1965) <doi:10.1080/01621459.1965.10480851>, the
conditional two-sided p-values of Kulinskaya (2008) <arXiv:0810.2124v1>,
and probabilities for the nonparametric trend statistic of Lehmann (1975,
ISBN: 0-816-24996-1). For handling heteroskedasticity, in addition to the
new auxiliary variance model methods, there is a function
to implement various existing Heteroskedasticity-Consistent Covariance
Matrix Estimators from the literature, such as those of White (1980)
<doi:10.2307/1912934>, MacKinnon and White (1985)
<doi:10.1016/0304-4076(85)90158-7>, Cribari-Neto (2004)
<doi:10.1016/S0167-9473(02)00366-3>, Cribari-Neto et al. (2007)
<doi:10.1080/03610920601126589>, Cribari-Neto and da Silva (2011)
<doi:10.1007/s10182-010-0141-2>, Aftab and Chang (2016)
<doi:10.18187/pjsor.v12i2.983>, and Li et al. (2017)
<doi:10.1080/00949655.2016.1198906>.

Thomas Farrar

skedastic

Handling Heteroskedasticity in the Linear Regression Model

University of the Western Cape 

ppeak function

<dl><dt>k</dt>
<dd>An integer or a sequence of integers strictly incrementing by 1,
with all values between 0 and <code>n - 1</code> inclusive. Represents the
number of peaks in the sequence.</dd>
<dt>n</dt>
<dd>A positive integer representing the number of observations in the
sequence.</dd>
<dt>lower.tail</dt>
<dd>A logical. Should lower tailed cumulative probability be
calculated? Defaults to <code>FALSE</code> due to function being designed
primarily for calculating \(p\)-values for the peaks test, which is
by default an upper-tailed test. Note that both upper and lower tailed
cumulative probabilities are computed inclusive of <code>k</code>.</dd>
<dt>usedata</dt>
<dd>A logical. Should probability mass function values be
read from <code>dpeakdat</code> rather than computing them from
<code>dpeak</code>? This option will save significantly on
computation time if \(n &lt; 170\) but is currently only available
for \(n \le 1000\).</dd></dl>

Arguments

Cumulative distribution function of number of peaks in an i.i.d. random sequence — ppeak

<dl>

<dt>k</dt>
<dd>An integer or a sequence of integers strictly incrementing by 1,
with all values between 0 and <code>n - 1</code> inclusive. Represents the
number of peaks in the sequence.</dd>


<dt>n</dt>
<dd>A positive integer representing the number of observations in the
sequence.</dd>


<dt>lower.tail</dt>
<dd>A logical. Should lower tailed cumulative probability be
calculated? Defaults to <code>FALSE</code> due to function being designed
primarily for calculating \(p\)-values for the peaks test, which is
by default an upper-tailed test. Note that both upper and lower tailed
cumulative probabilities are computed inclusive of <code>k</code>.</dd>


<dt>usedata</dt>
<dd>A logical. Should probability mass function values be
read from <code>dpeakdat</code> rather than computing them from
<code>dpeak</code>? This option will save significantly on
computation time if \(n &lt; 170\) but is currently only available
for \(n \le 1000\).</dd>

</dl>

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ppeak: Cumulative distribution function of number of peaks in an i.i.d. random sequence

Description

Usage

Value

Arguments

References

See Also

Examples