Description Usage Arguments Value Note Author(s) References See Also Examples
This is an adapted version of activeSetLogCon from the logcondens package for computing the MLE of a logconcave density with known location of mode.
Given a vector of observations x_n = (x_1, …, x_n) with potentially distinct or
nondistinct entries, activeSetLogCon.mode
first
computes vectors x_m = (x_1,
…, x_m) and w = (w_1,
…, w_m) where w_i is the weight of each x_i s.t.
∑_{i=1}^m w_i = 1. The vector \bold{x}_m contains the
fixed location of the mode, mode
. Then,
activeSetLogCon.mode
computes a concave, piecewise
linear function \widehat φ_m^0 on
[x_1, x_m] with p knots only in {x_1, …, x_m} and with mode value, mode
, such
that
L(φ) = ∑_{i=1}^m w_i φ(x_i)  int_{∞}^∞ exp(φ(t)) dt
is maximal. To accomplish this, an active set algorithm is used.
1 2 
x 
Vector of independent and identically distributed numbers, not necessarily unique. 
xgrid 
Governs the generation of weights for observations. See

mode 
This is the constrained value for the location of the mode. 
print 

w 
Optional vector of weights. If weights are provided, i.e., if

prec 
Governs precision of various subfunctions, e.g., the NewtonRaphson procedure. 
xn 
Vector with initial observations x_1, …, x_n. 
x 
Vector of observations x_1, …, x_m that was used
to estimate the density, i.e., points that include all possible
knots of the estimate.
Note that \bold{x} always includes the mode
value 
w 
The vector of weights that had been used. Depends on the
chosen setting for 
L 
The value L(φ_m^0) of the loglikelihoodfunction L at the maximum \widehat φ_m^0. 
MI 
Numeric vector of length 2 giving the endpoints of the modal interval. 
IsKnot 
Vector with entries IsKnot_i = 1\{\widehat{φ}_m^0 has a kink at x_i}. 
IsMIC 
Analogous to IsKnot; stands for "Is Modally Inactive Constraint," i.e., denotes whether the modal constraints are active or inactive. It is a numeric vector of length 2, corresponding to whether the mode is a leftknot or a rightknot. Just as with IsKnot, a 1 denotes an inactive constraint and a 0 denotes an active one. Thus a 0 indicates that the constraint that the estimate be equal in value at the mode and the nearest knot to the left or to the right, respectively, is active. Note also that if max(IsMIC)==1 then the corresponding index in IsKnot is a 1 (i.e., IsKnot[dlcMode$idx] ==1 ). 
constr 
knots[constr] is equal to MI; that is, constr is a numeric (integral) vector of length two with values in 1, …, p indicating which of the p knots are the left and right of the modal interval. 
knots 

phi 
Vector with entries \widehat φ_m(x_i), i=1,…,m. Named "phi" not "phihat" for backwards compatibility. 
fhat 
Vector with entries \widehat{f}_m^0(x_i) = e^{\widehat{φ}_m^0(x_i)}, i=1,…, m. 
Fhat 
A vector (\widehat F_{m,i}^0)_{i=1}^m of the same size as x with entries \widehat F_{m,i}^0 = \int_{x_1}^{x_i} \exp(\widehat φ_m^0(t)) dt. 
H 
Numeric vector (H_1, …, H_{m})' where H_i is the derivative of t \to L(φ + tΔ_i) at zero and Δ_i(x) = \min(x  x_i, 0) if x_i is
less than Note that in the unconstrained problems the derivatives in the directions \min(x_i  x, 0) and \min(xx_i,0) are equal, but in the constrained problem these derivatives are not equal. 
H.m 
Vector (H.m_1, H.m_2)' where H.m_1 is the derivative of t \to L(φ + tΔ_i) at zero and Δ_1(x) = \min(x  a, 0) and Δ_2(x) = \min(ax, 0), where a is the mode. 
n 
Number of initial observations, i.e., length of 
m1 
Number of unique observations. This count excludes the mode
if the mode is not a data point (or if 
m 
Number of points used to compute the estimator, i.e., unique
observations as well as the mode, i.e., length of 
dlcMode 
A list, of class "dlc.mode", with components Note, when the mode is not an 
sig 
The standard deviation of the initial sample x_1, …, x_n. 
phi.f 
All outputs named "name.f" are functions corresponding to name. So,

fhat.f 
Is a function such that 
Fhat.f 
Is a function such that 
EL.f 
Note that this is not analogous to 
ER.f 

E.f 
Equals 
phiPL 
Numeric vector of length m with values (\widehat{φ}_m^0)'(x_i) 
phiPR 
Numeric vector of length m with values (\widehat{φ}_m^0)'(x_i+) 
phiPL.f 
Is a function such that 
phiPR.f 
Is a function such that 
Adapted from
activeSetLogCon
in the
package logcondens
.
Kaspar Rufibach, kaspar.rufibach@gmail.com,
http://www.kasparrufibach.ch
Lutz Duembgen, duembgen@stat.unibe.ch,
http://www.staff.unibe.ch/duembgen
Charles R. Doss, cdoss@stat.washington.edu,
http://www.stat.washington.edu/people/cdoss/
Duembgen, L, Huesler, A. and Rufibach, K. (2010) Active set and EM algorithms for logconcave densities based on complete and censored data. Technical report 61, IMSV, Univ. of Bern, available at http://arxiv.org/abs/0707.4643.
Duembgen, L. and Rufibach, K. (2009) Maximum likelihood estimation of a logconcave density and its distribution function: basic properties and uniform consistency. Bernoulli, 15(1), 40–68.
Duembgen, L. and Rufibach, K. (2011) logcondens: Computations Related to Univariate LogConcave Density Estimation. Journal of Statistical Software, 39(6), 1–28. http://www.jstatsoft.org/v39/i06
Doss, C. R. (2013). ShapeConstrained Inference for ConcaveTransformed Densities and their Modes. PhD thesis, Department of Statistics, University of Washington, in preparation.
Doss, C. R. and Wellner, J. A. (2013). Inference for the mode of a logconcave density. Technical Report, University of Washington, in preparation.
The following functions are used by activeSetLogCon.mode
:
J00
, J10
, J11
,
J20
,
Local_LL.mode
,
LocalLLall.mode
,
LocalCoarsen.mode
, LocalConvexity.mode
,
LocalExtend
, LocalF
,
LocalMLE.mode
,
LocalNormalize
,
MLE.mode
logConDens
(or activeSetLogCon
) can be used to estimate an unconstrained
logconcave density.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36  ## estimate gamma density
set.seed(1977)
n < 200
x < rgamma(n, 2, 1)
TRUEMODE < 1; ## (21)*1
res < activeSetLogCon.mode(x, mode=TRUEMODE, w = rep(1 / n, n), print = FALSE)
## plot resulting functions
par(mfrow = c(2, 2), mar = c(3, 2, 1, 2))
plot(res$x, res$fhat, type = 'l'); rug(res$xn)
plot(res$x, res$phi, type = 'l'); rug(res$xn)
plot(res$x, res$Fhat, type = 'l'); rug(res$xn)
plot(res$x, res$H, type = 'l'); rug(res$xn)
## Or can use the ".f" functions
xpts < seq(from=0, to=9, by=.01)
par(mfrow = c(2, 2), mar = c(3, 2, 1, 2))
plot(xpts, res$fhat.f(xpts), type = 'l'); rug(res$xn)
plot(xpts, res$phi.f(xpts), type = 'l'); rug(res$xn)
## these are not analogous to res$H.
plot(xpts, res$EL.f(upper=xpts), type = 'l'); rug(res$xn)
plot(xpts, res$ER.f(lower=xpts), type = 'l'); rug(res$xn)
## compute and plot function values at an arbitrary point
x0 < (res$x[100] + res$x[101]) / 2
Fx0 < evaluateLogConDens(x0, res, which = 3)[, "CDF"]
plot(res$x, res$Fhat, type = 'l'); rug(res$x)
abline(v = x0, lty = 3); abline(h = Fx0, lty = 3)
## compute and plot 0.9quantile of Fhat
alpha < .1
q < quantilesLogConDens(1alpha, res)[2]
plot(res$x, res$Fhat, type = 'l'); rug(res$x)
abline(h = 1alpha, lty = 3); abline(v = q, lty = 3)

Loading required package: logcondens
Attaching package: 'logcondens.mode'
The following objects are masked from 'package:logcondens':
activeSetLogCon, intF, logConDens
The following object is masked from 'package:base':
dir.exists
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