Given a sample of
$n$ observations,
${x}_{1},{x}_{2},\dots ,{x}_{n}$, from a distribution with unknown density function,
$f\left(x\right)$, an estimate of the density function,
$\hat{f}\left(x\right)$, may be required. The simplest form of density estimator is the histogram. This may be defined by:
where
${n}_{j}$ is the number of observations falling in the interval
$a+\left(j1\right)h$ to
$a+jh$,
$a$ is the lower bound to the histogram and
$b={n}_{s}h$ is the upper bound. The value
$h$ is known as the window width. To produce a smoother density estimate a kernel method can be used. A kernel function,
$K\left(t\right)$, satisfies the conditions:
The kernel density estimator is then defined as
The choice of
$K$ is usually not important but to ease the computational burden use can be made of the Gaussian kernel defined as
The smoothness of the estimator depends on the window width
$h$. The larger the value of
$h$ the smoother the density estimate. The value of
$h$ can be chosen by examining plots of the smoothed density for different values of
$h$ or by using crossvalidation methods (see
Silverman (1990)).
Silverman (1982) and
Silverman (1990) show how the Gaussian kernel density estimator can be computed using a fast Fourier transform (
fft). In order to compute the kernel density estimate over the range
$a$ to
$b$ the following steps are required.
(i) 
Discretize the data to give ${n}_{s}$ equally spaced points ${t}_{l}$ with weights ${\xi}_{l}$ (see Jones and Lotwick (1984)). 
(ii) 
Compute the fft of the weights ${\xi}_{l}$ to give ${Y}_{l}$. 
(iii) 
Compute ${\zeta}_{l}={e}^{\frac{1}{2}{h}^{2}{s}_{l}^{2}}{Y}_{l}$ where ${s}_{l}=2\pi l/\left(ba\right)$. 
(iv) 
Find the inverse fft of ${\zeta}_{l}$ to give $\hat{f}\left(x\right)$. 
To compute the kernel density estimate for further values of
$h$ only steps
(iii) and
(iv) need be repeated.
Jones M C and Lotwick H W (1984) Remark AS R50. A remark on algorithm AS 176. Kernel density estimation using the Fast Fourier Transform Appl. Statist. 33 120–122
Silverman B W (1982) Algorithm AS 176. Kernel density estimation using the fast Fourier transform Appl. Statist. 31 93–99
 1: $\mathbf{n}$ – IntegerInput

On entry: $n$, the number of observations in the sample.
Constraint:
${\mathbf{n}}>0$.
 2: $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the $n$ observations,
${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
 3: $\mathbf{window}$ – Real (Kind=nag_wp)Input

On entry: $h$, the window width.
Constraint:
${\mathbf{window}}>0.0$.
 4: $\mathbf{slo}$ – Real (Kind=nag_wp)Input

On entry:
$a$, the lower limit of the interval on which the estimate is calculated. For most applications
slo should be at least three window widths below the lowest data point.
Constraint:
${\mathbf{slo}}<{\mathbf{shi}}$.
 5: $\mathbf{shi}$ – Real (Kind=nag_wp)Input

On entry:
$b$, the upper limit of the interval on which the estimate is calculated. For most applications
shi should be at least three window widths above the highest data point.
 6: $\mathbf{ns}$ – IntegerInput

On entry: the number of points at which the estimate is calculated, ${n}_{s}$.
Constraint:
${\mathbf{ns}}\ge 2$.
 7: $\mathbf{smooth}\left({\mathbf{ns}}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the ${n}_{s}$ values of the density estimate,
$\hat{f}\left({t}_{\mathit{l}}\right)$, for $\mathit{l}=1,2,\dots ,{n}_{s}$.
 8: $\mathbf{t}\left({\mathbf{ns}}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the points at which the estimate is calculated,
${t}_{\mathit{l}}$, for $\mathit{l}=1,2,\dots ,{n}_{s}$.
 9: $\mathbf{usefft}$ – LogicalInput

On entry: must be set to .FALSE. if the values of
${Y}_{l}$ are to be calculated by
g10baf and to .TRUE. if they have been computed by a previous call to
g10baf and are provided in
fft. If
${\mathbf{usefft}}=\mathrm{.TRUE.}$ then the arguments
n,
slo,
shi,
ns and
fft must remain unchanged from the previous call to
g10baf with
${\mathbf{usefft}}=\mathrm{.FALSE.}$.
 10: $\mathbf{fft}\left({\mathbf{ns}}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: if
${\mathbf{usefft}}=\mathrm{.TRUE.}$,
fft must contain the fast Fourier transform of the weights of the discretized data,
${\xi}_{\mathit{l}}$, for
$\mathit{l}=1,2,\dots ,{n}_{s}$. Otherwise
fft need not be set.
On exit: the fast Fourier transform of the weights of the discretized data,
${\xi}_{\mathit{l}}$, for $\mathit{l}=1,2,\dots ,{n}_{s}$.
 11: $\mathbf{ifail}$ – IntegerInput/Output

On entry:
ifail must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit:
${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
If on entry
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
See
Jones and Lotwick (1984) for a discussion of the accuracy of this method.
g10baf is not thread safe and should not be called from a multithreaded user program. Please see
Section 3.12.1 in How to Use the NAG Library and its Documentation for more information on thread safety.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The time for computing the weights of the discretized data is of order
$n$, while the time for computing the
fft is of order
${n}_{s}\mathrm{log}\left({n}_{s}\right)$, as is the time for computing the inverse of the
fft.
Data is read from a file and the density estimated. The first $20$ values are then printed. The full estimated density function is shown in the accompanying plot.