exp.tilt {boot} | R Documentation

## Exponential Tilting

### Description

This function calculates exponentially tilted multinomial distributions such that the resampling distributions of the linear approximation to a statistic have the required means.

### Usage

    exp.tilt(L, theta=NULL, t0=0, lambda=NULL,
             strata=rep(1, length(L)))

### Arguments

`L` | The empirical influence values for the statistic of interest based on the observed data. The length of `L` should be the same as the size of the original data set. Typically `L` will be calculated by a call to `empinf`.
`theta` | The value at which the tilted distribution is to be centred. This is not required if `lambda` is supplied but is needed otherwise.
`t0` | The current value of the statistic. The default is that the statistic equals 0.
`lambda` | The Lagrange multiplier(s). For each value of `lambda` a multinomial distribution is found with probabilities proportional to `exp(lambda * L)`. In general `lambda` is not known and so `theta` would be supplied, and the corresponding value of `lambda` found. If both `lambda` and `theta` are supplied then `lambda` is ignored and the multipliers for tilting to `theta` are found.
`strata` | A vector or factor of the same length as `L` giving the strata for the observed data and the empirical influence values `L`.

### Details

Exponential tilting involves finding a set of weights for a data set to ensure that the bootstrap distribution of the linear approximation to a statistic of interest has mean `theta`. The weights chosen to achieve this are given by `p[j]` proportional to `exp(lambda*L[j]/n)`, where `n` is the number of data points. `lambda` is then chosen to make the mean of the bootstrap distribution, of the linear approximation to the statistic of interest, equal to the the required value `theta`. Thus `lambda` is defined as the solution of a nonlinear equation. The equation is solved by minimizing the Euclidean distance between the left and right hand sides of the equation using the function `nlmin`. If this minimum is not equal to zero then the method fails.

Typically exponential tilting is used to find suitable weights for importance resampling. If a small tail probability or quantile of the distribution of the statistic of interest is required then a more efficient simulation is to centre the resampling distribution close to the point of interest and then use the functions `imp.prob` or `imp.quantile` to estimate the required quantity.

Another method of achieving a similar shifting of the distribution is through the use of `smooth.f`. The function `tilt.boot` uses `exp.tilt` or `smooth.f` to find the weights for a tilted bootstrap.

### Value

A list with the following components :

`p` | The tilted probabilities. There will be `m` distributions where `m` is the length of `theta` (or `lambda` if supplied). If `m` is 1 then `p` is a vector of `length(L)` probabilities. If `m` is greater than 1 then `p` is a matrix with `m` rows, each of which contain `length(L)` probabilities. In this case the vector `p[i,]` is the distribution tilted to `theta[i]`. `p` is in the form required by the argument `weights` of the function `boot` for importance resampling.
`lambda` | The Lagrange multiplier used in the equation to determine the tilted probabilities. `lambda` is a vector of the same length as `theta`.
`theta` | The values of `theta` to which the distributions have been tilted. In general this will be the input value of `theta` but if `lambda` was supplied then this is the vector of the corresponding `theta` values.

### References

Davison, A. C. and Hinkley, D. V. (1997) Bootstrap Methods and Their Application. Cambridge University Press.

Efron, B. (1981) Nonparametric standard errors and confidence intervals (with Discussion). Canadian Journal of Statistics, 9, 139–172.

### See Also

`empinf`, `imp.prob`, `imp.quantile`, `optim`, `smooth.f`, `tilt.boot`

### Examples

    # Example 9.8 of Davison and Hinkley (1997) requires tilting the resampling
    # distribution of the studentized statistic to be centred at the observed
    # value of the test statistic 1.84.  This can be achieved as follows.
    data(gravity)
    grav1 <- gravity[as.numeric(gravity[,2])>=7,]
    grav.fun <- function(dat, w, orig)
    {    strata <- tapply(dat[, 2], as.numeric(dat[, 2]))
         d <- dat[, 1]
         ns <- tabulate(strata)
         w <- w/tapply(w, strata, sum)[strata]
         mns <- tapply(d * w, strata, sum)
         mn2 <- tapply(d * d * w, strata, sum)
         s2hat <- sum((mn2 - mns^2)/ns)
         as.vector(c(mns[2]-mns[1],s2hat,(mns[2]-mns[1]-orig)/sqrt(s2hat)))
    }
    grav.z0 <- grav.fun(grav1,rep(1,26),0)
    grav.L <- empinf(data=grav1, statistic=grav.fun, stype="w",
                     strata=grav1[,2], index=3, orig=grav.z0[1])
    grav.tilt <- exp.tilt(grav.L, grav.z0[3], strata=grav1[,2])
    boot(grav1, grav.fun, R=499, stype="w", weights=grav.tilt$p,
         strata=grav1[,2], orig=grav.z0[1])

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