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Generate Weights

Source: R/gen_weight.R
gen.weight.Rd

This function generates a set of normalized weights based on theta (ability) values to be used in functions such as est_score(), sx2_fit(), and covirt().

Usage

gen.weight(n = 41, dist = "norm", mu = 0, sigma = 1, l = -4, u = 4, theta)

Arguments

n

An integer specifying the number of theta (node) values for which weights are to be generated. Default is 41.

dist

A character string indicating the distribution type used to generate weights. Available options are "norm" for a normal distribution, "unif" for a uniform distribution, and "emp" for an empirical distribution.

  • If dist = "norm", either n or theta must be provided.

  • If dist = "unif", only n is applicable.

  • If dist = "emp", only theta must be specified.

mu, sigma

Mean and standard deviation of the normal distribution (used when dist = "norm").

l, u

Lower and upper bounds of the uniform distribution (used when dist = "unif").

theta

A numeric vector of empirical theta (node) values for which weights are generated.

Value

A data frame with two columns:

  • theta: The theta (node) values.

  • weight: The corresponding normalized weights.

Details

If theta is not specified, n equally spaced quadrature points and corresponding weights are generated from either the normal or uniform distribution:

  • When dist = "norm", Gaussian quadrature points and weights are computed using gauss.quad.prob() from the statmod package.

  • When dist = "unif", equally spaced points are drawn from the specified interval [l, u], and weights are proportional to the uniform density.

If theta is specified:

  • When dist = "norm", the weights are proportional to the normal density evaluated at each theta value and normalized to sum to 1.

  • When dist = "emp", equal weights are assigned to each provided theta value.

See also

est_score(), sx2_fit(), covirt()

Author

Hwanggyu Lim hglim83@gmail.com

Examples

## Example 1:
## Generate 41 Gaussian quadrature points and weights from the normal distribution
gen.weight(n = 41, dist = "norm", mu = 0, sigma = 1)
#>            theta       weight
#> 1  -1.161494e+01 2.257864e-30
#> 2  -1.064754e+01 8.308559e-26
#> 3  -9.843433e+00 2.746891e-22
#> 4  -9.123070e+00 2.326384e-19
#> 5  -8.456099e+00 7.655982e-17
#> 6  -7.826882e+00 1.220335e-14
#> 7  -7.226023e+00 1.077818e-12
#> 8  -6.647308e+00 5.769853e-11
#> 9  -6.086349e+00 1.994795e-09
#> 10 -5.539884e+00 4.667348e-08
#> 11 -5.005397e+00 7.658186e-07
#> 12 -4.480878e+00 9.058609e-06
#> 13 -3.964684e+00 7.894719e-05
#> 14 -3.455432e+00 5.158014e-04
#> 15 -2.951937e+00 2.561642e-03
#> 16 -2.453159e+00 9.777903e-03
#> 17 -1.958171e+00 2.893721e-02
#> 18 -1.466125e+00 6.684766e-02
#> 19 -9.762388e-01 1.211489e-01
#> 20 -4.877686e-01 1.728495e-01
#> 21 -2.512499e-16 1.945450e-01
#> 22  4.877686e-01 1.728495e-01
#> 23  9.762388e-01 1.211489e-01
#> 24  1.466125e+00 6.684766e-02
#> 25  1.958171e+00 2.893721e-02
#> 26  2.453159e+00 9.777903e-03
#> 27  2.951937e+00 2.561642e-03
#> 28  3.455432e+00 5.158014e-04
#> 29  3.964684e+00 7.894719e-05
#> 30  4.480878e+00 9.058609e-06
#> 31  5.005397e+00 7.658186e-07
#> 32  5.539884e+00 4.667348e-08
#> 33  6.086349e+00 1.994795e-09
#> 34  6.647308e+00 5.769853e-11
#> 35  7.226023e+00 1.077818e-12
#> 36  7.826882e+00 1.220335e-14
#> 37  8.456099e+00 7.655982e-17
#> 38  9.123070e+00 2.326384e-19
#> 39  9.843433e+00 2.746891e-22
#> 40  1.064754e+01 8.308559e-26
#> 41  1.161494e+01 2.257864e-30

## Example 2:
## Generate 41 theta values and weights from the uniform distribution,
## given a minimum value of -4 and a maximum value of 4
gen.weight(n = 41, dist = "unif", l = -4, u = 4)
#>    theta     weight
#> 1   -4.0 0.02439024
#> 2   -3.8 0.02439024
#> 3   -3.6 0.02439024
#> 4   -3.4 0.02439024
#> 5   -3.2 0.02439024
#> 6   -3.0 0.02439024
#> 7   -2.8 0.02439024
#> 8   -2.6 0.02439024
#> 9   -2.4 0.02439024
#> 10  -2.2 0.02439024
#> 11  -2.0 0.02439024
#> 12  -1.8 0.02439024
#> 13  -1.6 0.02439024
#> 14  -1.4 0.02439024
#> 15  -1.2 0.02439024
#> 16  -1.0 0.02439024
#> 17  -0.8 0.02439024
#> 18  -0.6 0.02439024
#> 19  -0.4 0.02439024
#> 20  -0.2 0.02439024
#> 21   0.0 0.02439024
#> 22   0.2 0.02439024
#> 23   0.4 0.02439024
#> 24   0.6 0.02439024
#> 25   0.8 0.02439024
#> 26   1.0 0.02439024
#> 27   1.2 0.02439024
#> 28   1.4 0.02439024
#> 29   1.6 0.02439024
#> 30   1.8 0.02439024
#> 31   2.0 0.02439024
#> 32   2.2 0.02439024
#> 33   2.4 0.02439024
#> 34   2.6 0.02439024
#> 35   2.8 0.02439024
#> 36   3.0 0.02439024
#> 37   3.2 0.02439024
#> 38   3.4 0.02439024
#> 39   3.6 0.02439024
#> 40   3.8 0.02439024
#> 41   4.0 0.02439024

## Example 3:
## Generate normalized weights from the standard normal distribution,
## given a user-defined set of theta values
theta <- seq(-4, 4, by = 0.1)
gen.weight(dist = "norm", mu = 0, sigma = 1, theta = theta)
#>    theta       weight
#> 1   -4.0 1.338370e-05
#> 2   -3.9 1.986656e-05
#> 3   -3.8 2.919618e-05
#> 4   -3.7 4.248019e-05
#> 5   -3.6 6.119330e-05
#> 6   -3.5 8.727271e-05
#> 7   -3.4 1.232282e-04
#> 8   -3.3 1.722657e-04
#> 9   -3.2 2.384209e-04
#> 10  -3.1 3.266985e-04
#> 11  -3.0 4.432074e-04
#> 12  -2.9 5.952835e-04
#> 13  -2.8 7.915854e-04
#> 14  -2.7 1.042146e-03
#> 15  -2.6 1.358366e-03
#> 16  -2.5 1.752919e-03
#> 17  -2.4 2.239567e-03
#> 18  -2.3 2.832848e-03
#> 19  -2.2 3.547640e-03
#> 20  -2.1 4.398583e-03
#> 21  -2.0 5.399371e-03
#> 22  -1.9 6.561915e-03
#> 23  -1.8 7.895417e-03
#> 24  -1.7 9.405386e-03
#> 25  -1.6 1.109265e-02
#> 26  -1.5 1.295242e-02
#> 27  -1.4 1.497351e-02
#> 28  -1.3 1.713773e-02
#> 29  -1.2 1.941959e-02
#> 30  -1.1 2.178633e-02
#> 31  -1.0 2.419830e-02
#> 32  -0.9 2.660988e-02
#> 33  -0.8 2.897063e-02
#> 34  -0.7 3.122698e-02
#> 35  -0.6 3.332415e-02
#> 36  -0.5 3.520832e-02
#> 37  -0.4 3.682889e-02
#> 38  -0.3 3.814072e-02
#> 39  -0.2 3.910626e-02
#> 40  -0.1 3.969727e-02
#> 41   0.0 3.989626e-02
#> 42   0.1 3.969727e-02
#> 43   0.2 3.910626e-02
#> 44   0.3 3.814072e-02
#> 45   0.4 3.682889e-02
#> 46   0.5 3.520832e-02
#> 47   0.6 3.332415e-02
#> 48   0.7 3.122698e-02
#> 49   0.8 2.897063e-02
#> 50   0.9 2.660988e-02
#> 51   1.0 2.419830e-02
#> 52   1.1 2.178633e-02
#> 53   1.2 1.941959e-02
#> 54   1.3 1.713773e-02
#> 55   1.4 1.497351e-02
#> 56   1.5 1.295242e-02
#> 57   1.6 1.109265e-02
#> 58   1.7 9.405386e-03
#> 59   1.8 7.895417e-03
#> 60   1.9 6.561915e-03
#> 61   2.0 5.399371e-03
#> 62   2.1 4.398583e-03
#> 63   2.2 3.547640e-03
#> 64   2.3 2.832848e-03
#> 65   2.4 2.239567e-03
#> 66   2.5 1.752919e-03
#> 67   2.6 1.358366e-03
#> 68   2.7 1.042146e-03
#> 69   2.8 7.915854e-04
#> 70   2.9 5.952835e-04
#> 71   3.0 4.432074e-04
#> 72   3.1 3.266985e-04
#> 73   3.2 2.384209e-04
#> 74   3.3 1.722657e-04
#> 75   3.4 1.232282e-04
#> 76   3.5 8.727271e-05
#> 77   3.6 6.119330e-05
#> 78   3.7 4.248019e-05
#> 79   3.8 2.919618e-05
#> 80   3.9 1.986656e-05
#> 81   4.0 1.338370e-05

## Example 4:
## Generate equal normalized weights for theta values
## randomly sampled from the standard normal distribution
theta <- rnorm(100)
gen.weight(dist = "emp", theta = theta)
#>            theta weight
#> 1    0.136221893   0.01
#> 2    0.407167603   0.01
#> 3   -0.069654813   0.01
#> 4   -0.247664342   0.01
#> 5    0.695550807   0.01
#> 6    1.146228357   0.01
#> 7   -2.403096215   0.01
#> 8    0.572739555   0.01
#> 9    0.374724407   0.01
#> 10  -0.425267722   0.01
#> 11   0.951012808   0.01
#> 12  -0.389237182   0.01
#> 13  -0.284330662   0.01
#> 14   0.857409778   0.01
#> 15   1.719627299   0.01
#> 16   0.270054901   0.01
#> 17  -0.422184010   0.01
#> 18  -1.189113295   0.01
#> 19  -0.331032979   0.01
#> 20  -0.939829327   0.01
#> 21  -0.258932583   0.01
#> 22   0.394379168   0.01
#> 23  -0.851857092   0.01
#> 24   2.649166881   0.01
#> 25   0.156011676   0.01
#> 26   1.130207267   0.01
#> 27  -2.289123980   0.01
#> 28   0.741001157   0.01
#> 29  -1.316245160   0.01
#> 30   0.919803678   0.01
#> 31   0.398130155   0.01
#> 32  -0.407528579   0.01
#> 33   1.324258630   0.01
#> 34  -0.701231669   0.01
#> 35  -0.580614304   0.01
#> 36  -1.001072181   0.01
#> 37  -0.668178607   0.01
#> 38   0.945184953   0.01
#> 39   0.433702150   0.01
#> 40   1.005159218   0.01
#> 41  -0.390118664   0.01
#> 42   0.376370292   0.01
#> 43   0.244164924   0.01
#> 44  -1.426257342   0.01
#> 45   1.778429287   0.01
#> 46   0.134447661   0.01
#> 47   0.765598999   0.01
#> 48   0.955136677   0.01
#> 49  -0.050565701   0.01
#> 50  -0.305815420   0.01
#> 51   0.893673702   0.01
#> 52  -1.047298149   0.01
#> 53   1.971337386   0.01
#> 54  -0.383632106   0.01
#> 55   1.654145302   0.01
#> 56   1.512212694   0.01
#> 57   0.082965734   0.01
#> 58   0.567220915   0.01
#> 59  -1.024548480   0.01
#> 60   0.323006503   0.01
#> 61   1.043612458   0.01
#> 62   0.099078487   0.01
#> 63  -0.454136909   0.01
#> 64  -0.655781852   0.01
#> 65  -0.035922423   0.01
#> 66   1.069161461   0.01
#> 67  -0.483974930   0.01
#> 68  -0.121010111   0.01
#> 69  -1.294140004   0.01
#> 70   0.494312836   0.01
#> 71   1.307901520   0.01
#> 72   1.497041009   0.01
#> 73   0.814702731   0.01
#> 74  -1.869788790   0.01
#> 75   0.482029504   0.01
#> 76   0.456135603   0.01
#> 77  -0.353400286   0.01
#> 78   0.170489471   0.01
#> 79  -0.864035954   0.01
#> 80   0.679230774   0.01
#> 81  -0.327101015   0.01
#> 82  -1.569082185   0.01
#> 83  -0.367450756   0.01
#> 84   1.364434929   0.01
#> 85  -0.334281365   0.01
#> 86   0.732750042   0.01
#> 87   0.946585640   0.01
#> 88   0.004398704   0.01
#> 89  -0.352322306   0.01
#> 90  -0.529695509   0.01
#> 91   0.739589226   0.01
#> 92  -1.063457415   0.01
#> 93   0.246210844   0.01
#> 94  -0.289499367   0.01
#> 95  -2.264889356   0.01
#> 96  -1.408850456   0.01
#> 97   0.916019329   0.01
#> 98  -0.191278951   0.01
#> 99   0.803283216   0.01
#> 100  1.887474463   0.01