ELSEVIER Computational Statistics & Data Analysis 27 (1998) 141-150

COMPUTATIONAL STATISTICS

& DATA ANALYSIS

Asymmetric quadratic loss adjustments for a predictive t variable

Michael Cain

Department of Economics, University of Salford, Salford M5 4WT, UK

Received 1 July 1997; accepted 1 October 1997

Abstract

The predictive distribution of the response variable in a linear model is often univariate t and in this paper the prediction of such a response is considered in the presence of asymmetric quadratic loss. It is shown that the optimal prediction is an additive adjustment to the predictive mean, the adjustment being the product of the scale of the predictive variable and an appropriate adjustment factor. An extensive table of adjustment factor values is presented for various degrees of freedom and measures of asymmetry. © 1998 Elsevier Science B. V. All rights reserved.

Keywords: Prediction; Asymmetric quadratic loss; Univariate t distribution

1. Introduction

Let F be the (posterior) predictive distribution function of a continuous random variable, Y, for which it is required to find the optimal prediction 33 = 33 (data), taking into account the loss ¢ ( y - 33) associated with the predictive error. Of particular interest is the case where the loss function is asymmetric. Raiffa and Schlaifer (1961) derived theoretical results for prediction under asymmetric linear loss and provided a table of values of a Normal linear-loss integral to aid the evaluation of the expected loss. Bracken and Schleifer (1964) produced extensive tables of cumulative probabilities and values of a linear-loss function for a Student's t variable, thereby providing a numerical method for predicting such a variable under asymmetric linear loss. Granger (1969) noted that even if both the loss (cost) function and the conditional density of Y are symmetric the mean is not necess- arily the optimal predictor, and he proved theorems giving simple conditions which ensure that the mean is optimal. He provided a small table of numerical

0167-9473/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved PI! S0 1 6 7 - 9 4 7 3 ( 9 8 ) 0 0 0 1 0 - 3

142 M. Cain~Computational Statistics & Data Analysis 27 (1998) 141-150

adjustments for predicting the value of a normal random variable with an asym- metric linear cost of error function.

There are few analytic solutions with asymmetric loss functions and most researchers seem to have persevered with quadratic loss. However, recently it appears that asymmetry of the loss function is beginning to be taken more seriously. Zellner (1986) considered (asymmetric) LINEX loss, introduced by Varian (1975), and derived optimal predictors and the Bayes risk in a number of situations. He called for further study of the properties of alternative estimators relative to LINEX and other asymmetric loss functions. Cain and Janssen (1992) used LINEX and both asymmetric linear and asymmetric quadratic loss in predic- ting the price of real estate based on a standard linear model with normal errors. Harris (1992) considered non-symmetric loss in a general problem of optimal control where typically minimum variance controllers are used. L6on and Wu (1992) discussed the Taguchi problem of parameter design and considered exten- sions to the case of asymmetric loss.

Although the more recent papers cited above essentially consider n o r m a l predic- tive distributions, in practice a Bayesian analysis often yields a predictive t distribu- tion. See Broemeling (1985) for the results in analysing a linear model; also Cain and Owen (1990). The purpose of this short paper is to briefly give the results for predicting a univariate t distributed response under asymmetric quadratic loss and provide an extensive table of adjustment factor values, of practical use.

2. Optimising predictive loss

With the general asymmetric loss function

~E1 (y), y > 0 , (Y)=(E2(y), y < 0 ,

where ~1(0) = 0 = ~2(0), f~(y) > 0 for y > 0 and f~(y) < 0 for y < 0, the expected predictive loss in using 33 to predict Y is

~(y - j3) dF(y) = f z ( Y - j3) dF(y) + EI(y - ))dF(y) - - O 0

and the optimal prediction satisfies the first-order condition:

; f; ¢3(Y -- ~)dF(y) + ~i(Y - j~) dF(y) = 0. (1) - - O 0

The second-order condition for m i n i m a l expected predictive loss is

(o ) - + - ; ) d F ( y ) + i'(y - ; )dr (y ) > 0 (2) - - 0 0

dF(y) where f is the probability density function of Y ; f ( y ) - - -

dy

M. Cain/Computational Statistics & Data Analysis 27 (1998) 141-150 143

2.1. Asymmetric quadratic loss

With the asymmetr ic quadrat ic loss function

~ay 2, y > O, f(Y) = [by 2, y < O,

where a, b > 0, a ~ b, (1) reduces to

b (y - 33) dF(y) + a (y - 13)dF(y) = 0, (3) - - C O

and the second-order condit ion (2): a ~ ( Y > Y) + b ~ ( Y < Y) > 0 is clearly satisfied.

2.2. Predictive t distribution

If the predictive distr ibut ion of Y is univariate t with parameters d (degrees of freedom), , (location) and p (precision), the distr ibution of p l / 2 ( y _ #) is the Student 's t with d degrees of freedom. The scale is then a = p - 1/2 so that

E I Y - - ~ - ~ ] = E ( p l / 2 ( Y - . ) ) = O and

V = E ( p ( Y - , ) 2 ) - d _ 2 , d > 2 .

Lett ing Td denote the (cumulative) dis tr ibut ion function of a Student 's t r a n d o m variable with d degrees of f reedom and ta the corresponding probabili ty density function, the opt imal predict ion from (3) satisfies (for d > 2)

( , -- Y){a + (b - a)Ta(pl/2(~ - ,))}

(a - 1 -~ p(9 - It) ta(pl/2(9 - ,)) = 0. (4) + pl/2 (d ) d

See the appendix for details. The solut ion of (4) is

)3 = . + p - 1/26" : . + 0"3*,

an additive adjus tment to the predictive mean. , . where the adjus tment factor 3" is a solut ion 6 = 6" (a. b) of

( d - l ) 1 + ~ ta (6)+6 a - - b ) +Ta(6) = 0 . (5)

Eq. (5) has a unique solution which may be obta ined numerically but note immediately that 3" < 0 i fa < b and 3" > 0 i fa > b. Tables 1 and 2 give values of the adjustment factor 3" for a range of degrees of freedom, d, and values of a/b > 1

144 M. Cain/Computational Statistics & Data Analysis 27 (1998) 141-150

Table 1 Tables for various degrees of freedom, d, of adjustment factor, 6*, for given values of a/b > 1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Degrees of ~eedom= 2 1 - - 0.0674 0.1291 0.1861 0.2391 0.2887 0.3354 0.3796 0.4216 0.4617 2 0.5000 0.5367 0.5721 0.6061 0.6390 0.6708 0.7017 0.7316 0.7606 0.7889 3 0.8165 0.8434 0.8696 0.8953 0.9204 0.9449 0.9690 0.9925 1.0157 1.0384 4 1.0607 1.0826 1.1041 1 .1253 1.1461 1.1667 1.1869 1.2068 1.2264 1.2458 5 1.2649 1.2838 1.3024 1.3207 1.3389 1.3568 1.3745 1.3920 1.4093 1.4265 6 1.4434 1.4601 1.4767 1.4931 1.5094 1.5254 1.5414 1.5571 1.5728 1.5882 7 1.6036 1.6188 1.6338 1.6488 1.6636 1.6783 1.6929 1.7073 1.7217 1.7359 8 1.7500 1.7640 1.7779 1.7917 1.8054 1.8190 1.8325 1.8459 1.8593 1.8725 9 1.8856 1.8987 1.9116 1.9245 1.9373 1.9500 1.9627 1.9752 1.9877 2.0001

Degrees of ~eedom= 3

1 0.0526 0.1006 0.1448 0.1859 0.2242 0.2602 0.2940 0.3261 0.3565 2 0.3855 0.4131 0.4396 0.4650 0.4894 0.5129 0.5355 0.5574 0.5786 0.5991 3 0.6190 0.6383 0.6570 0.6753 0.6931 0.7104 0.7273 0.7438 0.7599 0.7756 4 0.7910 0.8061 0.8209 0.8354 0.8496 0.8635 0.8771 0.8905 0.9037 0.9166 5 0.9294 0.9419 0.9542 0.9663 0.9782 0.9899 1.0015 1.0129 1.0241 1.0351 6 1 .0461 1.0568 1.0674 1.0779 1.0882 1.0985 1.1085 1.1185 1.1283 1.1380 7 1 . 1 4 7 7 1.1571 1.1665 1.1758 1.1850 1.1941 1.2031 1.2119 1.2207 1.2294 8 1 . 2 3 8 1 1.2466 1.2550 1.2634 1.2717 1.2799 1.2880 1.2961 1.3040 1.3120 9 1 . 3 1 9 8 1.3276 1.3353 1.3429 1.3505 1.3580 1.3654 1.3728 1.3801 1.3874

Degrees of ~eedom= 4

1 - - 0.0477 0.0912 0.1313 0.1684 0.2031 0.2355 0.2661 0.2950 0.3223 2 0.3483 0.3731 0.3968 0.4195 0.4412 0.4622 0.4823 0.5018 0.5205 0.5387 3 0.5562 0.5733 0.5898 0.6059 0.6215 0.6367 0.6515 0.6659 0.6800 0.6937 4 0.7071 0.7202 0.7330 0.7456 0.7579 0.7699 0.7817 0.7932 0.8046 0.8157 5 0.8266 0.8373 0.8479 0.8582 0.8684 0.8784 0.8883 0.8980 0.9075 0.9169 6 0.9261 0.9353 0.9442 0.9531 0.9618 0.9704 0.9789 0.9873 0.9956 1.0037 7 1.0118 1.0197 1.0276 1.0354 1.0430 1.0506 1.0581 1 .0655 1.0728 1.0800 8 1.0872 1.0943 1.1013 1.1082 1.1150 1.1218 1.1285 1.1352 1.1417 1.1483 9 1.1547 1.1611 1.1674 1.1737 1.1799 1.1860 1.1921 1.1982 1.2041 1.2101

Degrees of ~eedom= 5

1 - - 0.0452 0.0865 0.1246 0.1598 0.1926 0.2234 0.2523 0.2796 0.3054 2 0.3300 0.3534 0.3757 0.3971 0.4176 0.4373 0.4562 0.4744 0.4921 0.5091 3 0.5255 0.5415 0.5570 0.5720 0.5866 0.6007 0.6146 0.6280 0.6411 0.6539 4 0.6664 0.6785 0.6904 0.7021 0.7135 0.7246 0.7355 0.7462 0.7567 0.7670 5 0.7771 0.7870 0.7967 0.8062 0.8156 0.8248 0.8339 0.8428 0.8516 0.8602 6 0.8687 0.8771 0.8853 0.8934 0.9014 0.9093 0.9170 0.9247 0.9322 0.9397 7 0.9471 0.9543 0.9615 0.9685 0.9755 0.9824 0.9892 0.9960 1.0026 1.0092 8 1 . 0 1 5 7 1.0221 1.0284 1.0347 1.0409 1.0471 1.0531 1.0591 1.0651 1.0710 9 1 . 0 7 6 8 1.0826 1.0883 0.0939 0.0995 1.1050 1.1105 1.1160 1.1214 1.1267

Note: The factor values for a/b < 1 can be obtained by using (6) and interpolation may be carried out linearly in log a/b. The tables are obtainable by anonymous ftp quouting filename Table 1 (or Table 2) in subdirectory rncc on the computer system aber.ac.uk.

M. Cain / Computational Statistics & Data Analysis 27 (1998) 141-150 145

Table 2 Values of adjustment factor, 6*, for given (extreme) values of a/b > 1 and degrees of freedom, d

Degrees of freedom a/b 2 3 4 5

1.001 0.0007 0.0006 0.0005 0.0005 1.01 0.0070 0.0055 0.0050 0.0047 1.02 0.0140 0.0109 0.0099 0.0094 1.03 0.0209 0.0163 0.0148 0.0140 1.04 0.0277 0.0216 0.0196 0.0186 1.05 0.0345 0.0269 0.0244 0.0232 1.06 0.0412 0.0321 0.0291 0.0277 1.07 0.0479 0.0373 0.0338 0.0321 1.08 0.0544 0.0424 0.0385 0.0365 1.09 0.0610 0.0475 0.0431 0.0409 1.10 0.0674 0.0526 0.0477 0.0452 1.11 0.0738 0.0576 0.0522 0.0495 1.12 0.0802 0.0625 0.0567 0.0538 1.13 0.0865 0.0674 0.0611 0.0580 1.14 0.0927 0.0723 0.0655 0.0622 1.15 0.0989 0.0771 0.0699 0.0663 1.16 0.1051 0.0819 0.0742 0.0704 1.17 0.1111 0.0866 0.0785 0.0745 1.18 0.1172 0.0913 0.0828 0.0786 1.19 0.1232 0.0960 0.0870 0.0826 1.20 0.1291 0.1006 0.0912 0.0865

10 2.0125 1.3946 1.2159 1.1320 100 7.0004 3.6389 2.8461 2.5098

1000 22.339 8.1242 5.4456 4.4310

6" d Note: lim - - -

a/b~ 1 + (a/b - 1) (d - 1) t~ (0).

obtained using MINITAB. The values for alb < 1 can be deduced by observing that

I 1 - - a +Td(b) -= -- + T , ( - b ) .

and hence, that for a. b > 0

6*(b, a) = - 6*(a, b). (6)

It also follows that the minimal expected predictive loss is (see the appendix for details)

[

which is less than ½ (a -4- b) V(Y), the corresponding expression for a symmetric loss function of the form: f (y) = ½(a + b) y 2, y ~ R.

146 M. Cain~Computational Statistics & Data Analysis 27 (1998) 141-150

The values of a/b considered in Table 1 are 1.1 in steps of 0.1 to 9.9 and in Table 2 are 1.01 in steps of 0.01 to 1.20 and the extreme values of 1.001, 10, 100 and 1000. Both tables consider the degrees of freedom, d = 2, 3, . . . , 130, 150, 200, 250, 300, 350, 400 and 500. Note that the case d = 2 is not really applicable since the expected predictive loss is then not finite. However, the relevant first-order equa- tion (5) can still be solved and its solution is given in Tables 1 and 2. This provides a point of reference for the corresponding adjustment factor values for non-integer degrees of freedom between 2 and 3. Complete versions of Tables 1 and 2, involving all 136 values of d, may be obtained over the internet and only reduced versions are presented here.

As examples in the use of the tables, with an asymmetric quadratic loss function of the form:

~0.2y 2, y_>0, ~0.54y 2, y_>0, (i) f (y)=[0 .5y2 , Y < 0 , and (ii) d(y)=[O.5y2, y < 0 ,

a predictive t variable with d = 5 degrees of freedom, mean 10 and precision ~ , the optimal prediction for the variable of interest would be (i) 10 - (0.4373)~]-g = 8.2508 and (ii) 10 + (0.0365)v/~ = 10.1460, respectively.

A plot of 6" against In (a/b) for the data of Tables 1 and 2 proves to be quite linear, particularly for large degrees of freedom (d > 10); see Fig. 1. This forms the basis of interpolation to provide 6" for other values of a/b.

"ID

This short paper has presented a solution to the problem of the prediction of a univariate t response variable under asymmetric quadratic loss. Extensive tables of adjustment factor values are given as an aid to the practical implementation of

0 . 5 - -

1 , 0 - -

0 . 0 - -

I I I I I I I I I I

1 2 3 4 5 6 7 8 9 10

a/b degrees of freedom = 10

3. Conclusion

Fig. l(a). Plot of delta* against a/b.

" O

0.5

1 . 0 - -

0.0 I I I

0 1 2

In(a/b)

degrees of freedom = l 0

M. Cain / Computational Statistics & Data Analysis 27 (1998) 141-150 147

Fig. l(b). Plot of delta* against ln(a/b).

such analysis in situations where the usual assumption of symmetric quadratic loss is questionable.

Appendix

1. Der iva t ion o f ( 4 )

The probability density function of Y is

[ = p l /2 t d ( p l / 2 ( y _ #))

(d > 0, p > 0) and (3) may be expressed as

b(# - Y) + (a - b) I °~ (y - ~ ) fd (y )dy = O. d~

Also, for d > 1

- ( d + D 2

f ~ xtn (x) dx =

, - - o o < y < 0 o

(7)

(d- i) ¢2

(8)

148 M. Cain/Computational Statistics & Data Analysis 27 (1998) 141-150

and for d > 2

f f x2ta(x)dx = f f {d[l +X---d ] - d } t d ( x ) dx

- -~ iZ -2~ [1 -Td-2 (c~ / -~ ) ] -d [ l -Ta(c ) ] ; or alternatively,

( d + 1)

(9)

x2ta(x)dx = xak(d) 1 + dx

d + l d (where k (d)= F ( ~ ) / F ( - ~ ) v / - ~ )

( a - I) ( d - I)

- ,d d l > k ( d > c I l + ~ l - - w - + I f k ( d > ( T - l ) I I + - ~ l - 2 dx

c

d k(d) d ta- 2 (z) dz + (d - 1~ k (d - 2) 2

= C td-2 C + 1 - T a - 2 c , (10)

noting that k(d)/k(d - 2) = (d - 1 ) / x / ~ - 2), and it follows from (9) and (10) that for any c e R

Ta-2 c = T a ( c ) - - ~ 4 - ~ z - ~ t a _ 2 ~ c 4 ~ ). (11)

Now, using (8),

;; L o (y -- ~)fa(y)dy = (y - #)fa(y)dy + (# - ~)~'(Y > ~)

f; = p-~/Exta(x)dx + (# _ ~)~(p l /2 (y _ #) > pl/2(~ _ #))

= p - 1/2 d p(~ /~) ( d - 1) 1 -~ ta(pl/2(f -/~)) + (/a - :9)[1 - Ta(pl/2(:~ -- #))]

and substituting this expression into (7) gives (4).

M. Cain~Computational Statistics & Data Analysis 27 (1998) 141-150 149

2. Derivation of the minimal expected predictive loss

The expected predictive loss in using 33 to predict Y is

b(y -- 33)2 dF(y) + a(y - 33)2dF(y) - - O O

f oO

= b {(y - U) / + 2 ( # - 3 3 ) ( y - # ) + ( # - 3 3 ) Z } d F ( y ) - - 0 1 3

+ (a - b) tl ~ {(y - #)2 + 2(# - 33)(y - #) + (~ 33)2} dF(y)

+ (a - b){A + B + (/z - 33)2 ~ ( y > 33)} (12)

where, using (8) and (9), for d > 2 f ; ~p r ° X 2

- - t~ (x) d x A = (y -- kl)2fa(y)dy = "~(;-u) P

- l ~ d ( d - 1 ) [ 1 - T a _ 2 ( p l / 2 ( 3 3 - 1 ~ ) N / ~ - d - 2 ) ] - d [ l - T a ( p l / 2 ( 3 3 - # ) ) ] }

and

x B = 2(# - 33) (Y - #fin(Y)dy = 2(# - Y) "~(~-.)p772 ta(x)dx

2(/~-33) d [1 + P 0 3 - # ) 2 1 = pl/2 (d 1~-~ d t " ( p ' 2 ( Y - ~ ) )

The minimal expected predictive loss occurs when 33 = I~ + p-~/23" and then

= d ( d - l ) ,

and, using (5),

B- p~-l)2a~* 1 + - Z j t"(~*) = p L(a - b) + r . (~*) .

The minimum of (12) is thus

P + + (a -- b) A - B + - - [ l p - - Td(~*)]

1 (d T + (a - b) Ta-2 6" dTa(6*) = p 2) (d - 2) +

+ Ta(6*) = p ( d - 2 )

150 g. Cain~Computational Statistics & Data Analysis 27 (1998) 141-150

where, using (5),

d(d - 1) . d - 2

apply ing (11) to the expression for D gives

D = (d - 2----~ r d - 2 ~*

and the min imal expected predictive loss is thus

p ( d - 2)

which is less t han (d(a + b ) / 2p (d - 2)).

References

Bracken, J., Schleifer A., 1964. Tables for Normal Sampling with Unknown Variance. Division of Research, Harvard Graduate School of Business Administration, Boston.

Broemeling, L.D., 1985. Bayesian Analysis of Linear Models, Marcel Dekker, New York. Cain, M., Janssen, C., 1992, Real estate pricing decisions under asymmetric loss, Research Report No.

92-6, Research Papers in Management Science, University of Alberta. Cain, M., Owen, R.J., 1990. Regressor levels for Bayesian predictive Response. J. Amer. Statist. Assoc.,

85, 228-231. Granger, C.W.J. 1969. Prediction with a generalized cost of error function. Oper. Res. Quart., 20,

199-207. Harris, T.J., 1992. Optimal controllers for nonsymmetric and nonquadratic loss functions, Techno-

metrics 34, 298-306. Le6n, R.V., Wu, C.F.J., 1992. A theory of performance measures in parameter design. Statistica Sinica

2, No. 2, 335-358. Raiffa, H, Schlaifer, R., 1961. Applied Statistical Decision Theory. Division of Research, Harvard

Graduate School of Business Administration, Boston. Varian, H.R., 1975. A Bayesian approach to real estate assessment. In: Studies in Bayesian Econo-

metrics and Statistics in Honor of Leonard J. Savage, Fienberg, S.E., Zellner, A. North-Holland, Amsterdam, 195-208.

Zellner, A., (1986). Bayesian Estimation and Prediction Using Asymmetric Loss Functions, J. Amer. Statist. Assoc. 81(394), 446-451.