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Gini Lorenz curve

Lorenz curve is a case of concentration from that measured. These statistics apply to socio-economic variables are likely to be spread, such as the level of income, wages and profits. The distribution of the variables can be based on geographic areas, locations or categories.

The objective is to observe and mediate as it divides the total of the variable of interest:

between individuals in the population.

Lorenz curve

Lorenz curve is the curve passing through point pairs where:

• 

with

• We can see that the cumulative relative frequency .
can also see that the distribution of individual values \u200b\u200bor the percentage of the total accumulated by individuals with an observed value

. Another feature that strikes us is that both measures take values \u200b\u200bbetween 0 and 1.
Properties:


Lorenz curve always starts at point (0.0) and ends at the point (1.1)

is a rising curve.
is always less than or equal to the diagonal that passes through the points (0.0) and (1.1).
3. Therefore to make the curve representing the first line we represent (0.0) and (1.1) and then we represent each pair of points
.

Lorenz curve has the following structure:




So the closer the curve to imply a more equal distribution digonal. And the more distant is less equal distribution in the distribution of the total.

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Simple Random Sampling: Sample size

In first problem we encounter when implementing a given sample design is the choice of sample size, since most of the time we are subject restrictions of some kind, such as economic or accuracy. In the case of simple random sampling the sample size is given by:

where the value varies depending on the parameter you want to estimate. If you want half

estimate is given by:

When we are interested in estimating the total

:

And if what we estimate is a ratio

we have:

Where:

k is the abscissa
• likely leaving your right is a value set in advance by the investigator or the study requerimento.
and

is the maximum permissible error
• you want to be considered, like k value is a value predetermined by the researcher. Since
n

have to be a natural number, we will take the nearest integer value par excellence that obtained in the formula.

As can be seen trying to get the sample size we find that in the formulas are parameters that are unknown before the study as is the case of quasivariable or the population proportion. In these cases what we do is the following:



If the sample or the study was conducted with a value

aterioridad we can use it to obtain an approximate value of

n
• .
If

then calculate n by the poorer of

, which will have the largest sample size for this variance.
• 
If we know the range of the variable

then it can be estimated by
• 

If you do not have any of this information then cojeremos a pilot sample to estimate.

the case of a similar proportion do not have information from a previous study

consider P = Q = 0.5

, we provide the largest possible sample size for a proportion.

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Simple Random Sampling: Estimators and Variances

ESTIMADORES DE LOS PARÁMETROS

Nos vamos a restringir a tres tipos de parámetros: el total
poblacional de una variable X:

la
media
poblacional de esta misma variable:

y la
proporción
de la población que pertenece a un grupo determinado:

donde
es una variable indicadora that takes value 1 if the unit has selected the property and is set to 0 if you do not have.

Estimates are:
-

TOTAL is the total sample:

-

MEDIA, is the average meustral:

-
SHARE
is the sampling rate:

These estimators are unbiased for the population parameters.

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ESTIMATES OF VARIANCE


Here we show the expression of the variances for the parameters shown above.

variance of the sample mean:

where is the sampling fraction and the quasivariable

sample.
the total sample variance:

Variance ratio:

Therefore the sampling error for each parameter is the square root of the above variances. Addition the estimators are unbiased estimators for the population variances.

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Simple Random Sampling

Simple random sampling is the sampling design in which

n distinct units are selected from the N population units, so that each possible combination of units is same probability of being selected. So call U the population, this sampling design is defined by the pair , where the design stand consists of all sample sizes n que se pueden obtener de U , y la distribución de probabilidad asociada es la uniforme. Así pués el muestreo aleatorio simple que suele denotarse por

, se puede definir como:

tal que

.

Por otra parte la probabilidad de que una unidad cualquiera

pertenezca a la muestra es:

y la probabilidad que tienen dos unidades de pertenecer simultáneamente a la muestra viene dada por:

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